Calculate Internal Energy Change for Gas Heated at Constant Volume

To calculate the change in internal energy of the gas, we follow these step-by-step calculations:

1. Given Data:

• Number of moles, $n = 5\text{ mol}$
• Initial temperature, $T_1 = 10\,^{\circ}\text{C}$
• Final temperature, $T_2 = 20\,^{\circ}\text{C}$
• Molar specific heat capacity at constant pressure, $C_p = 8\text{ cal/mol}\cdot^{\circ}\text{C}$
• Universal gas constant, $R = 8.36\text{ J/mol}\cdot^{\circ}\text{C}$

2. Conversion of $R$ into calories: Using the standard conversion factor $1\text{ cal} = 4.18\text{ J}$: $R = \frac{8.36\text{ J/mol}\cdot^{\circ}\text{C}}{4.18\text{ J/cal}} = 2\text{ cal/mol}\cdot^{\circ}\text{C}$

3. Molar Specific Heat at Constant Volume ($C_v$): Using Mayer's relation: $C_p - C_v = R$ $C_v = C_p - R = 8 - 2 = 6\text{ cal/mol}\cdot^{\circ}\text{C}$

4. Change in Temperature ($\Delta T$): $\Delta T = T_2 - T_1 = 20\,^{\circ}\text{C} - 10\,^{\circ}\text{C} = 10\,^{\circ}\text{C}$

5. Change in Internal Energy ($\Delta U$): The change in internal energy for an ideal gas is given by: $\Delta U = n C_v \Delta T$

Substituting the values: $\Delta U = 5 \times 6 \times 10 = 300\text{ calories}$

Final Answer: The change in the internal energy of the gas is $300\text{ calorie}$.