When a diatomic gas (rigid) undergoes adiabatic change, its pressure $(\mathrm{P})$ and temperature $(\mathrm{T})$ are related as $P \propto T^c$. The value of $c$ is

For an ideal gas, the density of the gas is $\rho_0$ when temperature and pressure of the gas are $\mathrm{T}_0$ and $P_0$ respectively. when the temperature of the gas is $2 \mathrm{~T}_0$, its pressure becomes $3 \mathrm{P}_0$. The new density will be

A centigrade and Fahrenheit thermometer are dipped in boiling water. The water temperature is lowered until the Fahrenheit temperature observed is $140^{\circ} \mathrm{F}$. At that time the temperature registered by the centigrade thermometer is

An engine operating between temperatures $T_1$ and $T_2$ has efficiency $\frac{1}{5}$. When $T_2$ is lowered by 45 K , its efficiency becomes $\frac{1}{2}$. Temperatures $T_1$ and $T_2$ are respectively