5 moles of unknown gas is heated at constant volume from 10 \, °C...

Given $n = 5$ moles, $\Delta T = 20^{\circ}\text{C} - 10^{\circ}\text{C} = 10^{\circ}\text{C}$.

The molar specific heat at constant pressure is $c_p = 8 \text{ cal/mol}\cdot^{\circ}\text{C}$.

The universal gas constant is $R = 8.36 \text{ J/mol}\cdot^{\circ}\text{C}$.

Converting $R$ into calories, we use $1 \text{ cal} = 4.18 \text{ J}$:

$R = \dfrac{8.36}{4.18} \text{ cal/mol}\cdot^{\circ}\text{C} = 2 \text{ cal/mol}\cdot^{\circ}\text{C}$

Using Mayer's relation, the molar specific heat at constant volume is:

$c_v = c_p - R = 8 - 2 = 6 \text{ cal/mol}\cdot^{\circ}\text{C}$

The change in internal energy of the gas is given by:

$\Delta U = n c_v \Delta T$

$\Delta U = 5 \times 6 \times 10 = 300 \text{ cal}$

Answer: $300$