Following figure shows on adiabatic cylindrical container of volume {V

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11.Thermodynamics

hard

Following figure shows on adiabatic cylindrical container of volume ${V_0}$ divided by an adiabatic smooth piston (area of cross-section = $A$ ) in two equal parts. An ideal gas $({C_P}/{C_V} = \gamma )$ is at pressure $P_1$ and temperature $T_1$ in left part and gas at pressure $P_2$ and temperature $T_2$ in right part. The piston is slowly displaced and released at a position where it can stay in equilibrium. The final pressure of the two parts will be (Suppose $ x$ = displacement of the piston)

${P_2}$

${P_1}$

$\frac{{{P_1}{{\left( {\frac{{{V_0}}}{2}} \right)}^\gamma }}}{{{{\left( {\frac{{{V_0}}}{2} + Ax} \right)}^\gamma }}}$

$\frac{{{P_2}{{\left( {\frac{{{V_0}}}{2}} \right)}^\gamma }}}{{{{\left( {\frac{{{V_0}}}{2} + Ax} \right)}^\gamma }}}$

Solution

It is given that displacement of piston be in final state $x$ and if $A$ is the area of cross-section of the piston.

Hence the final volumes of the left and right part finally can be given by figure as

${V_L} = \frac{{{V_0}}}{2} + Ax$ and ${V_R} = \frac{{{V_0}}}{2} – Ax$

As it is given that the container walls and the piston are adiabatic in left side and the gas undergoes adiabatic expansion and on the right side the gas undergoes adiabatic compressive.

Thus we have for initial and final state of gas on left side

${P_1}{\left( {\frac{{{V_0}}}{2}} \right)^\gamma } = {P_f}{\left( {\frac{{{V_0}}}{2} + Ax} \right)^\gamma }$ ….. $(i)$

Similarly for gas in right side, we have ${P_2}{\left( {\frac{{{V_0}}}{2}} \right)^\gamma } = {P_f}{\left( {\frac{{{V_0}}}{2} – Ax} \right)^\gamma }$ ….. $(ii)$

From eq. $(i)$ and $(ii)$

$\frac{{{P_1}}}{{{P_2}}} = \frac{{{{\left( {\frac{{{V_0}}}{2} + Ax} \right)}^\gamma }}}{{{{\left( {\frac{{{V_0}}}{2} – Ax} \right)}^\gamma }}}$ ==> $Ax = \frac{{{V_0}}}{2}\frac{{\left[ {P_1^{1/\gamma } – P_2^{1/\gamma }} \right]}}{{\left[ {P_1^{1/\gamma } + P_2^{1/\gamma }} \right]}}$

Now from equation $(i)$

${P_f} = \frac{{{P_1}{{\left( {\frac{{{V_0}}}{2}} \right)}^\gamma }}}{{{{\left[ {\frac{{{V_0}}}{2} + Ax} \right]}^\gamma }}}$

Standard 11

Physics

During the adiabatic expansion of $2$ moles of a gas, the internal energy of the gas is found to decrease by $2$ joules, the work done during the process on the gas will be equal to ……. $J$

easy

The adiabatic Bulk modulus of a diatomic gas at atmospheric pressure is

hard

A gas at $NTP$ is suddenly compressed to one-fourth of its original volume. If $\gamma $ is supposed to be $\frac{3}{2}$, then the final pressure is…….. atmosphere

medium

If during an adiabatic process the pressure of mixture of gases is found to be proportional to square of its absolute temperature. The ratio of $C_p / C_v$ for mixture of gases is ………

medium

For two different gases $X$ and $Y$, having degrees of freedom $f_1$ and $f_2$ and molar heat capacities at constant volume $C_{V1}$ and $C_{V2}$ respectively, the ln $P$ versus ln $V$ graph is plotted for adiabatic process, as shown

normal