Transcription
Hello, beginning with the first chapter of the program, which is the biophysics of solutions, there are liquid solutions, gaseous solutions, and solid solutions. For the program, it's liquid solutions, and tanukis are a state among other states of matter. We therefore begin by studying the most classic states of matter: the liquid state, the solid state, and the gaseous state. The liquid state, such as water; the solid state, such as salt crystals or aluminum, copper, iron; and the gaseous state, such as hydrogen gas or oxygen gas, all the gases we know. There are also other states, among others, called intermediate states. Intermediate means a state between two states, such as liquid crystals, which are used in mobile telephony or to manufacture microcomputer screens. Liquid crystals, as their name indicates, are made up of solid crystals and conventional liquids. Leaving aside the properties, they combine the properties of conventional liquids with those of a crystallized solid. We will also mention in passing, and I recall other states as well, that we will not discuss all of them here. In passing, we will mention plasma, which is an ionized gas and is called the fourth state of matter, an ionized gas. So, we start with a solid, we supply it with energy to liquefy it, we obtain a liquid. We supply it with more energy to turn it into a gaseous state, and we supply even more energy to this gas to ionize it. Ionizing means tearing off electrons. So, this ionized gas will therefore be composed of electrons that have been torn off, of ions that result from having torn off these electrons, and of neutral atoms or molecules. Of course, not all constituents of this gas are necessarily ionized; some become ions, and others retain their structure. The energy required to obtain this plasma is very high, and generally, the temperature is above 10 to the power of 4 Kelvin. The name plasma was used in physics by analogy with blood plasma, which is the liquid component of blood. It is a liquid in which blood cells, namely red blood cells, white blood cells, and platelets, are suspended, and which constitutes 55% of blood volume.
We will now begin with the first state, which is the liquid state. A liquid state possesses a proper volume, first of all, yes, of course, we speak of a liter of milk, a liter of water, and so on. Does it possess a proper shape? No, it takes on the shape of the container that holds it. It has no proper shape of its own; as soon as we place it in a container, it takes on the shape of that container. A liter of milk can be placed in a bottle, or it can be put in a cylindrical saucepan. Let's observe these three liquids: green, orange, and dark. These three liquids have this particularity: their free surface. What is the free surface? It is the surface that separates the liquid from the air. So, the surface is the surface that is in contact with the air. This free surface is flat and horizontal. This is a front view; if we could see from above, it would be a flat surface, and it is horizontal. It is flat, as opposed to vertical. Of course, provided that all liquids have a flat and horizontal free surface, provided they are at rest, of course. A liquid that is agitated should not be expected to have a flat and horizontal free surface. The liquid state is also condensed. Condensed is the opposite of dispersed. For example, gas is in a dispersed state; we will study it later. That is to say, we have a hydrogen molecule here, a hydrogen molecule far away on the other side. On the other hand, liquids are condensed, meaning that the molecules or atoms in the liquid, when it is an atomic liquid, will clump together, one next to the other, around each other. They will form a mass, they will form something condensed. There will not be, for example, water molecules here with a free distance and a void between one molecule there and another molecule on the other side. No, there will be a condensation of molecules. That is why we say it is condensed. And they are weakly bound. Indeed, we can explain this if we consider a monoatomic liquid, for example, by melting copper, and we will see a liquid composed of copper atoms. We will call 'r' the distance separating two atoms, each having a radius 'r0'. So, 'r' is the distance separating two radii, two atoms of r0. So, this distance 'r' will be the distance between the centers of the atoms. We will consider the case where this distance is greater than the sum of the two radii, meaning 'r' is greater than two times r0. So, this atom is separated from the other atom by the distance 'r', which is greater than r0 plus r0. So, what happens? It will happen that these atoms will attract each other. Among the forces of attraction, of course, we know electrostatic forces between components because between electrical components of atoms, we know that each atom is composed of protons and electrons, and therefore there is a net charge of positive and negative charges that can attract other charges. So, the positive will attract the negative of the other atom. But also the force of Newton, the fundamental force of Newton, called the gravitational force. We know very well that if we have a mass m1 and a mass m2, these two masses attract each other by Newton's force, which is written, if we recall this, or rather, it is G times m1 times m2 over the distance separating them squared. G is the constant, which is 6.67 times 10 to the power of -11 in the international system. There is also the case where these atoms can attract each other to the point where the distance between the two centers of these nuclei, of these atoms, becomes less than two times r0. It is less than the sum of the two radii. So, these atoms attract each other to the point where their electron clouds merge, and their nuclei get so close that there will be repulsion because positive charges, when they get close, repel each other. And we see repulsion forces. There is a third case, which is the case of equilibrium, which is when the atoms touch each other. They touch each other, and therefore the distance between the two nuclei is exactly equal to the sum of the two radii, so r is equal to two times r0. And this position is an equilibrium position. And at that moment, we will say that the atoms are packed, they are packed against each other, forming what is called short-range order. Let's explain this notion of short-range order. Here, we see this illustration. And we take an atom at random where it is mentioned "order," "short-range order." So, we see that if we take this short-range order, what is it? It means that if we take an atom at random, the distance between this atom and its first neighbors, I repeat, the first neighbors, is well-defined. So, if we look at the figure, we have, where it is mentioned "short-range order," we have an atom at the center, and the first neighbors, for this case, we have six first neighbors: 1, 2, 3, 4, 5, 6. So, these six first neighbors have formed a layer around the reference atom. It's not always the case; it depends on the dimensions. If we have large atoms, there will be fewer than six; if we have small atoms, there will be more than six. So, this distance is well-defined. Indeed, we have a hexagon, so it is a polygon with six sides. This hexagon is formed by segments, and the segments connect the center of one atom to the center of another. So, we have six equal line segments. So, this hexagon is well-defined. And so, we take the distance between a randomly chosen atom and its first neighbors; it is well-defined. Now, we have a layer around the reference atom, chosen at random. Now, if we try to form another layer, a second layer around this atom, it will be impossible. That is why we speak of short-range order, so an order between the atom and its first neighbors. Now, we have atom A and atom B. So, there is no, if A and B are first neighbors, there is a well-defined distance. Now, if we want to go beyond the first layer, it will be impossible. That is why we speak of short-range order, not for solids, it's a bit like long-range order. For example, in solid NaCl, the atoms, the ions Na+ and Cl-, are well arranged. The distance between a Na+ atom and its first neighbor is known, between it and the second neighbor, it is also known, it is well established. But with liquids, it is only between the atom and its first neighbors.
We will now study the gaseous state. A gas possesses neither shape nor proper volume. That is to say, a gas has no shape; it's not like a liquid that takes on the shape we give it. A gas has no shape, it has no proper volume. If we have, for example, a 3-liter gas cylinder, and we open this cylinder in a large room, this gas will spread everywhere, and therefore there will no longer be 3 liters of gas; it will be the volume of the room. So, a gas has no shape and no proper volume. It tends to occupy all available volume, as we just said. In the cylinder, the gas goes in all directions. It is dispersed. We said earlier that a liquid was condensed, the molecules clumped together, close to each other, next to each other. Well, for a gas, it is dispersed; the molecules each go in a direction. It is completely disorganized. This is total disorder; there is no order, rather, there is no order like in liquids, no order like in solids. It is total disorder, and the particles are almost practically independent. A gas particle is independent of another, except, of course, when they get very close to each other, when they bounce off each other, they move away. Otherwise, they are independent of each other. And a gas is made up only of atoms or molecules. We did not say that in gases, a gas is made up only of atoms or molecules.
We will now talk about energy and interactions. What is energy? It is the property of a system that is capable of doing work. A simple example: when we are tired, we cannot work, so we have no energy to do work. So, what is energy? It is the ability to do work. The units of energy are known: the Joule. In the international system, the international system is the meter, kilogram, second, and Ampere. So, m, kg, s, and A. A Joule is equal to a Newton times a meter. We know from high school that the work of a force of one Newton that moves an object over a distance of one meter, for example, the work will be the force multiplied by the distance, so it will be one Newton times one meter, and we get one Joule. There are other units, such as the erg. The erg is used in the CGS system (centimeter, gram, second). One erg is equal to 10 to the power of -7 Joules. We also know calories; one calorie is 4.184 Joules. And finally, we know electronvolts; one electronvolt is 1.602 times 10 to the power of -19 Joules.
We will now talk about interaction. An interaction is an action between two things. An interaction occurs when they exert a force on each other, an action called an interaction force. So, A acts on B, and B acts on A. A applies a force to B, and B applies a force to A. So, in interaction, there is the principle of reciprocity. It is reciprocal: if A exerts a force on B, B exerts a force on A. For example, the electric force: a negative charge attracts a positive charge, and a positive charge attracts a negative charge. Gravitational forces also: A attracts B, and B attracts A.
Now, we will talk about interatomic interaction between two atoms. It is also called primary interaction. The first interaction is called primary interaction. It can be covalent, for example, in the hydrogen molecule. We have a hydrogen atom with a free electron, so an electron on the last shell, on the outermost shell, only one shell. And another hydrogen atom. These two atoms bond with a covalent bond, meaning they share their electron clouds. They form what is called an electron pair, and so the two electrons belong to both hydrogen atoms in the molecule. So, the two electrons are on the shell of the first hydrogen and on the shell of the second hydrogen. This is a covalent bond. There is also the ionic bond in the NaCl molecule. How does this bond happen? We have an atom. When a sodium atom encounters a chlorine atom, the sodium atom has one electron on its outermost shell. The chlorine atom has one free electron on its outermost shell. So, the sodium atom will give its electron to the chlorine atom. It will become a sodium ion. It will gain stability because it will acquire the structure of the shell just below its outermost shell. And the chlorine atom will saturate its outermost shell. And so, we obtain an ion, a positive ion, Na+, and a negative ion, Cl-. They are attracted and form the NaCl molecule. And this bond is an ionic bond, a bond between a positive ion and a negative ion. There is also the metallic bond. The metallic bond occurs between metal atoms, as its name indicates. For example, let's take copper atoms. What happens? For example, let's take this copper plate. What happens is that these metals share one or more electrons, which are free electrons and are responsible for the electrical conductivity of metals. How to explain this? In metals, we know that metals tend to be reducing agents; they give electrons. So, if we enlarge this copper plate, we will see a copper ion with an electron next to it. How to explain this? We don't have ions in the copper plate. So, we must explain this story of free electrons. Metals have electrons on their outermost shell. These electrons are called free electrons. Why do they tend to lose them easily? So, what happens? In the copper plate, we have the free electron on the outermost shell of the copper atom. It moves freely between the atoms. So, easily, since each atom contributes one electron, the copper atom has a free electron. So, we write Cu+ and electron. This is not to say we have an ion, but to say that this electron is easily and quickly released from the atom. And so, when it moves away a little, we obtain a copper ion. The other atom has a free electron, and so we obtain a copper ion. And so, we tend to have free electrons. These atoms share these free electrons. These electrons that move easily. So, thanks to these free electrons, current flows easily. What is current? It is a movement of electric charge. And in metal, we have electric charges that move easily. This is called a metallic bond. So, a metallic bond is when atoms share their free electrons.
Intermolecular interaction is called secondary interaction. Interatomic is primary, intermolecular is secondary. So, it is responsible for the cohesion of liquids and molecular solids. Cohesion means adhesion. Why do we have, for example, if we take water, why is there not one molecule here and another molecule there, but there is adhesion, there is cohesion, adhesion? Water molecules clump together; they form water droplets. Why? Because there is an intermolecular force, there is an intermolecular interaction between the molecules that causes the molecules to gather, to form a gathering. And so, this intermolecular force, this adhesion, this intermolecular interaction, is governed by electrostatic forces of attraction between molecules. So, we have water molecules; they attract each other. One molecule attracts another, and the other attracts another. Why? Because there are forces of electrostatic origin. Where do these forces come from? Why are they electrostatic forces? It is because molecules are composed of electrons and positrons, so by the interaction between the charged particles, electrons and positrons, that compose them, we have protons and electrons in molecules. And these molecules, the negative charges of one molecule attract the positive charges of another molecule. This attraction creates this cohesion.
Now, we will talk about saturated vapor pressure. We will start by giving definitions. We have a system. This system is composed of water, and above it, there is oil. This system is composed of a water phase and an oil phase. It is composed of two liquids that are immiscible; we cannot mix water and oil. After a certain time, there is decantation. What is decantation? It is because the oil floats and the water settles at the bottom. So, we have two different phases, while we have two different phases and only one liquid state. Oil is a liquid, in the liquid state. Water is also in the liquid state, but we have two different phases. Why do we say we have two different phases? Because what is a phase? It is a region of space where the parameters are the same, the same chemical composition, the same physical properties, and the same state of matter. Comparing oil and water, do we have the same chemical composition of oil as water? No, oil molecules are not water molecules. So, we do not have the same phase. Do we have the same state of matter? Yes, it is the same liquid state, but it is not the same chemical composition. Do we have the same physical properties? No, for example, does water solidify at 0°C? Oil solidifies at another temperature. So, to have the same phase, one must have the same state of matter, the same chemical composition, and the same physical properties. Let's take another example: frozen water and liquid water. Do we have the same phase? No, even though we have the same chemical composition, it is not the same state of matter, because frozen water is in the solid state, and liquid water is another state. So, let's note that, for example, now we will talk about a small definition here. We have an open system. Why is it open? Because it exchanges energy and matter with the surrounding environment. This glass of milk, it exchanges matter. After a certain time, it will evaporate; it is an exchange of matter. It can also exchange energy, for example, it is hot, it will cool down. This is called an open system. So, let's observe this jam jar. If the jam is hot, after a certain time, it will cool down. So, between this jam jar and the surrounding environment, there will be an exchange of heat, so an exchange of energy. But the matter, the jam, will not evaporate because it is closed. So, the only exchange that will occur with the surrounding environment is an exchange of energy, and the system will be called a closed system. Now, for this thermos bottle, if we have coffee, it will stay hot for a certain time, of course. So, with the surrounding environment, there will be no exchange of matter, as it is closed. So, there will be no evaporation of coffee. There will also be no exchange of energy; it will not cool down. So, since this system exchanges neither energy nor matter with the surrounding environment, it will be called an isolated system.
Let's move on to the concept of saturated vapor pressure. Let's observe this enclosure. In this enclosure, there is water, and above this water, it is a vacuum. We have removed all the air. So, this enclosure is hermetically sealed, and it is at a fixed temperature. So, we maintain a fixed temperature. Well, since there is a vacuum, what is a vacuum? It is the absence of matter. So, there will be no pressure. The pressure of a vacuum is zero. Let's recall the concept of pressure. Pressure is force over surface. Force is an action. So, if there is no matter, if there is no mass, there is no action, so no force. Let's imagine pressure like this. If there were matter, matter in the form of molecules, necessarily, these molecules, if they existed, would hit the walls of the vacuum bottle. So, they would hit the walls and create pressure. Hitting means applying force on the surface, which will be pressure. There is no matter, there is no pressure. Nature, we know, abhors a vacuum. So, what will happen is that a certain number of water molecules will pass into the vapor state. They will pass into the vapor state until saturation. Let's look at the first molecule, then two others follow, and so on. And it continues, it continues until saturation. So, saturation, how do we imagine this? In the vacuum, there are empty spaces for the molecules. So, only a certain number of molecules are allowed to turn into vapor. Once all the spaces are occupied, we can no longer have molecules passing into the vapor state. When the empty space is saturated, when there is no more room, we can no longer have molecules passing into the vapor state. Now, when this saturation is reached, we will say that the saturated vapor pressure or saturation pressure has been reached. How to explain this? First, there is a vacuum. The first molecule passes into the vapor state, so there is matter. Now, this molecule that has passed into the vapor state will hit the walls, and so instead of zero pressure, we will have a certain pressure. The second molecule that joins it will do the same; it will hit the walls and create pressure. And we see that each time a molecule passes into the vapor state, the pressure increases compared to that of the vacuum. The pressure of the vacuum is zero. So, we will see the pressure climb little by little. Once no molecule can pass into the vapor state, we have reached saturation, so the pressure will no longer increase. The first one arrives, it creates pressure. The second molecule will create another pressure that will be added to the pressure of the first. The third, the fourth. At a certain number of molecules, let's say 50, let's assume that after 50, it is saturated. We can no longer have molecules evaporating. And so, the 50 molecules will create a pressure, and this pressure will be called the saturation pressure, also called the saturated vapor pressure. So, this saturated vapor pressure has a particularity: it depends only on the temperature.
Let's revisit the definition of saturation pressure. What is saturation pressure? It is the pressure of the vapor of a substance in the gaseous phase that is in equilibrium with the liquid phase of the same substance at a given temperature. So, saturation pressure is when we have an equilibrium between the phase, for water, for example, when we have an equilibrium between the gaseous phase of water and the liquid phase of water. What is this notion of equilibrium? We will explain it. We indicated that 50 is the maximum number; it is an arbitrary number. We took 50 and said that 50 is the maximum number of particles that can turn into vapor. A number like that. So, let's assume that beyond 50, two other molecules venture to pass into the vapor state. They were in the liquid state and will join the first five molecules. They have no space. And the molecules that have just arrived, they have no space. So, where will they fit? So, what will happen is that since two molecules arrive and 50 molecules are already in the vapor state, then only two molecules that are already in the vapor state will transform, they will go down, they will transform into the liquid state to make room for the two molecules that have just arrived. This is how we explain the notion of equilibrium. That is to say, equilibrium means that the flux of molecules passing from the liquid state to the gaseous state, we indicated earlier that two molecules, there were 50, it's finished, it's saturated. Two molecules arrive. So, the flux of molecules transforming from the liquid state to the gaseous state is equivalent, meaning equal, over a given time interval, to the flux of molecules passing from the gaseous state to the liquid state. So, two molecules pass from the liquid state to the gaseous state. So, two molecules, parallel to this, two molecules will pass from the gaseous state to the liquid state to make room for them and to maintain this number of 50.
For a volatile substance, we have a saturation pressure greater than atmospheric pressure. Indeed, if we have, for example, gasoline spilled on the ground, after a certain time, all the gasoline will volatilize, meaning it will disappear completely. So, why? If we have a liquid in contact with the surrounding pressure, which is atmospheric pressure, we have a liquid. So, what will happen? The gasoline molecules will pass into the vapor state to increase, and by increasing the pressure, earlier, when we made the pressure in the closed enclosure, it was water above which there was vacuum. The pressure was zero. Each time a molecule passed into the vapor state, the pressure increased compared to the vacuum. The same for gasoline, which is, of course, in the open air and at atmospheric pressure all around. So, each time a gasoline molecule passes into the vapor state, it is in order to increase the pressure compared to atmospheric pressure. And when will it stop? It will stop when it reaches saturation, when we can no longer evaporate the gasoline. And at that moment, we will have the saturation pressure. So, we started from atmospheric pressure with the goal of reaching saturation pressure. That is why saturation pressure is higher than atmospheric pressure. Earlier, we started from zero pressure to reach saturation pressure, of course. So, volatile substances like perfume, gasoline, disappear completely because they cannot saturate the surrounding space, unless it is something enormous, something impossible, if the universe were so vast.
Now, to calculate the saturated vapor pressure of water, a scientist has developed an experimental formula. This formula is valid only when the temperature is between 100 and 200 degrees Celsius. It is written: P_saturation = (T/100)^4. T is the temperature expressed in degrees Celsius, and the saturation pressure is obtained in bars. Let's do some reminders about the idea of pressure, because the bar is a unit of pressure. So, one bar is equal to 100,005 Pascals. One atmosphere is 101,325 Pascals. On the other hand, let's recall how to calculate the pressure exerted by a column of liquid. We start with a column, and then we will get rid of it completely in the formula. We will see a formula where the section of the column does not appear.
So, we have a column. We have a tube in which there is a blue liquid, and above it, there is a pink liquid. The tube has a section S. The pink liquid is the one for which we will calculate the pressure it exerts on the blue liquid. It has a height h, and the force exerted by this pink liquid on the blue liquid is nothing other than the force of weight, the weight, because it has a mass, and therefore it has a weight, and weight is mass multiplied by g. So, let's recall the formula from the beginning: pressure is force over surface, as known from high school. The force, we just said, is weight. So, weight is mg. So, over S. m is mass, g is acceleration due to gravity. Rho is density, multiplied by volume. And let's substitute everything slowly. The volume of this column of pink liquid is a certain shape, it is cylindrical, so it is the section multiplied by the height. We substitute, and we see S appearing in the numerator and denominator, and we simplify, and we obtain a very simple formula, which is the pressure of a liquid at height h, which is equal to rho * g * h. And this formula can be applied to calculate the pressure exerted by a column of mercury. We will apply it to a column of mercury whose density is 13.6 grams per cubic centimeter and whose height is 760 millimeters. So, we blindly apply the formula: Pressure = rho_mercury * g * h. And we substitute rho = 13.6 g/cm³, we convert it to kg/m³. g = 9.8, and h = 760 mm, we convert it to meters, 0.760 m. And we obtain 101,325 Pascals. It is not by chance that we calculated for 760 mm of mercury. We often use this information and simply write 760 mm of mercury is approximately 101,325 Pascals. It is the same pressure. Now, to find the conversion in millimeters of mercury, so 760 divided by 101,325, we get that one Pascal is 7.5 x 10^-3 millimeters of mercury. This is for conversion if we need to use Pascals or millimeters of mercury.
Now, let's do an application exercise. We place one liter of pure water in a pressure cooker, we close the pressure cooker, and we place it on the fire. The saturated vapor pressure of water is 1 bar. Calculate the boiling temperature of water. To answer, we will use the Dupré formula to calculate the boiling temperature, knowing the pressure value. The relation we know is the Dupré relation, which we will apply: (T/100)^4. We develop and extract T. So, T/100 is raised to the power of 1/4. So, T will be 100 * P_saturation^(1/4). We substitute P by 1, and we obtain 100 degrees Celsius. So, this result was predictable because we all know that water boils at 100 degrees Celsius, and the purpose of this exercise is to apply the Dupré formula.
Exercise number 2: We introduce a mass of water equal to 4 grams into a container of volume 10 liters, initially empty, and we bring it to a temperature of 80 degrees Celsius. So, for water, the saturation pressure at 80°C and at 100°C. The first question: will the saturated vapor pressure of water be in the enclosure? Deduce then the mass of water that will remain in the liquid state. Then, we are told that we bring the container to a temperature of 100°C. Earlier it was 80°C, now it will be 100°C. What is the nature of the new equilibrium state? What will be the vapor pressure in the enclosure? To answer this exercise, we will start by analyzing the 4 grams of mass that were placed in the enclosure, meaning we will calculate the number of moles.
He corresponds and the volume that corresponds to him too, we start with the volume. The volume, as we know, the density, the density, we know it, it was given to us, it's a grand per cubic centimeter. The volume will therefore be four cubic centimeters, that is to say 0.004 liters. This volume is very small compared to the volume of the enclosure, and this is important information. We will therefore neglect this volume, and this is important for what follows. We will come back to it. For the number of moles, it's the mass over the molar mass, and we find 0.22 moles. Now, for what follows, to continue the answer to this exercise, because if we have an equilibrium state, let's try to answer the question. So, we start first by calculating the number of moles that will saturate the water vapor, the water vapor, the enclosure, independently of the mass that was given to us. We consider that we were given nothing, and we will calculate on our own how many moles we need to saturate the enclosure with saturated vapor. So, let's assume that we manage to saturate this enclosure with saturated vapor. This vapor, it acts as an ideal gas. We will attribute the qualifier of ideal gas to it to do the calculations. We are in the process of doing it, so they said it's an ideal gas, and we apply the formula, the ideal gas law, PV = nRT, knowing that P is the saturation pressure, since we are saturating the enclosure with vapor. The volume is 10 liters. The saturation pressure, we also have it, it's 0.466 bar. You calculated n from the temperature. It was given to us. So, n = P_saturation * V / RT. We convert the saturation pressure to pascals by multiplying by 10^5. The volume, we convert it to cubic meters by multiplying by 10^-3. And the temperature, we convert it to Kelvin by adding 273. And we get 0.16 moles. So now, we go back to the data. We have 0.22 moles of water, and we only need 0.16 for all of it to turn into vapor. What does that mean? It means that we will indeed have saturated vapor, but there will remain a quantity of water that will not transform. This quantity of water that will not transform into vapor is the difference. So, we have 0.22 moles, and we have 0.22 - 0.16 left, that is to say 0.06 moles of water that will remain in the liquid state. Why? Because 0.22 moles will turn into vapor, since we only need 0.16 moles to saturate the enclosure. So, we take the 0.16 moles, and the rest remains in the liquid state. So, for this quantity of water that will remain in the liquid state, the number of moles is 0.06. So, the mass is the number of moles times the molar mass, and we find 1.08 grams. Now, we go back to the information from earlier that I said we would come back to. The volume of water is very, very small compared to the volume of the enclosure, so we neglect it. If we look at what we had, what was the answer we gave? We said we have 0.22 moles, 0.16 turns into vapor, and 1.08 grams remains in the liquid state. And the 0.16 moles that turned into vapor, we attributed the entire volume of 10 liters to them, meaning that these 0.16 moles will occupy the entire volume of 10 liters without taking into account in the calculations the volume of water that remained in the liquid state. Normally, what is 10 liters? It would be a volume for the 0.16 moles and a volume for the 1.08 grams. 1.08 grams is 1.08 cubic centimeters. No, no, no. The calculation should have started from there. The total volume is the volume of the vapor plus the volume of the liquid quantity. Here, we didn't do that. We didn't do that because the volume is truly negligible. Even 1.08 cubic centimeters is negligible compared to 10 liters. And that's why we considered that these 0.16 moles would occupy the volume of 10 liters. Otherwise, the calculations would have had to be redone differently, taking into account both volumes, and this will be dealt with in the tutorial. And so, what we wrote, we said, "Note that it is because the volume of water is very small compared to the volume of the enclosure that we considered that 0.16 moles occupy the entire volume of 10 liters." Otherwise, that is to say, if this were not the case, so if it were a volume of water that was not negligible, then otherwise, one would have had to take into account the volume of liquid water that does not turn into vapor. For the second question, we are told that the temperature is 100 degrees Celsius, and we are given the corresponding saturation pressure, and we are told what the new equilibrium state will be. So, we will start. We will do the same thing as we did for the first question, that is, we will calculate the number of moles of water that will be sufficient to saturate the enclosure with vapor, but of course, for the considered temperature, that is, 100 degrees Celsius, independently of the mass. We have a temperature, we have a volume, so we will calculate the number of moles that will be necessary to saturate the enclosure with vapor. So, we start from the relation PV = nRT, and we find by calculation n = P_saturation * V / RT. We replace P_saturation with 1 bar, we convert it to pascals by multiplying by 10^5. The volume, we convert it from liters to cubic meters. And the temperature, so we add 273 to it, 100 + 273 = 273. We get 0.22 moles, or rather 0.32 moles. So, will we have vapor, and will this vapor be saturated? That's the question. We have only 0.22 moles. We only have 0.22 moles. So, will these 0.22 moles saturate the enclosure? No, because to saturate it, we need 0.32 moles, and we only have 0.22 moles. So, we will say that since we only have 0.22 moles, these 0.22 moles will not be able to saturate the enclosure with vapor, and saturation, that is, equilibrium, so we will not have the equilibrium state corresponding to saturation. So, what will happen? The 0.22 moles will turn into vapor, by obligation. Why not? So, we have 0.22 moles. They will therefore turn into vapor, and each time a molecule turns into vapor, the pressure increases. So, once the 0.22 moles are entirely transformed into vapor, they will have created a certain pressure. This pressure will not be the saturation pressure, because to have the saturation pressure, we need 0.32 moles, and we only have 0.22 moles. So, it will be a pressure. We will calculate the pressure, but this pressure will not be the saturation pressure. It will be the vapor pressure, or partial vapor pressure. So, 0.32 moles gives us a saturation pressure. Below 0.32 moles, it will be a vapor, and therefore there will be a pressure that we will call the partial vapor pressure. So, we calculate this vapor pressure: P_vapor * V = nRT. We replace P_saturation with 1 bar, and we get P_vapor = nRT / V. And so, we calculate 0.22 * 8.31 * (100 + 273) and we divide by the volume, and we get 68250 pascals, that is, 0.68 bar. So, let's note together that this 0.68 bar is lower. It's the vapor pressure. It's lower than 1 bar, which is the saturation pressure. And therefore, the vapor pressure is lower than the saturation pressure, and so evaporation, what does it do? It continues. The process of evaporation persists as long as the vapor pressure remains below the saturation pressure. It continues. So, we need 0.32 moles, I only have 0.22 moles. So, it starts from the first molecule. It continues, it continues, as long as the saturation pressure is not reached. It continues, of course. Once all the molecules are used up, there's nothing left to evaporate. But for the number of moles available, it continues to evaporate until the saturation pressure is reached. As long as the vapor pressure is lower than the saturation pressure, once this saturation pressure is reached, that is, equilibrium is obtained. So, the equilibrium between the liquid phase and the gaseous phase is reached when the vapor pressure becomes equal to the saturation pressure. And from then on, no more molecules of vapor can be formed, because it is already saturated. So, roughly speaking, the vapor pressure is always less than or equal to the saturation pressure. It becomes equal to the saturation pressure when we have enough moles to achieve this saturation. But still, the vapor pressure, evaporation occurs as long as the saturation pressure is not reached. Now, a medical application of saturated vapor. We know that the human body eliminates 2.4 liters of water per day. These 2.4 liters are distributed among sweat, respiration, tears, etc. And among these 2.4 liters of water, 300 to 400 grams are eliminated. This elimination is also ensured by the lungs, by the alveoli. Let's look at the diagram for a small reminder of the alveoli. So, we have the trachea, then the bronchi, a main bronchus, then a smaller bronchiole, the main bronchus, and the alveoli are at the end. They are at the end of the bronchiole, these small sacs where gas exchange with the blood takes place. This elimination of 300-400 grams is done in the form of evaporation. Let's recall this. So, as an indication, this table shows the composition of inspired air and the composition of expired air. So, we have oxygen, nitrogen, CO2, and water vapor. Let's observe the water vapor. We inspire a variable quantity of water vapor, but we expire a very abundant quantity. So, this 300 to 400 grams of expiration, this elimination of water, however, depends greatly on the humidity of the inspired air. How? First, to explain the increase, to give an illustration, if you bring a mirror close and breathe on it, you exhale water vapor which forms. So, that's why it's water that comes out, it's evaporation. If we inspire, if the inspired air is saturated with water vapor, so you have brought air saturated with water vapor into your lungs, it means there is no room to form more water molecules. So, can we eliminate water from the body through the alveoli? Impossible, because there is no room, it is already saturated with water vapor. So, the water from the body that will exit as vapor through the alveoli cannot exit because there is no room, it is saturated with water vapor. So, the elimination of water from the body through the alveoli will be zero. On the other hand, if the inspired air is dry, meaning there is room to form water molecules, then the body's water can be eliminated via the alveoli, and therefore the elimination will be maximal. That's why for asthmatics, it is difficult, so one must avoid humid places because their breathing will become even more difficult. Humidity makes breathing difficult because the expiration of water from the body will be impossible if the inspired air is humid. Let's now consider phase changes. What is a phase change? For any pure substance, it would be passing from one phase to another, liquid, gaseous, or solid, and this is illustrated by what is called the phase diagram, whose coordinates, as seen in the figure, are temperature and pressure. That is, we can go from the solid phase to the gaseous phase and vice versa, solid to liquid phase and vice versa. And this will be illustrated on this diagram called the phase diagram. Let's say this diagram, the gray part represents the solid state of any pure substance, for example, for water. So, the entire gray part represents the solid state. That is, we take any point in this gray area, we can find the coordinates of this point, that is, the abscissa and the ordinate. We project it and make a projection on the x-axis, which is temperature, we get a temperature, and we make a projection on the y-axis to get a pressure. And so, we will say that for this temperature and for this pressure, the state is solid. The same applies to the gaseous state, which is of course a disordered gas, or rather a disordered state. For example, in this part, any point belonging to this zone, its temperature and pressure correspond to a gaseous state. And the liquid state. Similarly, we see very well that we go from the liquid state to the solid state by solidification, meaning it hardens. We harden the liquid by solidification. We go from the gaseous state to the liquid state by liquefaction. The gas liquefies, becomes a liquid. And we go from the gaseous state to the solid state by condensation. So, the gaseous state becomes the solid state. The reverse path, we go from the solid state to the liquid state by fusion, because we melt any metal, for example, we melt it, we get the liquid state for this metal. Then, from the liquid state to the gaseous state by vaporization. We vaporize it. And similarly, from the solid state to the gaseous state is by sublimation. That is, in this sublimation, we do not pass through the liquid state directly from the solid state to the gaseous state. The solid directly becomes gas without passing through the liquid state. Now, let's observe the points. First, the triple point. We call it the triple point because this point belongs to the three phases. It's a common point to the three phases: the liquid phase, the gaseous phase, and the solid phase. So, at the triple point, the three phases coexist at the same time. So, the temperature corresponding to the triple point and the pressure corresponding to the triple point, so the coordinates of this point, for these values, we have the liquid phase, the gaseous phase, and the solid phase at the same time. They coexist at the same time. It's a common point. Now, for the critical point. The critical point is why it is critical. First, observing the red curve, at the critical point, the curve connecting the points stops. From the triple point to the critical point, we have the red line, it stops here. Beyond the critical point, we ask ourselves, what state do we have? It's called a supercritical point. Beyond the critical point, we can no longer distinguish between the gaseous phase and the liquid phase. It's called a fluid phase. So, beyond that, if we can no longer distinguish between liquid and gas, it will be a single fluid phase. Of course, fluids are liquids or gases, but here it's not the same meaning. For any point beyond the critical point, we cannot say it's a liquid or a gas. We can no longer distinguish them. There is no discontinuity between the liquid and the gas. It's called a fluid phase. Now, let's look at the curve connecting the triple point to the critical point. Let's take any point on this red curve. There is a temperature and a pressure. First, this curve represents the equilibrium states between the liquid phase and the gaseous phase. And any point on this curve represents a state of saturation. So, we choose any point on this red curve. Its abscissa is a temperature, and its ordinate will be a special pressure that we know, it's the saturation pressure. So, the saturation pressure is read on this curve. Above the red curve is the liquid state. Below is the gaseous state. And on this curve, it's an equilibrium between liquid and gas. And the curve allows us to obtain the saturation pressure for any temperature. We choose any temperature that can give us this phase, this equilibrium. So, for any temperature, we get the corresponding pressure that will give saturation. And also, we will conclude by saying that any phase change, if we change the phase, meaning we change the state, if we go from the solid state to the liquid state, it's fusion. If we go from the liquid state to the solid state, it's solidification. Fusion and solidification are not without energy transfer. They are always accompanied by heat transfer, meaning either energy is consumed or energy is released. And so, any phase change will be accompanied by heat transfer. We distinguish between endothermic changes, meaning changes that consume energy. These are vaporization, sublimation, and fusion. They are represented by the green arrows on the phase diagram. For example, to melt a metal, heat must be supplied to it, it must be heated so that it can melt. So, fusion is an endothermic change. Exothermic changes release energy. They are represented by the pink arrows on the phase diagram, and these are liquefaction, solidification, and condensation. For example, we melt a metal, we supply energy to it. Now, we leave it in the open air so that it can solidify. At that moment, as it cools down, it will give off energy itself, so it will provide, release energy, and it's an exothermic change. In these endothermic and exothermic changes, we always observe heat transfer. So, there are heat transfers. Heat transfers are thermodynamics. So, we will give some notions of thermodynamics, knowing that thermodynamics, as its name indicates, "thermos" means heat and "dynamics" means movement. It studies the thermal behavior of bodies, meaning it studies heat transfers. We will give some definitions for thermodynamic concepts. We will not do chemical thermodynamics, just some notions. We will start by defining calorific energy, also called calorie or heat. The symbol for heat is Q, and the variation is delta Q. So, delta Q is the energy that flows from one body to another, of course, from the hotter to the colder. So, when heat flows from one body to another, there is a variation of heat. Now, let's give an alternative notation for delta Q. Let's take a body that loses energy, meaning it loses heat. A hot body that cools down. So, its initial heat is Q1, its final heat is Q2. Let's take an example: Q1 = 40 calories, Q2 = 10 calories. This body has lost heat. So, the variation delta Q will be negative because it's energy released. Delta Q = final - initial = 10 - 40 = -30 calories. So, it's energy released. On the other hand, if we have a body that consumes heat, for example, we heat milk. Let's assume it already has a certain heat Q1 = 10 calories, and we heat it, meaning we supply heat to it. So, it will go to 40 calories. So, the variation is from 10 to 40. So, final is 40 - initial is 10. So, it becomes something consumed, it's positive. So, we will say that when delta Q is negative, it means energy is released, and when delta Q is positive, it means energy has been absorbed. Now, if delta Q were zero, it would mean no heat transfer has occurred, and therefore no energy transfer. This happens when we have the same temperature. Bodies, when there is heat transfer from one body to another, when there is no transfer, it means that both bodies are at the same temperature. This is called thermal equilibrium. Now, let's define the internal energy of a system. Let's take any system, for example, a gas made of molecules. This is a system. So, what is the internal energy? First, it's symbolized by U. And by definition, since we're talking about internal, inside the system, it's the sum of all microscopic energies of the constituents of this system. So, let's take the gas as a system. We will sum all the microscopic energies of the constituents of this gas, which are, let's assume it's a gas of molecules. So, it's the sum of all microscopic energies of these molecules. What are these energies? We will talk about kinetic energy, of course, the molecules move, they move with a certain speed, so kinetic energy. Chemical energy, what is it? It's the energy required to dissociate the molecule into atoms. So, molecules have chemical energy. Electrical energy, because we know that molecules are made up of positive and negative charges. So, if we talked about intermolecular interactions, it will interact with other molecules. So, it's an energy between negative and positive charges that constitutes the molecule from one molecule to another. And nuclear energy, because in the atom, there is the nucleus. The nucleus is made of nuclear energy. This is the energy that holds the nucleons of an atom together. The nucleons are the constituents of the nucleus: neutrons and protons. Why are they united? Why is there cohesion, adhesion between them? It's because there is energy. And so, this is nuclear energy. The sum of all these energies gives the internal energy of our system. Now, the first principle of thermodynamics. For a closed system, energy is conserved. A closed system is a system that exchanges only energy with the external environment, and energy is conserved. We translate this into a formula by saying that the variation of internal energy of a body is the sum of the variation of heat and the work done by or on this body or system. So, the sum of the variation. So, if we have a variation of heat, if we have a variation of work, it will give a variation of the internal energy of this body. We have defined delta Q, we have defined delta U, we must now define delta W. We have defined work. So, the variation delta W is what? It's the work done by the system in order to vary or affect the thermodynamic state of this system. What is the thermodynamic state? It's defined by the state variables, which are volume, temperature, and pressure. For example, let's take a gas. We compress this gas to vary its volume or its pressure. So, we will act, and the system will be subjected to work. So, as we thought about the sign of delta Q, when it's positive it has a meaning, when it's negative it has another. In the same way, we will say that if delta W is negative, it means the system does work. If, on the other hand, it is positive, it means the system undergoes work. Let's go back to the example of the compressed gas. The system, the gas, is undergoing work to be compressed. So, it receives this work, in a way. So, the variation is positive. Now, in the first principle of thermodynamics, we said that energy is conserved. But we also said that heat flows from a hot body to a cold body. However, can we also think that it could go in the opposite direction, from a cold body to a hot body? Of course, this doesn't really happen. We cannot transfer heat from a cold body to a hot body. So, we must establish a second principle of thermodynamics for this principle. It defines evolution. It's a principle of evolution to say that thermal energy goes in a specific direction, from A to B, for example, and not from B to A. This is the principle of evolution, because so far, we know that heat goes from a hot body to a cold body, but without excluding the fact that it could happen or we could think it could happen from a cold body to a hot body. The second principle of thermodynamics precisely states that this is not possible, because it determines the principle of evolution, the evolution of heat transfer, it's from A to B, and it's also irreversible. So, this principle establishes the irreversibility of thermal exchanges. If there is a thermal exchange from A to B, it will not be from B to A. So, it's a principle of evolution. Of course, the second principle of thermodynamics, compared to the first, let's talk about the second. There are several other more detailed definitions, enthalpy, etc. This is a very simple definition, just a simple one. Let's do an application exercise. It says, during a process, 8000 calories of heat are supplied to a system. The system receives this heat, so we don't give a sign, but we must think that the system receives this heat, so it's positive. While this system performs a work of 6 kilojoules. So, the system performs work, it doesn't receive it, it doesn't undergo it, so it will be negative. How much does the internal energy of the system vary during the process? So, we will answer, not forgetting that the system receives heat and gives work. So, we first convert the heat from 8000 calories to joules to have uniform units, since the work is given in joules. 8000 * 4.1995 = 33500 joules, so 33.5 kilojoules. We give it the plus sign because it's received. The work, we are given the sign minus because it's performed by the system. Delta W is directly assigned the minus sign, 6 kilojoules. Delta U, which is equal to delta Q + delta W, will therefore be 33.5 - 6 = 27.5 kilojoules. Let's do another exercise. Calculate the variation of internal energy of a gas that, during an adiabatic process, we explain what an adiabatic process is: it's a process without heat transfer. Immediately, we think: there is no heat transfer, which means delta Q is equal to zero. So, for this gas, it expands, or rather, it expands, performing a work of 5 joules, or it is compressed, undergoing a work of 80 joules. So, we have the case of expansion, where it performs work, and therefore it expands, meaning its volume increases, or compression, where it undergoes work of 92 joules. So, to answer, we will first state the formula: the variation of internal energy is the variation of heat plus the variation of work. Delta Q is equal to zero because the process is adiabatic. In the case of expansion, we are told it performs work, so it will be negative. Delta U will be delta W, because delta Q is equal to zero. In the case of gas expansion, the work is supplied, it performs work, so delta W will be -5 joules. And so, delta U is equal to -5 joules. And in the case of compression, the work, since we are told it undergoes it, it receives the work, so delta W will be +80 joules. And therefore, the variation of internal energy is +92 joules. Now, let's define the latent heat of phase change, or enthalpy of phase change, symbolized by the letter L. So, L is the energy per unit mass. The unit is kilograms, or grams. If we find the unit in the form of energy per gram, it's generally per kilogram. So, this energy that must be supplied or extracted, of course, to change the initial state of a body, meaning it's in the liquid state, solid state, gaseous state. If we want to change this state, how much must be supplied or removed from this body, considering a unit mass, one kilogram of this body, how much does it need to change its state? So, the unit of L is necessarily joules per kilogram. Now, if we take the body, if we consider the entire body of mass m, we can obtain the relation. We will write that it is L * m. It's simple to derive this formula. Let's use a small rule of three. For example, L = 3 joules per kilogram. Let's do a small rule of three: 3 joules for 1 kg, how many joules is Q for m? And so, Q will be 3 * m, so L * m. So, this energy to change the state of the body will be written as Q = L * m. Q is therefore the energy that accompanies the phase change of the body of mass m. Let's do an application exercise. We are asked to calculate the heat to be supplied to 10 grams of water to change it into vapor under atmospheric pressure, and we are given the latent heat of vaporization of water at 100 degrees Celsius, which is 2265 kilojoules per kilogram. We are given L, we are given m, and we are asked to calculate Q. The relation is very simple: Q = L * m. We just need to convert grams to kilograms. So, 2265 kilojoules per kilogram. 10 grams becomes 10 * 10^-3 kilograms. And we get 22.65 kilojoules. So, we have answered the exercise. Now, let's go back to the definition of the quantity of energy. We talked about heat. We talked about heat, a small definition for heat, positive delta Q, negative delta Q. Now, let's define exactly how delta Q is written. The quantity of energy transferred in the form of heat. So, delta Q is the quantity of heat received or given, and which is necessary to increase or decrease the temperature of mass m by delta T. And we have a mass m which has an initial temperature T. We want to vary the temperature of this mass by delta T. So, how much heat must be given or removed from this body to obtain this variation? So, it is written: delta Q = m * c * delta T. Delta T, since we want to vary the temperature of this body, will be the final temperature T - the initial temperature T. m is the mass. c remains to be defined. So, c is the specific heat capacity. That is, it's the quantity of heat that one gram of matter must absorb for its temperature to increase by 1 degree Celsius. c is a constant, so these are known values that are given to us to do the calculations. That is, we have a substance, water, or anything else. We want to change it. We take 1 gram of this substance and we measure or find out how much heat must be given to this 1 gram for the temperature of this 1 gram to increase by 1 degree Celsius. The unit of c will therefore be joules per degree Celsius per gram. And let's do an application exercise. It says, a pan containing 800 grams of water is heated on a hot plate. If the temperature of the water goes from 20 to 85 degrees Celsius.
degrees Celsius. What quantity of energy was absorbed? And we are given that it is for water. To answer, we start from the formula delta q equals m c delta T. And knowing that delta T is 85 minus 12, which is 65 degrees Celsius. So we have m in grams. We will leave it in grams since it is per gram. And we have c. And therefore delta T, delta T that we have just calculated. We substitute and we obtain 2,217,1880 joules.
Now we will consider the temperature of a mixture of two liquids. How to obtain the temperature of a mixture of liquids? The first, with mass m1, specific heat capacity c1, and temperature T1. And the second, with mass m2, specific heat capacity c2, and temperature T2. With the first, T1 is greater than T2. So T1 > T2. What will happen? If they are mixed together, it is obvious that the hotter liquid will give energy, and the colder liquid will gain energy. So what will happen is that the first liquid will give delta Q1, and the second liquid will receive delta Q2. The heat given by the first liquid will be received by the second liquid. Except that we have a sign problem. The heat given is negative, the heat received is positive. So we will say, for example, the first liquid will give 4 joules, so delta Q1 will be minus 4 joules. The second liquid will receive these 4 joules, so it will be plus 4 joules. So in absolute value, it's the same thing. Absolute value of delta Q1 equals absolute value of delta Q2. Otherwise, we will get rid of the absolute value and say delta Q1 equals minus delta Q2, since there is a minus sign between the two. So we mix two liquids. The first gives to the second. And then we will obtain, of course, a mixture, and we will have a temperature of the mixture at the end. We will have a final temperature which will be the temperature of the mixture, regardless of whether the mixture is homogeneous or heterogeneous. For example, if we mix hot oil with water. So at the end, we will have the temperature of the oil, it will be a final state, and the temperature of the water will be the final temperature, even if they are two different phases. It doesn't matter. So initially, it's T1 for the first, T2 for the second. At the end, it will be T final.
So let's go back to delta Q1 and delta Q2 in terms of formulas. For the first liquid, its temperature goes from T1 to the final temperature T final. So the temperature variation will be T final minus T1. So it will be, how the liquid loses energy, it will cool down. So delta T1 will be negative. And delta Q1, by definition, is m1 c1 delta T1. For the second, its temperature goes from T2 to T final. Since it received heat, it means that relative to it, the final temperature is greater than the temperature it had, T2. So T final minus T2. It will therefore be positive, since liquid 2 was not as hot as liquid 1, so it received temperature, and therefore its temperature increased. Delta Q2 will therefore be m2 c2 delta T2. We reuse delta Q1 equals minus delta Q2, and we substitute. So it will be m1 c1 delta T1 equals minus m2 c2 delta T2. We substitute delta T1 and delta T2 with their values, and we start the calculations, and we obtain T final.
Now, if it is the same substance, if c1 equals c2 and is equal to c, for example, we mix hot oil with cold oil. So it will be the same c. So what have we done? We will simplify c1 and c2, so they will disappear. So instead of having m1 c1 delta T equals minus m2 c2 delta T2, it will simply be m1 delta T1 equals minus m2 delta T2. Now we substitute delta T1 with its expression, T final minus T1, and delta T2 with its expression. We expand. We expand. And then in these calculations, we develop. Then we transpose T1 and T2 to one side, and T final to the other side. So we obtain m1 T1 plus m2 T2 equals m1 plus m2 times T final. And therefore T final will be m1 T1 plus m2 T2 divided by m1 plus m2. We will call m1 plus m2 the total mass, the mass of the mixture. And so it will be T final, which will be m2 T1 plus m2 T2 divided by the total mass.
Let's do an application exercise. To cool 250 milliliters of hot chocolate at a temperature of 80 degrees Celsius, we add 75 milliliters of milk at 20 degrees Celsius. The specific heat capacity of milk is 3.97 joules per gram per degree Celsius, while that of hot chocolate is the same as that of water, which we already had in a previous exercise. If we consider that the densities of hot chocolate and milk are both equal to that of water, 1 gram per milliliter, what will be the final temperature of the mixture?
So we start from the relation. First, the hot chocolate will be m1. We set the information: m1, c1, and T1. c1 is that of water, so it's 4.19 joules per gram per degree Celsius. We set the information concerning the milk: m2, T2, and c2. Then we apply the relation delta Q1 equals minus delta Q2. Substituting, that is, m1 c1 (T final minus T1) equals minus m2 c2 (T final minus T2). Since we are asked to calculate T final in this equation, we have m1, we have everything except T final. The only unknown is T final. So we substitute, we do the correct transpositions, and we obtain the final temperature of 66.72 degrees Celsius.
And our exercise on the same theme, the same subject. We mix 100 milliliters of cold water with 150 milliliters of hot water. So here it's the same c. We measure the temperature of the mixture obtained, and the thermometer indicates 25 degrees Celsius. Knowing that the initial temperature of the cold water was 7.5 degrees Celsius, what was that of the hot water at the start? Always the same small relation. So cold water, we put the information m1. Hot water, we put the information m2. We don't know T2, the initial temperature. The relation we know is T final equals m1 T1 plus m2 T2 divided by the total mass of the mixture, that is, m1 plus m2. The only unknown is T2. We know the final temperature, which is equal to 25 degrees Celsius. And therefore, we obtain the initial temperature T2, which is equal to 36.7 degrees.