Consider a P-V diagram in which the path followed by one mole of a perfect gas in a cylindrical container is shown in the figure.

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Consider a P-V diagram in which the path followed by one mole of a perfect gas in a cylindrical container is shown in the figure.

a) find the work done when the gas is taken from state 1 to state 2

b) what is the ratio of temperature T₁/T₂ if V₂ = 2V₁

Answer:

According to the question, PV^1/2 = K = constant

a) Expression for the work done for the process 1 to 2 is:

${{W}_{1\to 2}}=\int\limits_{{{V}_{1}}}^{{{V}_{2}}}{PdV}=\int\limits_{{{V}_{1}}}^{{{V}_{2}}}{\frac{K}{\sqrt{V}}dV}$

Solving the above equation, we get

${{W}_{1\to 2}}=2{{P}_{2}}{{V}_{2}}^{\frac{1}{2}}\left( \sqrt{{{V}_{2}}-{{V}_{1}}} \right)$

b) We know that the ideal gas equation is

PV = nRT

Or, T = PV/nR

$T=\frac{K\sqrt{V}}{nR}$

$T\alpha \sqrt{V}$

$\frac{{{T}_{2}}}{{{T}_{1}}}=\sqrt{\frac{{{V}^{2}}}{{{V}_{1}}}}=\sqrt{\frac{2V}{{{V}_{1}}}}=\sqrt{2}$

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