Question

When an ideal gas at pressure p, temperature T and volume V is isothermally compressed to V/n, its pressure becomes $p_i$. If the gas is compressed adiabatically to $\frac{V}{n}$, its pressure becomes $p_a$. The ratio $p_i/p_a$ is $(\gamma =\frac{C_p}{C_V})$

• A. 1
• B. n
• C. $n^{\gamma }$
• D. $n^{1-\gamma }$

Answer 1

Answer: (D)

$n^{1-\gamma }$

For isothermal process,

$pV=$ constant.

Therefore $p_i{V}_i=pV$

or $p_i\frac{V}{n}=pV$

or $p_i=np$ .....(i)

For adiabatic process, $p{V}^{\gamma }=$ constant.

Therefore

$p_a(V_a{)}^{\gamma }=p{V}^{\gamma }$

or $p_a{\left(\frac{V_a}{n}\right)}^{\gamma }=p{V}^{\gamma }$

or $p_a=n^{\gamma }p$

From Eqs. (i) and (ii), we get

$\frac{p_i}{p_a}=\frac{n}{n^{\gamma }}=n^{(1-\gamma )}$