Question

A diatomic molecule has translational, rotational and vibrational degrees of freedom. The ${{\text{C}}_{\text{p}}}\text{/}{{\text{C}}_{\text{v}}}$ ratio is:
A. 1.67
B. 1.4
C. 1.29
D. 1.33

Answer 1

Hint: A diatomic molecule has two atoms, which can show movement like translational motion, rotational motion and vibrational motion. Vibrational is present or dominant at high temperatures.

Complete step-by-step answer:
The specific heat at constant volume is defined as the energy or heat required to raise the temperature of a body by a unit temperature at constant volume. It is denoted by ${{\text{C}}_{\text{v}}}$, which is expressed mathematically as,
${{\text{C}}_{\text{v}}}={{\left( \dfrac{dU}{dT} \right)}_{\text{v}}}$
The specific heat at constant pressure is defined as the energy or heat required to raise the temperature of a body by a unit temperature at constant temperature. It is denoted by ${{\text{C}}_{\text{p}}}$, which is expressed mathematically as,
${{\text{C}}_{\text{p}}}={{\left( \dfrac{dU}{dT} \right)}_{\text{p}}}$
For a diatomic molecule containing translational, rotational and vibrational degrees of freedom will comprise a total of 6 degrees of freedom (3 translational, 2 rotational, 1 vibrational).
So the specific heat capacity at a constant volume of a diatomic molecule is given by \[{{C}_{v}}=\dfrac{n}{2}R=\dfrac{6}{2}R\]
Where, n is the degrees of freedom.
The specific heat at a constant pressure of a diatomic is given by \[{{C}_{p}}=\left( 1+\dfrac{n}{2} \right)R=\dfrac{8}{2}R\]

So the ratio between ${{\text{C}}_{\text{p}}}\text{ and }{{\text{C}}_{\text{v}}}$ is,
${{C}_{p}}/{{C}_{v}}=\dfrac{\left( 8/2 \right)}{\left( 6/2 \right)}=\dfrac{8}{6}$
$\therefore \dfrac{{{C}_{\text{p}}}}{{{C}_{\text{v}}}}=1.33$
So the answer to the question is option (D)- 1.33.

Note: If you know the specific heat capacity at constant volume or pressure of any gas, we can calculate the other specific heat at constant pressure or volume by using the relation, ${{\text{C}}_{\text{p}}}-{{\text{C}}_{\text{v}}}=\text{R}$, where R is the gas constant.
The Energy associated with a gas whether it be a monatomic, diatomic or triatomic is given by the formula, $U=\dfrac{n}{2}RT$, where n is the degrees of freedom associated with the molecule.