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

হানি নাটস

For isothermal process,
$\displaystyle pV=$ constant.
Therefore $\displaystyle p_iV_i = pV$
or $\displaystyle p_i \frac{V}{n} = pV$
or $\displaystyle p_i =np$ .....(i)
For adiabatic process, $\displaystyle pV^{\gamma} =$ constant.
Therefore
$\displaystyle p_a(V_a)^{\gamma} = pV^{\gamma}$
or $\displaystyle p_a \left( \frac{V_a}{n} \right)^{\gamma} = pV^{\gamma}$
or $\displaystyle p_a= n^{\gamma} p$
From Eqs. (i) and (ii), we get
$\displaystyle \frac{p_i}{p_a} = \frac{n}{n^\gamma} = n^{(1- \gamma)}$

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