An ideal gas undergoes a quasi static, reversible process in which its molar heat capacity $$C$$ remains constant. If during this process the relation of pressure $$P$$ and volume $$V$$ is given by $$P{V^n} = $$ constant, then $$n$$ is given by (Here $${C_p}$$ and $${C_v}$$ are molar specific heat at constant pressure and constant volume, respectively:

A pendulum clock loses $$12$$ $$s$$ a day if the temperature is $${40^ \circ }C$$ and gains $$4$$ $$s$$ a day if the temperature is $${20^ \circ }C.$$ The temperature at which the clock will show correct time, and the co-efficient of linear expansion $$\left( \alpha \right)$$ of the metal of the pendulum shaft are respectively :

$$'n'$$ moles of an ideal gas undergoes a process $$A$$ $$ \to $$ $$B$$ as shown in the figure. The maximum temperature of the gas during the process will be :

Consider a spherical shell of radius $$R$$ at temperature $$T$$. The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume $$u = {U \over V}\, \propto \,{T^4}$$ and pressure $$p = {1 \over 3}\left( {{U \over V}} \right)$$ . If the shell now undergoes an adiabatic expansion the relation between $$T$$ and $$R$$ is: