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Question: The ratio \(\dfrac{{{C_p}}}{{{C_v}}} = \gamma \) for gas. Its molecular weight is \('M'\) . Its spec...

The ratio CpCv=γ\dfrac{{{C_p}}}{{{C_v}}} = \gamma for gas. Its molecular weight is ′M′'M' . Its specific heat capacity at constant pressure is:
A) Rγ−1\dfrac{R}{{\gamma - 1}}
B) γRγ−1\dfrac{{\gamma R}}{{\gamma - 1}}
C) γRM(γ−1)\dfrac{{\gamma R}}{{M\left( {\gamma - 1} \right)}}
D) γRM(γ−1)\dfrac{{\gamma RM}}{{\left( {\gamma - 1} \right)}}

Explanation

Solution

In order to solve this you have to know the concept of specific heat capacity for gases and their ratio. Also remember the relationship between the specific heat capacity at constant volume and the specific heat capacity at constant pressure.

Formula used:
The relationship between the specific heat capacity at constant pressure and the specific heat capacity at constant volume is given by
Cp−Cv{C_p}- {C_v} =RM \dfrac{R}{{M}}
Where, R is the universal gas constant
M is the molecular weight of the gas
Cp{C_p} and Cv{C_v} are the specific heat capacity at constant pressure and the specific heat capacity at constant volume respectively.

Complete step by step solution:
As the ratio of the specific heat capacity at constant volume and the specific heat capacity at constant pressure is given as:
CpCv=γ\dfrac{{{C_p}}}{{{C_v}}} = \gamma
⇒Cp=γCv\Rightarrow {C_p} = \gamma {C_v} ……….(i)
And we know that the relationship between the specific heat capacity at constant pressure and the specific heat capacity at constant volume is given by
Cp−Cv=RM{C_p} - {C_v} = \dfrac{R}{M} …………(ii)
Now, on putting value of Cp{C_p} from equation (i) in equation (ii), we have
⇒γCv−Cv=RM\Rightarrow \gamma {C_v} - {C_v} = \dfrac{R}{M}
On taking Cv{C_v} common, we have
⇒Cv(γ−1)=RM\Rightarrow {C_v}\left( {\gamma - 1} \right) = \dfrac{R}{M}
On further solving, we get the value of specific heat capacity at constant volume as
⇒Cv=RM(γ−1)\Rightarrow {C_v} = \dfrac{R}{{M\left( {\gamma - 1} \right)}}
Similarly, on putting this above value in equation (ii), we get the value of specific heat capacity at constant pressure as
Cp=γRM(γ−1){C_p} = \dfrac{{\gamma R}}{{M\left( {\gamma - 1} \right)}}

Therefore, the correct option is (C).

Note: Always remember that the specific heat of dry air varies with the change in pressure and temperature. The heat capacity of gases at constant pressure Cp{C_p} is greater than the heat capacity of gases at constant volume Cv{C_v}, as the substance or gases expands and works, when heat is added at constant pressure.