Question

An ideal gas is heated from \(50^{\circ} \mathrm{C}\) to $80^{\circ} \mathrm{C}(a)\( at constant volume and \)(b)$ at constant pressure. For which case do you think the energy required will be greater? Why?

Short answer

Answer: The energy required to heat the ideal gas under constant pressure (case b) will be greater than at constant volume (case a) due to the additional work done by the gas on the surroundings during the process.

Step-by-step solution

Case (a) - Heating at Constant Volume

When the gas is heated at a constant volume (isochoric process), no work can be done on or by the gas due to the lack of volume change. Thus, the only energy transfer occurring in this case is the heat (\(Q_V\)) absorbed by the system. According to the first law of thermodynamics, the internal energy change (\(\Delta U_V\)) is equal to the heat absorbed: \(\Delta U_V = Q_V\) For an ideal gas, the internal energy change depends on its heat capacity at constant volume (\(C_V\)) and the temperature difference: \(\Delta U_V = nC_V (T_2 - T_1)\) where \(n\) is the number of moles and \(T_1\) and \(T_2\) represent the initial and final temperatures.

Case (b) - Heating at Constant Pressure

When the gas is heated at a constant pressure (isobaric process), the gas can expand and do work on its surroundings. In this case, the energy transfer includes both heat absorbed by the system (\(Q_P\)) and the work done by the system on the surroundings (\(W_P\)). According to the first law of thermodynamics, the internal energy change (\(\Delta U_P\)) is equal to the heat absorbed minus the work done by the gas: \(\Delta U_P = Q_P - W_P\) For an ideal gas, the internal energy change depends on its heat capacity at constant pressure (\(C_P\)) and the temperature difference: \(\Delta U_P = nC_P (T_2 - T_1)\) The work done by the system during the process can be calculated using the equation: \(W_P = P \cdot \Delta V\) where \(P\) is the constant pressure and \(\Delta V\) is the change in volume. Now, let's compare the energy required for each case:

Comparison of Energy Requirements

We know that for an ideal gas, the heat capacities at constant volume and constant pressure are related by the following equation: \(C_P = C_V + R\) (R is the ideal gas constant) Since \(C_P>C_V\), it means the internal energy change for case (b) is higher than case (a): \(\Delta U_P > \Delta U_V\) However, we also need to take into account the work done by the gas on its surroundings (\(W_P\)) in case (b) which contributes to the total energy required. Since \(W_P > 0\), the energy required for case (b) will be greater than for case (a). In conclusion, the energy required to heat the ideal gas under constant pressure (case b) will be greater than at constant volume (case a) due to the additional work done by the gas on the surroundings during the process.