\(5\) moles of unknown gas is heated at constant volume from \(10 \, °C\) to \(20 \, °C\). The molar specific heat of this gas at constant pressure \(c_p = 8\) cal/mol.°C and \(R = 8.36\) J/mol.°C. The change in the internal energy of the gas is _______ calorie.

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: \(300\)