Formula for the Calculation of Heat Capacities , Specific Heat Capacity Of Gas And Ratio Of Specific Heat Capacities At Constant Pressure And Volume Of Gases With Derivations Of Relations

HEAT CAPACITIES OF GASES

Heat capacity is the amount of heat required to change the temperature of  substance , it is it is extensive property, it magnitude depend on the mass of the substance. It is defined as the amount of heat required to raise the temperature of a substance by 1⁰c, in thermodynamics quantities can be classified ad either extensive or intensive depending if their magnitude increase or decrease by change in the size or amount of the  substance under study.  Heat capacity of a substance varies as the size, mass, or quantity varies therefore it is an extensive property. It is measured in joules per kelvin J/K

mathematically, heat or thermal capacity = mass(m) X specific heat capacity(c)

SPECIFIC HEAT CAPACITY OF GAS

Specific heat capacity  is the quantity of heat required to raise the temperature of 1kg mass of substances by 1⁰C,   it is an intensive property as it magnitude is independent of the mass of the substances involved, specific heat capacity of a substances does not vary with the mass, its magnitude is independent of the mass. It is measured in joules per kilogram per Kelvin, J/kgK.  

the quantity of heat Q, added or removed from a body  of mass m over a temperature θ,  can be given by Q =mcθ where c is the specific heat capacity.

c =Q/mθ

 in thermodynamics the specific eat capacity at boundary point can be expressed as

C_x = (δQ/δT)_x

The symbol x can donate a stage in the process be it at constant , volume, temperature or pressure

Specific heat capacity for gas

 for gas the specific heat capacity can either be at constant pressure(C_P) or at constant volume (C_V),

Cp →CpδT  = δH and Cv →CvδT =δU  respectively.

Cp =δH/δT  and Cv =δU/δT

recall U =U (V,T)

 

RECALL

PV =RT

Cp =Cv + R dT/dT

Cp =Cv + R

Cp-R =Cv  _PROVED

RATIO OF SPECIFIC HEAT CAPACITIES OF GASES

Cp/Cv =γ

U = N_A X average kinetic energy of molecules

U = N_A X K_E

U =3/2 N_A X K_T

_WITH  K.E =3/2 KT

U =3/2RT

but  Cv =δU/δT

so at constant volume

Cv =3/2 R

recall

Cp= Cv + R

Cp =3/2 R + R

Cp =5/2 R

1.       For monoatomic gases  Cv = 3/2R and Cp =5/2R

γ =CP/CV =1.67

2.       For diatomic gases  U = 5/2 RT

Cv =5/2 R and Cp =Cv + R = 7/2 R

γ =1.40

3.       For polyatomic gases U = 6/2 RT

Cv =6/3 R and Cp =8/2 R

γ= 1.33

γ gives the number N of atoms in a molecules , for N>3 . It provides information about the atomicity of gases, this ration is very useful in adiabatic process and in determination of velocity of sound in gases