The equation of a certain gas can be written as: $\left( \frac{T^7}{P^2} \right)^{1/5} = \text{constant}$. The specific heat at constant volume of this gas is (in $\text{J/mol K}$): (in $R$)

Using the law of equipartition of energy, the specific heat (in $J\, kg^{-1}\, K^{-1}$) of aluminium at room temperature can be estimated to be (atomic weight of aluminium $= 27$).

To increase the temperature of $1$ mole of an ideal monatomic gas by $10^{\circ}C$ at constant pressure,$40 \, cal$ of heat is required. How much heat (in $cal$) is required to increase the temperature by the same amount at constant volume?

$c_P$ and $c_V$ are specific heats at constant pressure and constant volume respectively. It is observed that
$c_P - c_V = a$ for hydrogen gas
$c_P - c_V = b$ for nitrogen gas
The correct relation between $a$ and $b$ is

$Assertion:$ At a given temperature,the specific heat of a gas at constant pressure $(C_p)$ is always greater than its specific heat at constant volume $(C_v)$.
$Reason:$ When a gas is heated at constant volume,some extra heat is needed compared to that at constant pressure for doing work in expansion.

What is the relationship between the internal energy $U$ of an ideal gas and its pressure $P$ and volume $V$?