AECH 2131 Physical Chemistry for Chemical Engineers Topic 1 PDF

Topic 1 The properties of Gases AECH 2131 Physical Chemistry for Chemical Engineers 1A. The perfect (ideal) gases 2  1. Variables of State The physical state of a sample of a substance, its physical conditi...

Topic 1 The properties of Gases AECH 2131 Physical Chemistry for Chemical Engineers 1A. The perfect (ideal) gases 2  1. Variables of State The physical state of a sample of a substance, its physical condition, is defined by its physical properties. Two samples of the same substance that have the same physical properties are said to be ‘in the same state’. The variables of state, the variables needed to specify the state of a system, are the pressure, p A system at two different states the volume it occupies, V the temperature, T the amount of substance it contains, n. 3  Pressure and Volume 𝑭 The pressure, p, is defined as a normal force , F, exerted by a fluid per unit area: 𝒑= 𝑨 The SI unit of pressure is pascals (Pa). - Newtons per square meter (N/m2), The volume, V, of a gas is a measure of the extent of the region of space it occupies. The SI unit of volume is m3. 1 bar = 105 Pa 1 atm = 101.325 kPa 760 torr (or mmHg) = 101.325 kPa a) When a region of high pressure is separated from a region of low pressure by a movable wall, the wall will be pushed into the low pressure region until the pressures are equal. (b) When the two pressures are identical, the wall will stop moving. At this point there is mechanical equilibrium between the two regions. 4  Temperature Temperature is a property that describes the flow of energy. Energy flows as heat from a region at a higher temperature to one at a lower temperature if the two are in contact through a diathermic (able to conduct heat) wall. If the two regions have identical temperatures, there is no net transfer of energy as heat. Two regions being at thermal equilibrium Temperature Scales Celsius scale (℃) T(K) = T(℃) +273.15 Fahrenheit scale (℉) T(R) = T(℉) + 459.67 Kelvin scale (K) T(0F) = 1.8 T(℃) +32 Rankine scale (R) T(R) = 1.8 T(K) 5  Absolute Temperature The Kelvin temperature scale begins at absolute zero. Absolute zero, or 0 K, is equal to −273 °C. An increase of 1 K is the same change as an increase of 1 °C. It is not possible to have a temperature lower than 0 K. All particle motion would stop at absolute zero. A sample of gas would not take up any volume at absolute zero. absolute zero 6  Amount of Substance The amount of substance, n, of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles. The SI unit of amount of substance is the mole (mol). The amount of substance is related to the mass, 1 mol of a substance contains exactly 6.02214076×1023 m, of the substance through the molar mass, M, entities. which is the mass per mole of its atoms, its The number of entities per mole is called Avogadro’s molecules, or its formula units. constant, NA. It follows from the definition of the mole 𝒎 that NA 6.02214076×1023 mol-1. It is not correct to 𝒏= 𝑴 specify amount as the ‘number of moles’: the correct The SI unit of molar mass is kgmol−1 but it is more phrase is ‘amount in moles’. common to use gmol−1. 7  Intensive and extensive properties An extensive thermodynamic property is one whose value n m is equal to the sum of its values for the parts of the system. V e.g: mole, mass and volume T P An intensive thermodynamic property is one whose value 𝜌 does not depend on the size of the system. e.g: Density, pressure, and temperature ½n ½n extensive ½m ½m property ½V ½V The value of a property X divided by the amount n gives the T T P P intensive molar value of that property Xm: that is, Xm = X/n. 𝜌 𝜌 property All molar properties are intensive, whereas X and n are both extensive. molar volume 𝑽𝒎 = 𝑽/𝒏 𝑚3 /𝑚𝑜𝑙 molar enthalpy 𝑯𝒎 = 𝑯/𝒏 J/𝑚𝑜𝑙 8  2. Equations of state The state of a pure gas is described by an equation of state, an equation that interrelates four variables (V, p, T, n). The general form of an equation of state is p = f (T,V,n) This equation means that if the values of n, T, and V are known for a particular substance, then the pressure has a fixed value. One very important example is the equation of state of a ‘perfect gas’, which has the form p = nRT/V, where R is a constant independent of the identity of the gas. Generally, a gas behaves more like a perfect gas at higher temperature (above about ℃) and lower pressure (below about 1 atm). The ideal gas model tends to fail at lower temperatures or higher pressures, when intermolecular forces and molecular size become important. 9  The empirical basis of the perfect gas law The equation of state of a perfect gas was established by combining a series of empirical laws that arose from experimental observations. These laws can be summarized as: Boyle’s Law: Charles’ Law: Gay-Lussac’s Law: Avogadro’s principle: 10  Boyle’s Law Boyle investigated the relation between the pressure and volume of gases and found that, for a fixed amount of gas kept at a fixed temperature, P and V are inversely proportional: The pressure-volume dependence of a fixed amount of Straight lines are obtained when the pressure of gas that obeys Boyle’s law. Each curve is for a different a gas obeying Boyle’s law is plotted against 1/V at temperature and is called an isotherm; each isotherm constant temperature. These lines extrapolate to zero is a hyperbola (pV = constant). pressure at 1/V = 0. 11  Charles’ Law and Gay-Lussac’s Law Charles law states that the volume of an ideal gas is directly proportional to the absolute temperature at constant pressure. Gay-Lussac’s Law states the pressure exerted by a gas is proportional to the temperature of the gas when the mass is fixed, and the volume is constant. The volume-temperature dependence of a fixed amount of The pressure-temperature dependence of a fixed gas that obeys Charles’s law. Each line is for a different amount of gas that obeys Charles’s law. Each line pressure and is called an isobar. Each isobar is a straight line is for a different volume and is called an isochore. and extrapolates to zero volume at T = 0 K, corresponding to Each isochore is a straight line and extrapolates to T =−273.15 °C. zero pressure at T = 0 K. 12  Avogadro’s principle For a given mass of an ideal gas, the volume and amount (moles) of the gas are directly proportional if the temperature and pressure are constant. Avogadro's law states that, "equal volumes of all gases, at the same temperature and pressure, have the same number of molecules". 𝑽 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 (𝑎𝑡 𝑐𝑜𝑛𝑠𝑡. 𝑇 𝑎𝑛𝑑 𝑃) 𝒏 13  Ideal Gas Law The combination of Boyle’s law, Charles’ law and Avogadro's hypothesis gives the ideal gas law; 𝑷𝑽 = 𝒏𝑹𝑻 P: absolute pressure of a gas V: volume of the gas n: number of moles of the gas R: the gas constant; The gas constant is now T: absolute temperature of the gas defined as R = NAk, where NA is Avogadro’s constant and k is Boltzmann’s constant. For a flow system: 𝑷𝑽ሶ = 𝒏𝑹𝑻 ሶ 𝑉:ሶ volumetric flow rate of the gas 𝑛:ሶ molar flow rate of the gas Molar volume version: 𝑷𝑽𝒎 = 𝑹𝑻 14  Combined Gas Law The perfect gas law in the form pV/nT = R implies that, if the conditions are changed from one set of values to another, then, because pV/nT is equal to a constant, the two sets of values are related by the ‘combined gas law’: 𝑃1 𝑉1 𝑃2 𝑉2 = 𝑛1 𝑇1 𝑛2 𝑇2 15  Molecular Weight of a Perfect Gas 𝒎 The number of moles in a gas is 𝒏= 𝑴 and ideal gas is 𝑷𝑽 = 𝒏𝑹𝑻 𝑷𝑽 𝒎 𝒎𝑹𝑻 When we combine these equations = →𝑴= 𝑹𝑻 𝑴 𝑷𝑽 𝝆𝑹𝑻 Since, 𝒎 = 𝝆𝑽 then 𝑴= 𝑷 16  Mixtures of Ideal Gases Different gases are mixed up with each other completely by diffusion in the same container in a very short time. It was found that the total pressure of a gas mixture was the sum of the pressures of each gas in the mixture in the same container at constant temperature. Dalton’s Law: mol fraction, 𝑥𝑖 17  𝒏𝒊 amount of moles of a gas in the mixture Mol Fraction, 𝒙𝒊 = 𝒏𝑻 total amount of moles in the mixture 𝒏𝒊 𝑷𝒊 = × 𝑷𝑻 𝑷𝒊 = 𝒙𝒊 × 𝑷𝑻 𝒙𝑻 = 𝒙𝟏 + 𝒙𝟐 + ⋯ + 𝒙𝒏 = 𝟏 𝒏𝑻 Amagat’s Partial Volumes Law: Each gas in the ideal gas mixture, has its own partial volume in a total certain volume. This partial volume cannot be measured therefore, it can only be calculated by using the ideal gas law. 𝑽𝟏 = 𝒙 𝟏 × 𝑽𝑻 𝑽𝟐 = 𝒙 𝟐 × 𝑽𝑻 𝑽𝟑 = 𝒙 𝟑 × 𝑽𝑻 18  Mean Molecular Weights of Ideal Gas Mixtures: An ideal gas mixture has its own mean molecular weight. 𝑴𝒎𝒆𝒂𝒏 = 𝒙𝟏 × 𝑴𝟏 + 𝒙𝟐 × 𝑴𝟐 + 𝒙𝟑 × 𝑴𝟑 + ⋯ 19 20 Gas Properties – Interactive Simulation https://learncheme.com/simulations/thermodynamics/thermo-1/ideal-gas-law/ https://phet.colorado.edu/sims/html/gases-intro/latest/gases-intro_all.html 21 1B. The kinetic model 22  Development of “kinetic theory of gases” Everything described for ideal gases in the previous section are obtained from the macroscopic physical facts which can be determined experimentally by using thermometers, manometers balance and volumetric measurements. However, the results of such an experiment can be theoretically derived by applying the "kinetic theory of gases" by using Newtonian mechanics and calculus mathematics. Kinetic theory of gases is one of the most important scientific discovery in the history of science. The development of this theory first began in 1738 by the publications of Swiss mathematician Bernoulli. Clausius (German), Joule and Maxwell (British), Boltzmann (Austrian), Van der Waals (Dutch) were involved in the improvement of “Kinetic Theory” after 1830 and developed it to its present form in 1890’s. The validity and accuracy of the kinetic theory was accepted in the scientific world after being discussed and tested intensively between 1890 and 1920. 23  Assumptions used to derive the kinetic gas theory: 1. Gases are made up of very small particles called “molecules’’. The same molecules of a gas have the same mass and volume. Molecules of different gases have different masses and volumes. There are very large number of molecules even in very small volume of a gas. The own sizes (i.e. the volumes) of the gas molecules are neglected in comparison with the volume of the container which occupies the gas. 2. The gas molecules in a container are not stagnant and make chaotic and linear flight movements continuously and constantly. They collide with each other and with the walls of the container during their flights. Molecular movements obey the classical Newtonian mechanics (i.e. velocity, force, energy equations). 24  Assumptions used to derive the kinetic gas theory: 3. “Gas pressure’’ is generated by the bombardment the walls of the container by the gas molecules with their momentums during their flights. The pressure of a gas is the average total force on the unit wall area in which the molecules collide with the wall. 3. All intermolecular collisions and all the collisions of gas molecules with the walls of the container are assumed to be “elastic”. Consequently, there is no energy loss in these collisions which can be converted into heat by friction. For this reason, the pressure of a gas on the walls of a container (when the temperature and pressure is constant) does not change over time. 25  Assumptions used to derive the kinetic gas theory: 5. At low pressures, the average distances between the gas molecules are much higher than the diameters of these gas molecules. Therefore “the attraction forces between these molecules’’, which are given by Newton's gravitation law and varying with the square of the distance between the molecules, are very small. Therefore, these gravitational attraction forces are assumed to be completely neglected. Likewise, because of the distance between them is too high, the molecular masses are considered to be “point mass’’ where the volume of the point mass is zero. 6. “The absolute temperature’’ is a magnitude directly proportional to the “average kinetic energies” of the gas molecules. The temperature measured by the thermometer is actually an indication of the average kinetic energies of the molecules. It is assumed that the average kinetic energies of all gases are the same at the same temperatures. In kinetic theory, all the energy of the gas is considered to be the sum of the motion energy of the molecules. 26  Pressure and molecular speeds n : the number of gas molecules After derivations we observed: M: molar mass of the molecules Basic equation of the Kinetic Theory: 27  Mean values With the Maxwell–Boltzmann distribution we can The Maxwell–Boltzmann distribution can be used to calculate: evaluate the mean speed, vmean, of the molecules in a gas: The mean relative speed, vrel, the mean speed with which one molecule approaches another of the same kind, can also be calculated from the distribution: 𝑷𝑽 = 𝒏𝑹𝑻 28  The collision frequency and The mean free path Although the kinetic model assumes that the molecules are point- like, a ‘hit’ can be counted. The collision frequency, z, is the number of collisions made by one molecule divided by the time interval during which the collisions are counted. σ : collision cross-section k : Boltzmann’s constant 29 1C. Real gases 30  The perfect gas equation of state is an approximation! It works well under some conditions—generally speaking, at temperatures above about 0°C and pressures below about 1 atm— but at other conditions its use may lead to substantial errors. Here is a useful rule of thumb for when it is reasonable to assume ideal-gas behavior. Let Xideal be a quantity calculated using the ideal-gas equation of state [ X = p(absolute), T(absolute), n or V ] and ε be the error in the estimated value, An error of no more than about 1% may be expected if the quantity RT=P (the ideal specific molar volume) satisfies the following criterion 31  Real gases Real gases do not obey the perfect gas law exactly except in the limit of p → 0. Real gases show deviations from the perfect gas law because molecules interact with one another. Repulsive forces: assist expansion of gas significant when molecules are close to one another operative at high pressures, when intermolecular distances are near a single molecular diameter Attractive forces: assist compression of gas The dependence of the potential energy and of the can have influence over a long distance (close but not force between two molecules on their internuclear touching) separation. The region on the left of the vertical dotted line indicates where the net outcome of the operative at moderate pressures intermolecular forces is repulsive. At large separations (far to the right) the potential energy is zero and there is no interaction between the molecules. 32  The compression factor An ideal gas obeys the equation of state; pVm = RT (Vm, molar volume Vm=V/n) As a measure of the deviation from ideality of the behavior of a real gas, we define the compressibility factor or compression factor Z of a gas as pVm = ZRT where Z(p, T) – as a function of p and T For an ideal gas, Z = 1 for all temperatures and pressures. 33  Note that; Z = Vm/Vmid perfect (Vmid is the molar volume of an ideal gas at the same T and gas P as the real gas) Z = P/Pid, (Pid is the pressure of an ideal gas at the same T and Vm as the real gas) → Ideal gas: Z = 1 → Intermediate Pressure: Z < 1 perfect gas Compression is favored, due to dominance of attractive forces perfect gas → High Pressure: Z > 1 Expansion is favored, as repulsive forces come into play 34 An algebraic formula for the equation of state of a real gas is more convenient to use than numerical tables of Z. 1. Virial equation of state 2. Cubic equations of state a) van der Waals equation of state b) Redlich–Kwong equation of state (RK) c) Soave-Redlich–Kwong equation of state (SRK) d) Peng-Robinson (PR) 35  Virial equation of state At large molar volumes and high temperatures the real-gas isotherms do not differ greatly from perfect-gas isotherms. The coefficients B, C,..., the values of which depend on the temperature, are the second, third,...virial coefficients. The first virial coefficient is 1. The third virial coefficient, C, is usually less important than the second coefficient, B, in the sense that at typical molar volumes C/Vm Tc may be much denser that we normally consider typical of gases, and the name supercritical fluid is preferred. An isotherm slightly below T < Tc ( T1 or T2 ) behaves as we have already described: at a certain pressure, a liquid condenses from the gas and is distinguishable from it by the presence of a visible surface. The temperature, pressure, and molar volume at the critical point are called: ▪ critical temperature, Tc ▪ critical pressure, pc ▪ critical molar volume, Vc 43  The principle of corresponding states A related fundamental property of the same kind and to set up a relative scale on that basis is necessary. The critical constants are characteristic properties of gases, so it may be that a scale can be set up by using them as yardsticks and to introduce the dimensionless reduced variables of a gas by dividing the actual variable by the corresponding critical constant: The compression factors of four gases plotted using reduced variables. The curves are labelled with the Tr reduced temperature Tr = T/Tc. The use of reduced variables organizes the data on to single curves. pr 44 Generalized compressibility chart at low pressures 45 Generalized compressibility chart at medium pressures 46 Generalized compressibility chart at high pressures 47 The van der Waals equation w/ reduced variables This equation has the same form as the original, but the coefficients a and b, which differ from gas to gas, have disappeared. 48 AECH 2131 Physical Chemistry for Chemical Engineers