Question

The amount of heat required to raise the temperature of a diatomic gas by ${1^0}C$ at constant pressure is ${Q_p}$ and at constant volume is ${Q_V}$. The amount of heat which goes as internal energy of the gas in the two cases is nearly
(A) ${Q_p}$ and ${Q_V}$
(B) $0.71{Q_p}$ and $0.71{Q_V}$
(C) $0.71{Q_p}$ and ${Q_V}$
(D) $0.7{Q_p}$ and $0.9{Q_V}$

Answer 1

The amount of heat required to raise the temperature of a diatomic gas by ${1^0}C$ at constant pressure and constant volume is given as ${Q_p}$ and ${Q_V}$. Some amount of heat will be converted into the internal energy of the gas in both cases. We have to find the amount of heat that is converted as the internal energy of the gas.

Formula used:
${Q_p} = \dfrac{7}{2}R$
${Q_V} = \dfrac{5}{2}R$
where $R$ stands for the universal gas constant.

Complete Step by step solution:
Here we are considering a diatomic gas.
The specific heat capacity of a diatomic gas at constant pressure is given by,
${Q_p} = \dfrac{7}{2}R$
The specific heat capacity of a diatomic gas at constant volume is given by,
${Q_V} = \dfrac{5}{2}R$
where $R$ stands for the universal gas constant.
In the case of gases, the external heat applied will be used for increasing the volume of the gas and to increase the internal energy of the gases.
In the second case, we know that the volume is kept constant. Since there is no change in the volume, the heat supplied will be completely used for increasing the internal energy of the molecules. Therefore, we can write
$U = {Q_V} = \dfrac{5}{2}R$
This internal energy can be written as,
$U = {Q_V} = \dfrac{5}{7}\left( {\dfrac{7}{2}R} \right)$
We know that ${Q_p} = \dfrac{7}{2}R$
Substituting this value in the above equation, we get
${Q_V} = \dfrac{5}{7}{Q_p}$
This will be
${Q_V} = 0.71{Q_p}$
Therefore the answer is: Option (C): $0.71{Q_p}$ and ${Q_V}$.

Note:
The specific heat at constant volume is defined as the quantity of heat required to increase the temperature of a unit mass of gas through one kelvin keeping the volume constant. And if the pressure is kept constant then the heat required to raise the temperature is called specific heat at constant pressure.