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(A) The process is adiabatic because the cylinder walls and the piston are heat insulators.For an adiabatic process,the relation between pressure and

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$(2)^{7/5}$

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(A) The process is adiabatic because the cylinder walls and the piston are heat insulators.For an adiabatic process,the relation between pressure and volume is given by $P_1 V_1^\gamma = P_2 V_2^\gamma$.Here,$V_2 = \frac{V_1}{2}$,so $\frac{V_1}{V_2} = 2$.For hydrogen (a diatomic gas),the adiabatic index is $\gamma = \frac{C_p}{C_v} = \frac{7/2 R}{5/2 R} = \frac{7}{5}$.Substituting these values into the adiabatic equation:$\frac{P_2}{P_1} = \left( \frac{V_1}{V_2} \right)^\gamma = (2)^{7/5}$.Thus,the pressure increases by a factor of $(2)^{7/5}$.

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