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0:05
[Music]The three physical states of matter are solid, liquid, and gas. Chapter one is about the properties of gases. The first half of the chapter covers an idealized gas model, called, “perfect gas”. We will learn the general properties of gases. The second half of the chapter is about real gases. We will learn what deviates the real gases from the behavior of perfect gas. Gas is a collection of molecules or atoms in continuous random, chaotic motion with speed that increases as
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temperature is raised. It is different from a liquid in that, except during collisions, the molecules of a gas are widely separated from each other and move largely independently of each other. The states of a gas are specified by its volume, pressure, moles, and temperature.
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What is pressure? I mean, how do you define the pressure? It is force per unit area or F over A. It is directly proportional to force but inversely proportional to area. Here is force acting on Area 1 and the same force acting on Area 2. Which one would produce greater pressure? The answer is the force acting on Area 2 because pressure is inversely proportional to area and Area 2 is smaller than Area 1. Consequently, p2 is greater than p1. SI units of pressure is pascal.
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Pascal is equivalent to newton per meter squared. Take note that newton is the SI unit of force
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and meter squared is the SI unit of area. Pascal is also equivalent to kilogram per meter per second squared. Bar is a non-SI unit. One bar is equal to 100 kPa. atm is another non-SI unit. One atm is equal to 101.325 kPa. Pressure allows us to determine mechanical equilibrium. What does that mean? Check this animated illustration. A gas is trapped in the cylinder and exerts internal pressure, p in. The pressure outside the cylinder or external pressure is denoted by p ex.
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What do you expect to happen if external pressure is greater than internal pressure? The gas will be compressed until the two pressures become equal. What would happen if external pressure is less than internal pressure? The gas will expand until the two pressures become equal. If the two pressures are equal, then the piston won’t move due to the mechanical equilibrium between inside and outside of the system. This is how we learn whether the system
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is in mechanical equilibrium or not. The next factor that specifies the physical states of gas is temperature. It determines whether the system reaches thermal equilibrium or not. And it also determines the direction of heat flow. This is a system split into two compartments by a diathermic divider, which allows heat transfer between the two compartments. The whole system is isolated from the surroundings by an adiabatic wall, which does not allow heat transfer between the
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system and the surroundings. Suppose the temperature of the left compartment is higher than the temperature of the right compartment. Which direction do you think heat would flow? The answer is obvious. Heat would flow from the left to the right. This is because heat always flows from an object at a higher temperature to an object at a lower temperature until the temperatures at both sides become equal. The state of being at equal temperatures on both sides is called, “thermal equilibrium”.
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The zeroth law of thermodynamics describes well about the thermal equilibrium. It states, "If A is in thermal equilibrium with B, and B is in thermal equilibrium with C, then A is in thermal equilibrium with C. It provides the operational principle of a thermometer. Here are Solution A and Solution C. Thermometer B is placed in Solution A until both solution and thermometer are in thermal equilibrium, which is indicated by persisting constant height
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of the red liquid inside Thermometer B. Now, the same thermometer is placed in Solution C. If the red liquid inside Thermometer B remains the same height with the one at the previous step, then we can say that both Solution A and Solution C are in thermal equilibrium. Consequently, both Solution A and C are at the same temperature. Solution A can be replaced by a reference liquid like water with known freezing temperature and boiling temperature.
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You can calibrate and scale your new thermometer based on the freezing and boiling temperatures of a reference liquid prior to measuring the temperature of unknown Solution C. However, here is an issue. The thermometers that we normally use in laboratories have a flaw that is associated with inconsistent thermal expansion of thermometer liquid. I mean the degree of expansion of thermometer liquid from 10 to 20 degree
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C is likely to be different from the degree of expansion of it from 50 to 60 degree C. As a result of the inconsistency, temperature readings are not reliable. A perfect gas thermometer is the most reliable thermometer because it does not have such an issue. Boyle’s law states under a low pressure condition, pressure times molar volume or pVm always makes a constant value. This constant value is no longer constant if temperature changes, which is
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denoted by function of theta. Be aware that theta is temperature in Celsius. You may plot the function, f with respect to temperature, theta. You make pick any temperature on the x-axis like 100 Celsius, its corresponding value on the y-axis is always fixed. It illustrates that as long as temperature is fixed, the value of pVm always remains constant. Take note when temperature decreases, the value of function, f also decreases. Remember, the function, f is the same with PVm.
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The plotted line can be extended and it will cross the x-axis at negative 273.15 Celsius. This temperature is called, “absolute zero” or or zero Kelvin. Take note that temperature below absolute zero cannot be reached. This is because either pressure or molar volume should be negative to make negative PVm, which makes no sense. SI unit of temperature is Kelvin, which is equal to Celsius plus 273.15. Celsius and Fahrenheit are non-SI units of temperature.
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Celsius and Fahrenheit are related by this equation.
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Like I said in the previous slide, the physical state of a gas can be described by four variables, volume, pressure, moles, and temperature. There are three gas laws that show the relationships among those four variables. The three gas laws are Boyle’s law, Charles’s law, and Avogadro’s principle. Each gas law describes how two of the four variables are related while the remaining two variables are kept constant.
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Suppose moles and temperature of gas are kept constant. Check the equation of state for perfect gas, pV equals nRT. The right side of the equation, nRT is a constant as a whole because multiplying three constants n, R, and T makes another constant value. It implies that pressure times volume always produces a single constant value regardless of the states of gas as long as moles and temperature of gas are fixed. It further implies that pressure and
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volume of gas are inversely proportional to each other. This is Boyle’s law. Suppose moles and pressure of gas are held constant. Check the perfect gas equation again. Keep the variables on the left side of the equation and keep the constants on the right side. As you can see, nR over P is a constant as a whole. It implies volume over temperature always produces a single constant value regardless of the states of gas as long as moles and pressure of gas are fixed. It further implies that
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volume and temperature of gas are directly proportional to each other. This is Charles’s law. Finally, suppose temperature and pressure of gas are held constant. Keep the variables on the left side of the perfect gas equation and keep the constants on the right side like we did before. As you can see, RT over P is a constant as a whole. It implies volume over moles always produces a single constant value regardless of the states of gas as long as temperature and pressure of gas are
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fixed. It further implies that volume and moles of gas are directly proportional to each other. This is Avogadro’s principle. The three gas laws can be combined by utilizing the direct and inverse proportional relationships between the four variables like this, pV over nT equals a constant. The constant is called, “Gas constant” and denoted by Capital R. The equation can be rearranged to pV equals nRT or pVm equals RT, which brings the perfect gas equation back. The gas
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constant, R is 8.314 joules per mole per Kelvin or 0.08206 liters per atm per mole per Kelvin. Let’s use the perfect gas equation and construct a 3D plot of perfect gas with axes of pressure, temperature, and volume. Be aware that moles of gas remain constant. Charles’s law states that volume and temperature of gas are directly proportional to each other, which is represented by this straight blue line. Relocate the axis of temperature like
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this. Boyle’s law states that pressure and volume of gas are inversely proportional to each other, which is represented by these two red curves. Perfect gas equation also exhibits that pressure and temperature of gas are directly proportional to each other, which is represented by this green line. All possible states of perfect gas in any pressure, volume, and temperature are present on the orange surface of the plot. The pressure-volume plot from the
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Boyle’s law is called, “Isotherm” because it is under a constant temperature condition. The volume-temperature plot from Charles’s law is called, “isobar” because it is under a constant pressure condition. The pressure-temperature plot is called, “isochore” because it is under a constant volume condition. Empirical data values are most likely to vary if laboratory conditions change. So we need a reference state that we can compare our data from, which is called,
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“standard state”. STP is standard temperature and pressure with temperature of 0 Celsius and pressure of 1 atm. The molar volume of perfect gas at STP can be calculated simply by plugging the standard pressure and temperature into the perfect gas equation, which is 22.414 liters per mole. SATP is standard ambient temperature and pressure with temperature of 25 Celsius and pressure of 1 bar. The molar volume of perfect gas
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at SATP can be calculated simply by plugging the standard ambient pressure and temperature into the perfect gas equation, which is 24.789 liters per mole. Dalton’s law of partial pressure states, “The pressure exerted by a mixture of perfect gas is the sum of the pressures exerted by the individual gases occupying the same volume alone.” The law can be expressed in a mathematical equation like this. The pA, pB, and pC are the partial pressures of gas components A, B, and C in the mixture.
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Mole fraction is the number of moles of a specific component in a mixture divided by the total number of moles in the mixture. It is denoted by x sub J and J represents component, J in the mixture. The mole fraction of J can be written as n sub J over n sub total. The n sub J is the moles of component J, and n sub total is the total moles of the gas mixture. Use the perfect gas equation to replace both n sub J and n sub total with pV over
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RT. Take note that gas J has the same volume and temperature of the mixture because it shares the same container and has the same thermal condition with other gas components. The Volume and RT of both numerator and denominator are cancelled, which is simplified by P J over P total. Now the equation can be re-written as P J equals x J times p total. This equation is very useful because if you know the mole fraction of a
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component in a gas mixture and the total pressure exerted by the mixture, you can easily determine the partial pressure of the component with this equation. Take note that the sum of mole fractions of all components in a mixture is always one. Sum the partial pressures of all gas components. Each partial pressure can be replaced with mole fraction times p total. P total can be factored out. Since the sum of all mole fractions is one, the sum of individual partial pressures
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in a gas mixture is p total. And this is the Dalton’s law of partial pressures. Perfect gas is an idealized model gas that obeys the perfect gas equation. There are no intermolecular interactions, I mean, intermolecular attractions and repulsions. However, real gases are quite different from the perfect gas in that they deviates significantly from the behavior of perfect gas. When gas molecules or atoms are separated from one another by relatively
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longer distance, the gas particles attract one another. This is because positively charged nuclei attracts negatively charged electrons. This long range attraction reduces gas pressure lower than the pressure of perfect gas at the same condition and it leads to the compression of the real gas. When gas molecules or atoms are close to one another, the gas particles repel one another. This is because positively charged nuclei repel one another and negatively
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charged electrons also repel one another. This short range repulsion raises gas pressure greater than the pressure of perfect gas at the same condition and it leads to the expansion of the real gas. Compression factor is a tool to determine the degree of deviation of real gases from the behavior of perfect gas and it is denoted by Capital Z. The Capital Z is defined by p Vm over RT. It is obvious that the compression factor of perfect gas is one because p Vm equals RT.
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Repulsive intermolecular interactions of a real gas expand the gas due to the increase of gas pressure. This would increase the numerator value of the compression factor and as a result of it, Z value becomes greater than one. Attractive intermolecular interactions of a real gas compress the gas due to the reduction of gas pressure. This would decrease the numerator value of the compression factor and as a result of it, Z value becomes less than one.
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Virial equation of state describes how a real gas deviates from the behavior of perfect gas. The virial coefficients, B’, C’ exhibit the degree of deviation. Suppose all virial coefficients are zero, then only the first term, one remains and this makes the perfect gas equation. Because pressure is inversely proportional to molar volume or directly proportional to one over molar volume, the virial equation can be written like this. Here, B’, C’ in the 1st equation and B and C in the second
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equation are called, “virial coefficients.” Now focus on the second equation and move RT to the left side of the equation. Then, the equation is modified like this. Check the right side of the modified equation. What is it? It is the compression factor, Capital Z. Now the virial equation is modified for the compression factor. Under a certain condition, the real gas behaves like perfect gas. What is that condition? Focus on the equation of compression factor
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and take the first derivative of Capital Z with respective to one over Vm. This makes the right side of the equation, “B + 2C times one over Vm.” Take note that virial coefficients depend on temperature. Under the condition of high temperature, the molar volume of gas increases significantly, which approximates one over Vm to zero. This makes the second term and beyond negligible and as a result of it, the left side of the equation is simplified to B.
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You may adjust the temperature of gas until the virial coefficient B becomes zero. The temperature that makes B zero is denoted by T sub b. Now check the compression factor of virial gas equation. Since B is zero and the third term and beyond are negligible, the compression factor can be approximated to one, or PVm is approximated to RTB. It implies that real gas at TB behaves like a perfect gas. The temperature, TB is called, “Boyle temperature.” In other words, real gas at Boyle
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temperature behaves like a perfect gas. Here is a plot Capital Z versus p or one over Vm. Apparently, the compression factor of perfect gas is one regardless of pressure or volume. Under high temperature condition, the volume and pressure of real gas is greater than the volume and pressure of perfect gas. This would raise the Capital Z value beyond the ideal value, 1. The degree of deviation becomes greater as pressure or one over Vm increases. Be
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aware as pressure decreases, the value of Capital Z approaches to one, which implies that the real gas behaves like a perfect gas under a low pressure condition. Under a low temperature and moderate pressure condition, the Capital Z value of real gas becomes less than one. This is due to the long range intermolecular attractions that are more favorable over the short range intermolecular repulsions. At Boyle temperature, the curve at a low
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pressure has zero slope, which makes the real gas behave like a perfect gas.
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This is a generic phase diagram of a single component system. Regions of solid, liquid, or gas represents conditions of pressure and temperature at which each phase is stable. Lines between phases are called, “Phase boundary lines”. Each phase bounded line represents a set of temperatures and pressures at which the substance is in equilibrium between the two phases on either side of the line. Triple point is the unique set of
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conditions at which the three phases, solid, liquid, and gas are in equilibrium. Critical point represents the temperature and pressure above which a supercritical fluid exists. Beyond this point the liquid and gas phases become indistinguishable from each other. The pressure, volume, and temperature at the critical point are called, “critical pressure, critical volume, and critical temperature” and denoted by “Pc, Vc, and Tc.” They are also called, “critical constants”
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as a whole. Take note that a single phase is formed at and above the critical temperature. This single phase is called, “supercritical fluid”. The Van der Waals equation of state was derived by amending perfect gas equation and includes corrections for intermolecular attractions and repulsions. Keep in mind that real gases have non-negligible intermolecular interactions. Like I said in previous slide long-range intermolecular attractions of real gas cause its pressure lower than the
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pressure of perfect gas at the same condition. Here n squared a over V squared is added to pressure and it is the correction for the reduced pressure short range intermolecular repulsions of real gas caused the expansion of its volume. Here nb is subtracted from volume and it is the correction for the increased volume due to intermolecular repulsions the n squared a over v squared can be modified by dividing both numerator and denominator by n squared
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And it can be rewritten as a over Vm squared. Both V minus nb and nRT can be divided by n and they can be rewritten as Vm minus b and RT. This modified van der Waals equation can be rewritten with respect to pressure like this. We will use this form of van der Waals equation for the remaining slides. The a and b are van der Waals coefficients and they are independent of temperature. Individual gas has different values of a and b. The coefficient, a is a positive value
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and it represents reduced pressure due to intermolecular attractions. The coefficient, b is also a positive value and it represents molar volume occupied by gas molecules specified by intermolecular repulsions.
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Let’s look into the behavior of real gas on a pressure-volume plot. Boyle’s law states that the pressure of gas is inversely proportional to its volume under a constant temperature condition. So, you may expect an isotherm like this shape, which is what we expect from perfect gas. Actually, under a low pressure and high volume condition, real gas behaves like perfect gas. I mean, as the volume of real gas decreases, its pressure increases like
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the way that perfect gas does. However, when the real gas is compressed further below a certain point, its volume is reduced rapidly all of a sudden without increasing pressure. After this point, further compression requires enormous pressure. So, what happened to the real gas at the plateau region, A to B. Gas begins to condense to liquid at Point A and the transition to liquid is completed at Point B. Only single phase liquid is present
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after Point B. The region A to B is where both gas and liquid coexists. Be aware that the van der Waals equation at low temperature, T1 exhibits an abnormal trend in the pressure-volume plot. Check the region with positive slope. In this the region, as volume decreases, pressure also decreases, which does not make any sense at all. This is the greatest flaw of the van der Waals equation. The oscillations of the curve are called, “van der Waals loops”. But at the critical temperature and
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beyond, it fits well with empirical data. At temperature above the critical temperature, the gas behaves like a perfect gas. The critical point is an inflection point where the curvature changes. The van der Waals plot has the following features. First, perfect gas isotherms are obtained at high temperature and larger molar volume. Second, liquid and gas coexist when cohesive and dispersing effects are in balance.
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Third, the critical constants are related to the van der Waals coefficients. And I will show you how the critical constants are related to the van der Waals coefficients. Like I said previously, the critical point has the same properties of an inflection point. This is the van der Waals equation written with respect to pressure. Because the critical point is an inflection point, both the first derivative and the second derivative of pressure with respect to molar volume are zeros.
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the first derivative is negative RT over Vm minus b squared plus 2a over Vm cubed equals zero. And the second derivative is 2RT over Vm minus b cubed minus 6a over Vm to the fourth equals zero. We may replace pressure, molar volume, and temperature with Pc, Vc, and Tc because we are working on the critical point. These two equations are simultaneous equations and we can determine each of the Vc and Tc by manipulating the two equations. Once Vc and Tc are determined,
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you should plug them to the van der Waals equation to determine Pc. Modify the two equations like this. Combine the two equations by dividing the left side of the first equation by the left side of the second equation and dividing the right side of the first equation by the right side of the second equation. This would lead to Vc over 3 equals Vc minus b over 2. Now Vc is 3b. You may plug this into either the first equation or the second equation.
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Let's plug it into the first equation like this. Remember, you may do the same to the second equation and you will get the same Tc. Now, the Tc is 8a over 27Rb. Plug the Vc and Tc to the van der Waals equation. Finally, Pc is a over 27 b squared. Now, the critical constants, Pc, Vc, and Tc are expressed with respect to van der Waals coefficients, a and b. It implies that you can calculate the critical constants of any real gas if you are given with the van der Waals coefficients of the gas.
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We can also determine the compression factor of van der Waals gas at the critical point. Plug the Pc, Vc, and Tc to the equation for compression factor. The fraction is simplified by canceling a, b, R, and 27 in both numerator and denominator. The compression factor of a van der Waals gas at critical point is three over eight. This is less than one. Compression factor less than one implies that attractive intermolecular interactions are dominant.
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Critical point is an intrinsic property of a substance and can be used as a scale or yard stick. Reduced variables are the ones produced by utilizing critical constants as reference. Reduced variables are reduced pressure, reduced volume, and reduced temperature and denoted by Pr, Vr and Tr. The Pr, Vr and Tr of a gas are obtained simply by dividing pressure, volume, and temperature of gas by the critical pressure, critical volume, and critical temperature of the gas, respectively.
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Be aware that the reduced variables do not have units because units in the numerator and denominator are cancelled. The three variables, pressure, volume, and temperature can be re-written like this. Plug these into the van der Waals equation. Plug the Pc, Vc, and Tc that were determined previously. This would lead to this equation. This new equation does not have van der Waals coefficients, a and b which differ from gas to gas. In this plot, the compression factors of
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nitrogen, methane, propane, and ethane are plotted with respect to reduced pressure. Surprisingly, Capital Z plot exhibits exactly the same at each reduced temperature 1.0, 1.2, and 2.0 regardless of gases. The principle of corresponding states is, “real gases at the same reduced volume and reduced temperature exerts the same reduced pressure." This approximation works best for gases that consist of spherical molecules.
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