An expandable container holds 2.50 moles of He gas with an initial pressure of 750 kPa and an initial volume of 2.0 L. The gas expands isothermally to a final pressure of 375 kPa. How much heat is gained by the gas in this process?

Question

Quizplus · Accepted Answer

The heat gained by the gas in an isothermal expansion can be calculated using the formula $Q = nRT \ln\left(\frac{V_f}{V_i}\right)$, where $n$ is the number of moles, $R$ is the ideal gas constant (8.314 J/mol·K), and $V_f$ and $V_i$ are the final and initial volumes, respectively. Since the process is isothermal and the gas follows the ideal gas law ($PV = nRT$), the relationship between pressures and volumes can be used: $P_iV_i = P_fV_f$. Given $P_i = 750$ kPa, $P_f = 375$ kPa, and $V_i = 2.0$ L, we can find $V_f = 2V_i = 4.0$ L because the pressure is halved, so the volume must double to maintain the equation $PV = constant$. Substituting the given values into the formula for $Q$, we get:$$Q = 2.50 \times 8.314 \times \ln\left(\frac{4.0}{2.0}\right)$$$$Q = 20.785 \times \ln(2)$$$$Q \approx 20.785 \times 0.693$$$$Q \approx 14.4 \times 20.785$$$$Q \approx 1040 J$$Thus, the heat gained by the gas is approximately 1040 J.

An Expandable Container Holds 2