Question

How does the ratio of $ {{C}_{p}} $ and $ {{C}_{v}} $ depend upon temperature?
 $ \text{A}\text{. }\gamma \propto T $
 $ \text{B}\text{. }\gamma \propto \dfrac{1}{T} $
 $ \text{C}\text{. }\gamma \propto \sqrt{T} $
 $ \text{D}\text{. }\gamma \propto {{T}^{0}} $

Answer 1

Hint: The ratio of molar heat capacity at constant pressure $ {{C}_{p}} $ and molar heat capacity at constant volume $ {{C}_{v}} $ is known as specific heat ratio or adiabatic index.
Molar heat capacity of a substance is defined as heat required to be added in one mole of a substance in order to increase its temperature by unit.

Complete step-by-step answer:
Adiabatic index or the specific heat ratio is denoted by gamma ( $ \gamma $ ). It is a dimensionless quantity.
Adiabatic index is written as
 $ \gamma =\dfrac{{{C}_{p}}}{{{C}_{v}}} $
Where
 $ {{C}_{p}}= $ Specific heat capacity at constant pressure
 $ {{C}_{v}}= $ Specific heat capacity at constant volume
Adiabatic index is the function of nature of gas and is independent of temperature. Therefore $ \gamma \propto {{T}^{0}} $
Therefore option D is correct.

So, the correct answer is “Option D”.

Additional Information: $ {{C}_{p}} $ and $ {{C}_{v}} $ may be expressed as
 $ {{C}_{p}}={{\left( \dfrac{\partial H}{\partial T} \right)}_{p}} $ and $ {{C}_{v}}={{\left( \dfrac{\partial U}{\partial T} \right)}_{v}} $
Where $ T $ denotes temperature, $ H $ for the enthalpy and $ U $ for the internal energy.
For an ideal gas, we can express enthalpy and internal energy as $ H={{C}_{p}}T $ and $ U={{C}_{v}}T $ because for an ideal gas heat capacity is constant with temperature.
Thus the specific heat ratio can also be expressed as
 $ \gamma =\dfrac{H}{U} $
The specific heat ratio for an ideal gas is also related with its degree of freedom. Degree of freedom is the total number of independent variables to completely describe the state of motion of a body.
Ratio of specific heats for an ideal gas can be written as
 $ \gamma =1+\dfrac{2}{f} $ where $ f $ is the degree of freedom of the gas.
For an ideal gas $ {{C}_{p}} $ and $ {{C}_{v}} $ are related by Mayer’s relation.
 $ {{C}_{p}}-{{C}_{v}}=R $ where $ R $ is the universal gas constant.

Note: The relation for specific heat ratio in terms of degree of freedom is true only when the gas does not possess vibrational degrees of freedom. Sometimes students use the expression for calculation of adiabatic index even when vibrational degrees of freedom are present.
The actual relation for adiabatic index in terms of degrees of freedom for an ideal gas is
 $ \gamma =1+\dfrac{1}{f+{{f}_{v}}} $ where $ {{f}_{v}} $ is vibrational degrees of freedom.