A monoatomic gas at pressure $$\mathrm{P}$$ and volume $$\mathrm{V}$$ is suddenly compressed to one eighth of its original volume. The final pressure at constant entropy will be :

Sound travels in a mixture of two moles of helium and n moles of hydrogen. If rms speed of gas molecules in the mixture is $$\sqrt2$$ times the speed of sound, then the value of n will be :

Let $$\eta_{1}$$ is the efficiency of an engine at $$T_{1}=447^{\circ} \mathrm{C}$$ and $$\mathrm{T}_{2}=147^{\circ} \mathrm{C}$$ while $$\eta_{2}$$ is the efficiency at $$\mathrm{T}_{1}=947^{\circ} \mathrm{C}$$ and $$\mathrm{T}_{2}=47^{\circ} \mathrm{C}$$ The ratio $$\frac{\eta_{1}}{\eta_{2}}$$ will be :

A certain amount of gas of volume $$\mathrm{V}$$ at $$27^{\circ} \mathrm{C}$$ temperature and pressure $$2 \times 10^{7} \mathrm{Nm}^{-2}$$ expands isothermally until its volume gets doubled. Later it expands adiabatically until its volume gets redoubled. The final pressure of the gas will be (Use $$\gamma=1.5)$$ :