Question

The ratio of the specific heat of air at constant pressure to its specific heat constant volume is
(A) Zero
(B) Greater than one
(C) Less than one
(D) Equal to one

Answer 1

Hint: We first need to know about specific heat of air at constant pressure and at constant volume. Specific heat of air at constant pressure is defined as the amount of heat needed to raise the temperature of one mole of air by unit temperature at constant pressure. And, specific heat at constant volume is defined as the amount of heat needed to raise the temperature of one mole of the air by unit temperature at constant volume.

Complete step-by-step answer:
Firstly we will see how the heat is utilized by the air when it is supplied at constant pressure. At constant pressure, when heat is supplied to the air then, this supplied heat is absorbed by the air and two phenomenons happens i.e.
(a) Some amount of heat increases the internal energy of the air.
(b) And, the rest of the heat is used by air to do mechanical work i.e. expansion of air. The expression of work done is given by –
$W=P\Delta V$
Where, $\Delta V$ is the change in volume of the air.
Now secondly, at constant volume, when heat is supplied to the air then, this supplied heat is absorbed by air and only one phenomenon happens i.e.
(a) All the amount of heat is utilized by air to increase the internal energy as well as the temperature.
(b) As volume is constant so heat is not utilized by air to do some mechanical work.
$\because \Delta V=0$
$\therefore W=P\Delta V=0$
As heat is not wasted to do mechanical work by air at constant volume, so at constant volume lesser amount of heat is required to raise the temperature same as at constant pressure. Hence, the ratio of the specific heat of air at constant pressure to its specific heat constant volume is always greater than one.

So, the correct answer is “Option B”.

Note: We can write from the above discussion that ${{C} _ {P}} $is greater than ${{C} _ {V}} $ and, this provides evidence to the expression:
${{C} _ {P}} = {{C} _ {V}} +R$ where,
${{C} _ {V}} $ = specific heat of air at constant volume,
${{C} _ {P}} $ = specific heat of air at constant pressure,
$R=8.314\dfrac{1}{mol\times K} $ = universal gas constant.
The adiabatic constant of the air is defined as the ratio of the specific heat of air at constant pressure to its specific heat constant volume.