The Properties of Gases Kinetic gas theory Maxwell Boltzman distribution, Collisions Real (non-ideal) gases fugacity, Joule Thomson effect Mixtures of gases Entropy, Chemical Potential Liquid Solutions - Electrolyte activity, Henry s and Raoult s Law Thermodynamics of Mixtures - Colligative Properties
Readings Atkins - PC Mortimer PC Pitzer and Brewer Thermodynamics Kittel Kroemer - Thermal Physics http://www.tcd.ie/chemistry/staff/people/duesberg/asin/ 20web/2027-10-09/teaching.html Or also via my chemistry staff page - link to ASIN page
Chapter 1 The Properties of Gases
pressure = p = F / A Pressure force area Ar molecules/atoms of gas are constantly in motion
Kinetic theory and Gas Laws the pressure of a gas increases when it is compressed at constant temperature? Boyles Law When a gas is compressed at constant temperature, the molecules have less volume to move and hit the wall of the container more frequently. As a result, pressure will increases.
Kinetic theory and Gas Laws Boyle s Law pressure volume relationship (temperature is constant) Boyle (1627-1691) p 1/V
The volume of a gas increases when heated at constant pressure - Charles Law When a gas is heated, the gas molecules move faster and hit the wall of the container violently. The volume of gas must increase to keep the pressure constant. So that the gas molecules hit the wall less frequently. Kinetic theory and Gas Laws
Gay-Lussac s Law (also Charles law) temperature volume relationship (pressure is constant) Gay-Lussac (1778-1850) V T
Kinetic theory and Gas Laws ISOTHERMS p 1/V p = const/v => p V = const p 1 V 1 = const p 2 V 2 = const The pressure-volume dependence of a fixed amount of perfect gas at different temperatures. Each curve is a hyperbola (pv = constant) and is called an isotherm. p 1 V 1 = p 2 V 2
Kinetic theory and Gas Laws The pressure of a fixed volume of gas increases with temperature. As temperature rises, the molecules move faster The molecules will hit the walls of the container frequently and violently Hence, the pressure increases
Isobare V T V = const T V/T = const V 1 / T 1 = const V 2 / T 2 = const The variation of the volume of a fixed amount of gas with the temperature constant. Note that in each case they extrapolate to zero volume at -273.15 C. V 1 / T 1 = V 2 / T 2
Surface of states Isobare and Isotherm Chapter 1 : Slide 12
Avogadro s Law 2 H 2 (g) + O 2 (g) 2 H 2 O(l) R = 8.314 J / mol / K Avogadro (1776-1856) n V n 1 / V 1 = n 2 / V 2 k=1.38x10-23 J/K N A = 6.022 10 23 Avogadro number N k = A R
Avogadro principle: Volume of real gases At a given T and p, equal volumes of gases contain the same number of molecules, V m = V/n. Table below presents the molar volumes of selected gases at standard conditions (SATP 25 C and 100kPa) Gas V m /(dm 3 mol 1 ) Perfect gas 24.7896* Ammonia 24.8 Argon 24.4 Carbon dioxide 24.6 Nitrogen 24.8 Oxygen 24.8 Hydrogen 24.8 Helium 24.8 At STP V m = 22.414 m 3 /kmol at 0 C and 101.325 kpa dm 3 mol -1. At IUPAC = 22.711 m 3 /kmol at 0 C and 100 kpa m3/kmol p V n T 1 1 useful : = 1 1 p n 2 2 V T 2 2
IDEAL GAS EQUATION (1) Boyle Law p 1/V (2) Gay-Lussac s Law V T (3) Avogadro s Law n V V 1/p V T V n V T n / p N = nn A p V = const n T p V = k nn A T k=1.38x10-23 J/K p V = R n T R = 8.314 J / mol / K N k = R A N A = 6.022 10 23 Avogadro number
Application: Barometric formula: p as a function of height Consider a column of gas with unit cross sectional area. Variation of pressure with altitude
Barometric formula: p as a function of height Boundary condition: ground level pressure is p 0 so that p = p 0 exp(-mgh/rt) An exponential decrease of p with height. Equal Δh's always give the same proportional change in p. Note the assumptions: 1) Ideal gas behavior 2) Constant g 3) Isothermal atmosphere Mgh is the gravitational potential energy. We will often see properties varying in proportion to exp(-e/rt) = exp(-ε/k B T) where E is a form of molar energy (ε is a molecular energy) because these are examples of "Boltzmann distributions". Chapter 1 : Slide 17
Barometric Formula As elevation increases, the height of the atmosphere decreases and its pressure decreases. F P = = Write in differential form. dp mg F S = = ρ Vg = ρghs S ρhsg = ρgh = ρ gdh Therefore, density = ρ = mass volume Rewrite PV = nrt as = ρ kg 3 m PM W RT Check units. m x 2 s x m = = kg 2 m moles V n = V m 2 s N = 2 m ( M ) P RT W
Continue Derivation of Barometric Formula Substitute the expression for density into the differential eqn. dp = PM RT W g dh Divide both sides of the above equation by P and integrate. dp P MW g = dh RT Integration of the left side and moving the constants outside the integral on the right side of the differential equation gives, MW g MW g ln P = dh = h + RT RT lnc
Continue Derivation of Barometric Formula Evaluating the integral between the limits of P 0 at zero height and P h at height h, gives The constant of integration C can be determined from the initial condition P(h = 0) = P 0, where P 0 is the average sea level atmospheric pressure. ln M gh P = W + h RT ln P 0 P h = 0 P e M W RT gh
Sample calculation Calculate the pressure on Mount Carrauntoohil (1,038 m) under normal conditions? h = 1082 m Temperature as 25 C T = 298 K P 0 = 101.3 kpa = 1 bar (760 Torr) m (air) = 29 g/mol (N 2 = 28 amu, O 2 =32 amu) g = 9.81 ms -2 Standard gravity R = 8.314 J / mol / K p = p 0 exp(-mgh/rt) = 89.5 kpa = 671 Torr 21
Height distribution in a gas IG-09 Energy (E = Mgy) being considered is significantly higher than a quanta of energy. E is nearly continuous. Easier to think of probability density functions: (, ;, ;, ) e mgy kt P x x+ dx x y+ dy x z+ dz dxdydz P is the probability of finding a molecule between x & x + dx, y & y + dy and z & z + dz NB: dx, dy and dz are large compared to a molecule but small compared to the size of the system 22
( ) P y Probability density function e mgy kt dxdydz The directions parallel to the ground (x & z) do not contribute to the probability density function; only the height (y) above ground has an influence y dy x dx
Height distribution in a gas ( ) P y e mgy kt dxdydz For an ideal gas at constant temperature T, the probability density P(y) is related to the number density (# of molecules N per unit volume V ) n(y) : ( ) n y ( = 0) n y = e mgy kt
DALTON S LAW pure gases gas mixtures Dalton (1801) (atmospheres) the total pressure of a gas mixture, p, is the sum of the pressures of the individual gases (partial pressures) at a constant temperature and volume p = p A + p B + p C +.
p V = n R T p A = n A R T / V p B = n B R T / V p = p A + p B p = (n A + n B ) R T / V mole fraction x < 1 p A / p = n A /(n A + n B ) = x A p A = x A p n p = Σ p i i=1
Dalton s Law Suppose we have two gases in a container: n A moles of gas A and n B moles of gas B. We can define individual partial pressures p A = n A RT/V and p B = n B RT/V. Dalton s Law is that the measured total pressure p is the sum of the partial pressures of all the components: p = p A +p B + = (n A +n B + )RT/V. Mole fractions: define x J for species J as n J /n where n = (n A +n B + ). Then, x A + x B + = 1 and p J = p x J
Chapter 2 Kinetic gas theory
Kinetic Molecular Theory of Gases macroscopic (gas cylinder) microscopic Maxwell (1831-1879) (atoms/molecules) Boltzmann (1844-1906)
Kinetic Molecular Theory of Gases Physical properties of gases can be described by motion of individual gas atoms/molecules Assumptions: 1)each macroscopic and microscopic particle in motion holds an kinetic energy according to Newton s law 2)They undergo elastic collisions 3)They are large in number and are randomly distributed 4)They can be treated as points of mass (diameter<< mean free path)
Kinetic Molecular Theory of Gases: Assumptions 1)According to Newton's law of action reaction, the force on the wall is equal in magnitude to this value, but oppositely directed. 2.) Elastic collision with wall: v after = -v before - v v Δvelocity Force = mass = Δtime m 2v Δt
Kinetic Molecular Theory of Gases: Assumptions 3. Avogardo Number Brownian motion 4. Gases are composed of atoms/molecules which are separated from each other by a distance l much more than their own diameter d d = 10-10 m L = 10-3 m.. few m molecules are mass points with negligible volume
Collisions of the gas molecules with a wall L Small volume, v=la, adjacent to wall where L is less than the mean free path F reaction As a result of a collision with the wall the momentum of a molecule changes by
Kinetic Molecular Theory of Gases Pressure = Force total /Area P=F/A F total = F 1 collision x number of collisions in a particular time interval Only molecules within a distance ν x Δt with ν x > 0 can reach the wall on the right in an interval Δt. L = v x Δ t Assume that in a time Δt every molecule (atom) in the original volume, v=la, within the range of velocities will collide with the wall.
Collisions of the gas molecules with a wall This means that Δt is given by: The reaction force of a molecule on the wall is the negative of the average rate of change in the momentum of gas molecules in the volume v that collide with the wall in the time Δt. The total force on the wall is the sum of the average rate of momentum change for all molecules in the volume v=la that collide with the wall Here we have divided by 2 since only ½ of the molecules in our volume have a positive velocity toward the wall
Collisions of the gas molecules with a wall (cont.) We do the sum by noting that the total number of molecules in the volume v is (N/ V) v=la N/ V = density L Remembering Pascal s law dividing by A yields the pressure everywhere. P = F A
Kinetic theory: go from 1 to 3 dimensions L Velocity squared of a molecule: 2 2 x 2 y v = v + v + v 2 z The average of a sum is equal to the sum of averages All the directions of motion (x, y, z) are equally probable. Remember homogeneous and isotropic! Equipartition principle
Kinetic theory Combing these results yields From the ideal gas law And with c = <v> v 2 = c = 3kT 2 = 3kT 2m m Relation between the absolute temperature and average kinetic energy of a molecule.
Kinetic theory v rms of a molecule is thermal speed : The absolute temperature is a measure of the average kinetic energy of a molecule. Example: What is the thermal speed of hydrogen molecules at 800K?