Question

A gas has 3 translation and 2 rotational degrees of freedom. find $\dfrac{C_{p} }{C_{v} }$

Answer 1

ANSWER.

$\sf \to \: \dfrac{ c_{p} }{ c_{v} } = \dfrac{7}{5}$

option [ 2 ] is correct answer.

EXPLANATION.

$\sf \to \: a \: gas \: has \: 3 \: translation \: \: and \: \: 2 \: rotation \: degree \\ \\ \sf \to \: f \: = translation \: + rotation \\ \\ \sf \to \: f \: = 3 + 2 = 5 \\ \\ \sf \to \: \gamma = 1 + \frac{2}{f} \\ \\ \sf \to \: \gamma \: = 1 + \frac{2}{5} \\ \\ \sf \to \: \gamma \: = \frac{7}{5}$

$\sf \to \: as \: we \: know \: that \: \\ \\ \sf \to \: \gamma \: = \frac{ c_{p}}{c_{v} } \\ \\ \sf \to \: \frac{ c_{p} }{ c_{v} } = \frac{7}{5}$

Answer 2

Answer:

     $\large\bullet\quad \boxed{\sf\dfrac{C_{p}}{C_{v}}=\dfrac{7}{5}}$

Explanation:

$\rule{300}{1.5}$

Given, gas has 3 translational and 2 rotational degrees of freedom.so, let's find the total degrees of freedom.

$\longmapsto \textsf{Total degrees of freedom}= \textsf{Translational + Rotational}$

Therefore,

$\longmapsto\sf f= 3 + 2\\\\\\\longmapsto\sf f= 5\\\\\\\longmapsto\boxed{\sf f= 5}$

$\rule{300}{1.5}$

$\rule{300}{1.5}$

Now, From the formula we know that,

$\large\bigstar\;\underline{\boxed{\sf C_{v}=\dfrac{f}{2}\;R}}$

Here,

• f Denotes Total Degrees of freedom.
• R Denotes Universal Gas constant.

Substituting the values,

$\longmapsto\sf C_{v}=\dfrac{f}{2}\;R\\\\\\\\\longmapsto\sf C_{v}=\dfrac{5}{2}\;R\\\\\\\\\longmapsto\sf C_{v}=\dfrac{5\;R}{2} \dots\dots \sf (1)$

$\rule{300}{1.5}$

$\rule{300}{1.5}$

We know that,

$\large\bigstar\;\underline{\boxed{\sf C_{p} = C_{v} + R}}$

Here,

• R Denotes Universal gas constant.

Substituting the values,

$\longmapsto\sf C_{p}=\dfrac{5\;R}{2}+R\\\\\\\\\longmapsto\sf C_{p}=\dfrac{5R+2R}{2}\\\\\\\\\longmapsto\sf C_{p}=\dfrac{7\;R}{2}\dots\dots \sf (2)$

$\rule{300}{1.5}$

$\rule{300}{1.5}$

Calculating the ratio,

$\longmapsto\sf \dfrac{C_{p}}{C_{v}}=\dfrac{7/2\; R}{5/2\;R}\\\\\\\\\longmapsto\sf \dfrac{C_{p}}{C_{v}}=\dfrac{7/2}{5/2}\\\\\\\\\longmapsto\sf \dfrac{C_{p}}{C_{v}}=\dfrac{7}{5}\\\\\\\\\longmapsto\large{\underline{\boxed{\red{\sf \dfrac{C_{p}}{C_{v}}=\dfrac{7}{5}}}}}$

Therefore, the value of C_{p}/C_{v} is 7/5.

$\rule{300}{1.5}$