Question

Internal energy(U), pressure(P) and volume(V) of an ideal gas are related as U=2+3PV. The gas is
A)monoatomic
B)diatomic
C)polyatomic
D)none of them

Answer 1

gas can be heated in two ways. Accordingly, there are two specific heat capacities in case of a gas.

(a) Specific Heat Capacity at Constant Volume (cv)

Specific heat capacity at constant volume is defined as the amount of heat required to raise the temperature of 1 g of the gas through 1ºC keeping volume of the gas constant.

If we take 1 mole of gas in the barrel, the corresponding specific heat capacity is called Gram molar specific heat capacity at constant volume.

Molar specific heat capacity, at constant volume (Cv), is defined as the amount of heat required to raise the temperature of 1 mole of gas through 1ºC keeping its volume constant.

Cv= Mcv

(b)Specific Heat Capacity at Constant Pressure (cp)

Specific heat capacity, at constant pressure, is defined as the amount of heat required to raise the temperature of 1 g of gas through 1ºC keeping its pressure constant.

In case of 1 mole of the gas:

Gram molecular specific heat capacity of a gas (Cp), at constant pressure, is defined as the amount of heat required to raise the temperature of 1 mole of the gas through 1ºC keeping its pressure constant.

Cp = Mcp

It is self evident from the above discussion that ‘Cp’ is greater than ‘Cv’ by an amount of heat which is utilized in doing external work. 

Relation of Cv With Energy

From first law of thermodynamics,

(dQ)v = dU

Or (1/m) [(dQ)v/dT] = (1/m) (dU/dT)

By definition (1/m) [(dQ)v/dT] = Cv i.e., the heat required to raise the temperature of one mole of gas by  1ºC at constant volume.

Cv = 1/m (dU/dT)

(a) Mono-Atomic Gas (3 Degree of Freedom)

Total energy, U = mN 3 [(1/2) KT], Here m is the number of moles of the gas and N is the Avogadro’s number.

Cv =1/m (dU/dT) = (1/m) (m3N) (1/2 k) = (3/2) R

 and

Cp =Cv+R = (5/2) R

So, γ = Cp/ Cv = 5/3 = 1.67

(b) Diatomic Gas

(i) At very low temperature, the number of degrees of freedom (DOF) is 3.

U = (3/2) mRT

Cv = (3/2) R and Cp = (5/2) R

So, γ = Cp/ Cv = 5/3 = 1.67

(ii) At medium temperature, the number of degrees of freedom (DOF) is 5.

U = (5/2) mRT

Cv = (5/2) R and Cp = (7/2) R

So, γ = Cp/ Cv = 7/5 = 1.4

(iii) At high temperature, the number of degrees of freedom (DOF) is 7.

U = (7/2) mRT

Cv = (7/2) R and Cp = (9/2) R

So, γ = Cp/ Cv = 9/7 = 1.29

Difference Between two Specific Heat Capacities – (Mayer’s Formula)

(a) Cp - Cv = R/J

(b) For 1 g of gas, cp - cv = r/J

(c)Adiabatic gas constant, γ = Cp/ Cv = cp/ cv