Question

A diatomic gas undergoes adiabatic expansion against the piston of a cylinder. As a result, the temperature of the gas drops from 1150 K to 400 K. The number of moles of the gas required to obtain 2300 J of work from the expansion is .................

Hint:

For adiabatic expansion of a gas, use the relation \( W = n C_V \Delta T \) and the heat capacity for a diatomic gas to solve for the number of moles.

Answer 1

Correct Answer: 0.14

Step 1: Use the formula for work done in an adiabatic process.
For an adiabatic expansion, the work done \( W \) by the gas is given by: \[ W = n C_V \Delta T \] where \( n \) is the number of moles, \( C_V \) is the molar heat capacity at constant volume for a diatomic gas, and \( \Delta T \) is the change in temperature. For a diatomic gas, \( C_V = \frac{5}{2} R \), where \( R = 8.314 \, \text{J/mol·K} \) is the universal gas constant.
Step 2: Calculate the work done.
The temperature change is: \[ \Delta T = 400 \, \text{K} - 1150 \, \text{K} = -750 \, \text{K} \] Now, substitute into the work equation: \[ 2300 \, \text{J} = n \times \frac{5}{2} \times 8.314 \times (-750) \] Solving for \( n \): \[ n = \frac{2300}{\frac{5}{2} \times 8.314 \times (-750)} = 1.00 \, \text{mol} \]
Step 3: Conclusion.
Thus, the number of moles required to obtain 2300 J of work from the expansion is 1.00 mol.