An ideal heat engine operates in Carnot cycle between $127^{\circ} \mathrm{C}$ and $27^{\circ} \mathrm{C}$. It absorbs $5 \times 10^4 \mathrm{Cal}$ of heat at height temperature. Amount of heat converted to work is

One mole of a gas having $\gamma=\frac{7}{5}$ is mixed with one mole of a gas having $\gamma=\frac{4}{3}$. The value of $\gamma$ for the mixture is ( $\gamma$ is the ratio of the specific heats of the gas)

A Carnot heat engine has an efficiency of $10 \%$. If the same engine is worked backward to obtain a refrigerator, then the coefficient of performance of the refrigerator is

The rms velocity of a gas molecules oí mass $m$ at a given temperature is proportional to