- Home
- Standard 11
- Physics
One mole of an ideal gas $(C_p/C_v = \gamma )$ at absolute temperature $T_1$ is adiabatically compresses from an initial pressure $P_1$ to a final pressure $P_2$. The resulting temperature $T_2$ of the gas is given by
${T_{_2}} = {T_1}{\left( {\frac{{{P_2}}}{{{P_1}}}} \right)^{\frac{\gamma }{{\gamma - 1}}}}$
${T_{_2}} = {T_1}{\left( {\frac{{{P_2}}}{{{P_1}}}} \right)^{\frac{{\gamma - 1}}{\gamma }}}$
${T_{_2}} = {T_1}{\left( {\frac{{{P_2}}}{{{P_1}}}} \right)^\gamma }$
${T_{_2}} = {T_1}{\left( {\frac{{{P_2}}}{{{P_1}}}} \right)^{\gamma - 1}}$
Solution
Adiabatic process hence $\mathrm{PV}^{\mathrm{Y}}=\mathrm{constant}$
If $\mathrm{n}=1$ then $\mathrm{PV}=\mathrm{RT}$ hence $\mathrm{V}=(\mathrm{RT} / \mathrm{P})$
$\therefore \mathrm{P}(\mathrm{RT} / \mathrm{P})^ \mathrm{Y}=\mathrm{constant}$
$\therefore \mathrm{P}^{(1-\mathrm{y})} \cdot \mathrm{T}^{\mathrm{Y}}=\mathrm{constant}$
$\therefore P^{1-Y} \propto T Y^{Y}$
$\therefore\left(T_{2} / T_{1}\right) Y=\left(P_{2} / P_{1}\right)^{(1-Y)}$
$\therefore\left(T_{2} / T_{1}\right)=\left[\left(P_{2} / P_{1}\right)^{(1-y) \times(1 / Y)}\right]$
$T_2=T_1(p_2/p_2)^{(Y-1)/Y}$
