Question

The temperature of 1 mole of an ideal monoatomic gas is increased by \( 50^\circ {C} \) at constant pressure. The total heat added and change in internal energy are \( E_1 \) and \( E_2 \), respectively. If \( \frac{E_1}{E_2} = \frac{x}{9} \), then the value of \( x \) is:

Hint:

The heat added at constant pressure and the change in internal energy are related to the specific heat capacities \( C_p \) and \( C_v \), respectively. For a monoatomic ideal gas, \( C_p = \frac{5}{2} R \) and \( C_v = \frac{3}{2} R \).

Answer 1

The heat added at constant pressure is: \[ E_1 = n C_p \Delta T = 1 \times \frac{5}{2} \times 8.314 \times 50 = 1039.25 \, {J} \] The change in internal energy is: \[ E_2 = n C_v \Delta T = 1 \times \frac{3}{2} \times 8.314 \times 50 = 624.75 \, {J} \] We are given that: \[ \frac{E_1}{E_2} = \frac{x}{9} \] Substituting the values for \( E_1 \) and \( E_2 \): \[ \frac{1039.25}{624.75} = \frac{x}{9} \] Solving for \( x \): \[ x = 9 \times \frac{1039.25}{624.75} \approx 15 \] Thus, the value of \( x \) is \( \boxed{15} \).