In a thermodynamic process, $2 \, \text{moles}$ of a monatomic ideal gas obey $P \propto V^{-2}$. If the temperature of the gas increases from $300 \, K$ to $400 \, K$, find the work done by the gas in terms of $R$ (where $R$ is the universal gas constant).

An ideal gas is subjected to a thermodynamic process $PV^{2.5} = 0.40$,where $P$ is in $Pa$ and $V$ is in $m^3$. What is the slope of the $P-V$ curve with volume plotted against the $x-$axis at $V = 1 \, m^3$?

Work done to increase the temperature of one mole of an ideal gas by $30^{\circ} C$, if it is expanding under the condition $V \propto T^{2/3}$ is, $(R = 8.314 \ J/mol \cdot K)$ (in $J$)

$P_i, V_i$ and $P_f, V_f$ are initial and final pressures and volumes of a gas in a thermodynamic process respectively. If $PV^n = \text{constant}$, then the amount of work done is

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 $PV^n = \text{constant}$,then $n$ is given by (Here $C_p$ and $C_v$ are molar specific heat at constant pressure and constant volume,respectively):

$1 \, \text{mole}$ of an ideal gas at temperature $T_1$ expands according to the law $P/V = \text{constant}$. Find the work done when the final temperature becomes $T_2$.