Question

When an ideal diatomic gas is heated at constant pressure, the fraction of the heat energy supplied which increases the internal energy of the gas is?
A.$2/5$
B.$3/5$
C.$3/7$
D.$5/7$

Answer 1

Heat capacity at constant pressure: It is defined as the amount of heat energy absorbed or released by the substance with the change in temperature at constant pressure. It is represented as ${C_p}$.
Heat capacity at constant volume: It is defined as the amount of heat energy absorbed or released by the substance with the change in temperature at constant volume. It is represented as ${C_v}$.

Complete step by step answer:
Internal energy of a system: It is defined as the energy associated with the random movement of the molecules, is known as the internal energy of a system. It is represented by the symbol $U$.
Change in internal energy: It is defined as the sum of the heat transferred and the work done.
Total rise in internal energy is $n{C_V}\Delta T$, where ${C_V}$ is heat capacity at constant volume, $n$ is number of moles and $\Delta T$ is the temperature difference. And total energy supplied to raise the temperature of a diatomic gas at constant pressure is $n{C_P}\Delta T$, , where ${C_P}$ is heat capacity at constant pressure, $n$ is number of moles and $\Delta T$ is the temperature difference.
Now the fraction of the heat energy supplied which increases the internal energy of the gas is $= \dfrac{{n{C_V}\Delta T}}{{n{C_P}\Delta T}}$.
We know that the ratio ${C_V}$ and ${C_P}$ is $\dfrac{f}{{f + 2}}$ where $f$ is the number of degrees of freedom of the gas. For diatomic gases the value of $f$ is $5$.
So the fraction of the heat energy supplied which increases the internal energy of the gas is $\dfrac{5}{7}$.

Hence option D is correct.

Note:
Number of degrees of freedom of the gas is defined as the dimensions of the phase space. For diatomic gases the number of degree of freedom is $5$ and for polyatomic gases it is as $3n - 6$, where $n$ is total number of atoms in the compound, if the compound is non-linear in shape and $3n - 5$, where $n$ is total number of atoms in the compound, if the compound is linear in shape.