A monoatomic gas is suddenly compressed to $$(1 / 8)^{\text {th }}$$ of its initial volume adiabatically. The ratio of the final pressure to initial pressure of the gas is $$(\gamma=5 / 3)$$

A monoatomic ideal gas initially at temperature $$\mathrm{T}_1$$ is enclosed in a cylinder fitted with 8 frictionless piston. The gas is allowed to expand adiabatically to a temperature $$\mathrm{T}_2$$ by releasing the piston suddenly. $$\mathrm{L}_1$$ and $$\mathrm{L}_2$$ are the lengths of the gas columns before and after the expansion respectively. Then $$\frac{\mathrm{T}_2}{\mathrm{~T}_1}$$ is

For a monoatomic gas, the work done at constant pressure is '$$\mathrm{W}$$' The heat supplied at constant volume for the same rise in temperature of the gas is

$$[\gamma=\frac{C_p}{C_v}=\frac{5}{2}$$ for monoatomic gas]

An ideal gas with pressure $$\mathrm{P}$$, volume $$\mathrm{V}$$ and temperature $$\mathrm{T}$$ is expanded isothermally to a volume $$2 \mathrm{~V}$$ and a final pressure $$\mathrm{P}_{\mathrm{i}}$$. The same gas is expanded adiabatically to a volume $$2 \mathrm{~V}$$, the final pressure is $$\mathrm{P}_{\mathrm{a}}$$. In terms of the ratio of the two specific heats for the gas '$$\gamma$$', the ratio $$\frac{P_i}{P_a}$$ is