Question

In a process, 120 J heat is given to O2 gas and 180 J work is done on the gas. The molar heat capacity for this process is (Assume gas is ideal and gas constant is R

Answer 1

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Specific Heat Capacity and Its Relation with Energy

Table of Content

Specific Heat Capacity of Gases

Specific Heat Capacity at Constant Volume (cv)

Specific Heat Capacity at Constant Pressure (cp)

Relation of Cv With Energy

Mono-Atomic Gas (3 Degree of Freedom)

Diatomic Gas

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

Solved Examples

Related Resource

Specific Heat Capacity of Gases

Specific heat capacity of a substance is defined as the amount of heat required to raise the temperature of a unit mass of substance through 1ºC.

While defining the specific heat capacity, it was assumed that whole of the heat supplied to the substance is used in raising its temperature and in another way. This assumption holds good in the case of solids and liquids since expansion, in them due to 1ºC rise of temperature is negligible. Gases expand quite appreciably due to a rise in temperature. Therefore, heat required by the gas to do external work cannot be neglected.

A 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

Answer 2

Explanation:

q=120J

w=180J

Delta U = n*5/2R* delta T

n delta T =120/R

C= q/n deta T

=120/120/R

=R

R is the answer of this question