Kinetic Theory of Gases (KTG)

PV/nT =constant

R=8.314Jmol^-1K^-1 or 25/3 Jmol^-1K^-1 R=2 calmol^-1K^-1

PV=nRT=NkbT where kb=R/Na kb value =1.38*10^-23 J/K ,n=M/M° ,M is total mass of the gas M° is molar mass of gas P=(p/M°)RT( p is density) P=(p/m) Kb T( m is mass of one molecule)

· The gas is a collection of large number of molecules. The molecules are perfectly elastic rigid spheres and are different for different gases.
· The size of a molecule is negligible compared with the distance between the molecules. · The molecules are always in a random motion.
· During their motion, the molecules collide with each other and with the walls of the containing vessel. The collisions are elastic and hence the total K.E. and the total momentum of the colliding molecules before and after collisions are the same.
· Between two collisions a molecule moves in a straight path with a uniform velocity.
· There is no force of attraction or repulsion between molecules.

gas that obeys the ideal gas equation at all temperatures and pressures.
No real gas is truly ideal.

[Diagram]
IDEAL GAS = [Box: Obeys gas law] + [Box: Ideal gas equation] + KTG
at All Temperature and pressure

While deriving ideal gas equation the following two assumptions were used:

· The size of the gas molecules is negligibly small.
· There is no force of attraction amongst the molecules of the gas

Real gases approach ideal gas behaviour at low pressures and high temperatures.

The compressibility factor denoted by Z, is a dimensionless ratio quantifying how much a real gas deviates from ideal gas behavior. Z=actual molar volume/ideal molar volume =PV/nRT for one mole Z=PV/RT Z=1 ideal gas Z>1 positive deviation Z<1 negative deviation

real gases do not obey gas laws and gas equations Van der waal made two corrections in ideal gas equation 1. the real gases show considerable deviations from equation of ideal gas as (P+(n^2a/V^2))(V-nb)=nRT for n mole of gas

P=(Nm/V)*(vrms^2/3) =(pvrms^2)/3, p is density of gas

PV=(2/3)E ,E/N=(1/2)mvrns^2= (3/2)KbT

PTotal=P1+P2+... =(n1+n2+....)RT

r12=√(M2/M1)

vrms=√((v1^2+v2^2....vn^2)/n). vrms= √(3RT/M)= √(3PV/M)= √(3P/d)= √(3KbT/m)

V=√(gamma*RT/M) ,Vrms=√(3amma) * speed of sound of speed of gas

Vavg=|V1|+|V2|+...+|Vn|/n ,Vavg=√(8RT/πM)=√(8P/πd)=√(8KbT/πm)

Vmp=√(2RT/M)=√(2P/d)=√(2KbT/m)

Vrms>Vavg>Vmp Vmp:Vavg:Vrms=1:1.14:1.228

In the kinetic theory of gases, degrees of freedom (f) refer to the number of independent ways a molecule can move and store energy. These motions include translational (moving along X, Y, Z axes), rotational (spinning), and vibrational (oscillating).

f=3T T here is transnational Examples
He, Ne, Ar, Xe, Kr

For rigid diatomic
Examples
O_2, N_2, F_2, Cl_2, Br_2
f = 3T + 2R
f = 5 (R here is rotational degree of freedom). For non rigid diatomic
f = 3T + 2R + 1V

where
T here is Translational degrees of freedom
R here is Rotational degrees of freedom
V here is Vibrational degrees of freedom

Linear Triatomic
f = 3T + 2R
f = 5

Examples
CO_2, CS_2, BeCl_2

Non Linear Triatomic
f = 3T + 3R
f = 6

Examples
H_2O

1) the number of atoms forming a molecule. 2)the structure of the molecule. 3) the temperature of gas Temperature increase degree of freedom increases

According to law of equipartition of energy, for any system in thermal equilibrium, the total energy is equally distributed among its various degrees of freedom. Each degree of freedom is associated with an energy (1/2)kbT for a molecule energy per degree of freedom total energy of one molecule of gas =(f/2)kbT ,for a dree of freedom fv

1 molecule= (f/2)KbT. n molecule=(f/2)nKbT 1 mole =(f/2)RT. n mole =(f/2)nRT K.E=(3/2)KbT (transilatory motion only) Note : Each vibrational mode gives 2 (not 1) degrees of freedom (kinetic and potential energy modes), corresponding to the energy

Cp-Cv=R

gamma=Cp/cv

Cv=(3+f)R ,Cp=(4+f)R ,gamma=(4+f)/(3+f)

Cvmix=(n1Cv1+n2Cv2)/(n1+n2) Cpmix=(n1Cp1+n2Cp2)/(n1+n2) gamma(mix)=Cp(mix)/Cv(mix). n1+n2/(gamma(mix)-1)=(n1/(gamma1 -1)) +(n2amma2 -1)

Emix= n1E1+n2E2 =(f1/2)n1RT+(f2/2)n2RT

Tmix =(n1F1T1+n2F2T2+....)/n1F1+n2F2+....

C=∆U/∆T=3R ,(3R=6 cal mol^-1K^-1 =25 J mol^-1K-1 for most solids debye temperature is close to room temperature

C=∆U/∆T=9R

lamda=1/(√2 nπd^2) n is no of molecules per unit volume d is diameter of gas molecule lamda=m/(√2 pπd^2) p is density lamda= KbT/√2Pπd^2 P is pressure of gas T is absolute temperature

=1/(nπ<v>d^2) here <v> is average speed