Question

The ratio of the specific heats $\dfrac{{{C}_{p}}}{{{C}_{v}}}=\gamma$ in terms of degree of freedom $(n)$ is given by:
$\begin{aligned} & A)\left( 1+\dfrac{2}{n} \right) \\\ & B)\left( 1+\dfrac{n}{2} \right) \\\ & C)\left( 1+\dfrac{1}{n} \right) \\\ & D)\left( 1+\dfrac{n}{3} \right) \\\ \end{aligned}$

Answer 1

Relation between specific heat at constant pressure and specific heat at constant volume is given by Mayer’s formula. Degree of freedom of a thermodynamic substance refers to the number of directions the molecules of the substance can move under thermodynamic variations. Internal energy of a substance is related to the specific heat of the substance at constant volume as well as degree of freedom of the thermodynamic substance.

Formula used:
$1){{C}_{p}}-{{C}_{v}}=R$
$2)U=\dfrac{n}{2}RdT={{C}_{v}}dT$

Complete step-by-step answer:
Relation between specific heat at constant pressure ${{C}_{p}}$ and specific heat at constant volume ${{C}_{v}}$ is given by Mayer’s formula, as expressed below:
${{C}_{p}}-{{C}_{v}}=R$
where
${{C}_{p}}$ is the specific heat capacity of a substance at constant pressure
${{C}_{v}}$ is the specific heat capacity of a substance at constant volume
$R$ is the gas constant
Let this be equation 1.
Coming to our question, we are required to determine the ratio of the specific heats $\dfrac{{{C}_{p}}}{{{C}_{v}}}=\gamma$ in terms of degree of freedom $(n)$.
We know that internal energy of a thermodynamic substance is given by
$U=\dfrac{n}{2}RdT={{C}_{v}}dT$
where
$U$ is the internal energy of a thermodynamic substance
$n$ is the degree of freedom of the substance
$R$ is the gas constant
${{C}_{v}}$ is the specific heat of the substance at constant volume
$dT$ is the change in temperature of the substance
Let this be equation 2.
Here, the degree of freedom of thermodynamic substance refers to the number of directions the molecules of the substance can move (vibrate), when heat is supplied to the substance. Also, from equation 2, it is clear that
${{C}_{v}}=\dfrac{n}{2}R$
Let this be equation 3.
Substituting equation 3 in equation 1, we have
${{C}_{p}}-{{C}_{v}}=R\Rightarrow {{C}_{p}}=R+\dfrac{n}{2}R=R\left( 1+\dfrac{n}{2} \right)$
where
${{C}_{p}}$ is the specific heat of substance at constant pressure
Let this be equation 4.
Diving equation 4 by equation 3, we have
$\dfrac{{{C}_{p}}}{{{C}_{v}}}=\dfrac{R\left( 1+\dfrac{n}{2} \right)}{R\left( \dfrac{n}{2} \right)}=\dfrac{2+n}{2}\times \dfrac{2}{n}=\dfrac{2+n}{n}=\dfrac{n}{n}\left( \dfrac{2}{n}+1 \right)=1+\dfrac{2}{n}$
Let this be equation 5.
Therefore, from equation 5, it is clear that ratio of specific heats $\dfrac{{{C}_{p}}}{{{C}_{v}}}=\gamma$ in terms of degree of freedom $(n)$ is given by $1+\dfrac{2}{n}$ and hence, the correct answer is option $A$.

So, the correct answer is “Option A”.

Note: Specific heat capacity of a substance is defined as the amount of heat supplied to unit mass of substance in order to bring a unit change in temperature of the substance. Heat can be supplied by keeping the pressure as well as volume of the substance, constant, and their corresponding specific heat capacities are termed as ${{C}_{p}}$ and ${{C}_{v}}$, respectively. The relation between them is given by Mayer’s formula as already discussed.