$1 \, \text{mole}$ of a gas expands with temperature as $V = KT^{2/3}$. What is the work done when temperature changes by $30^oC$ (in $R$)?

In a process,temperature and volume of one mole of an ideal monoatomic gas are varied according to the relation $VT = K$,where $K$ is a constant. In this process,the temperature of the gas is increased by $\Delta T$. The amount of heat absorbed by the gas is ($R$ is the gas constant).

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 $J$)

One mole of an ideal gas at temperature $T_1$ expands according to the law $\frac{P}{V^2} = a$ (constant). The work done by the gas until the temperature of the gas becomes $T_2$ is:

An ideal gas undergoes a polytropic process given by the equation $PV^n = \text{constant}$. If the molar heat capacity of the gas during this process is the arithmetic mean of its molar heat capacity at constant pressure $(C_P)$ and constant volume $(C_V)$, then the value of $n$ is ..............

$1 \, \text{mole}$ of an ideal monoatomic gas at temperature $T_0$ expands slowly according to the law $P = KV$,where $K$ is a constant. If the final temperature of the gas is $2T_0$,find the heat supplied to the gas.