Two Carnot engines $A$ and $B$ are operated in series. The engine $A$ receives heat from the source at temperature $T_1$ and rejects the heat to the sink at temperature $T$. The second engine $B$ receives the heat at temperature $T$ and rejects to its sink at temperature $T_2$. For what value of $T$ the efficiencies of the two engines are equal?

Two rigid boxes containing different ideal gases are placed on a table. Box $A$ contains one mole of nitrogen at temperature $T_0$ , while box $B$ contains one mole of helium at temperature $(7/3)\, T_0$ . The boxes are then put into thermal contact with each other, and heat flows between them until the gases reach a common final temperature (ignore the heat capacity of boxes), then the final temperature of gases $T_f$ , in terms of $T_0$ is

Gas obey $P^2V =$ constant. The initial temperature and volume are $T_0$ and $V_0$. If gas expands to volume $2V_0$, its final temperature becomes

An ideal gas heat engine operates in a Carnot's cycle between ${227^o}C$ and ${127^o}C$. It absorbs $6 × 10^4 \,J$ at high temperature. The amount of heat converted into work is ….

The coefficient of apparent expansion of mercury in a glass vessel is $153 \times 10^{-6}/\,^oC$ and in a steel vessel is $144 \times 10^{-6}/\,^oC$. If $\alpha $ for steel is $12 \times 10^{-6}/\,^oC$, then $\alpha $ that of glass is