Question

A fixed mass of an ideal gas is heated from 50 to \(80^{\circ} \mathrm{C}\) at a constant pressure of \((a) 1\) atm and \((b) 3\) atm. For which case do you think the energy required will be greater? Why?

Short answer

Answer: Both cases (a) 1 atm and (b) 3 atm require the same amount of energy to heat the fixed mass of the ideal gas from 50 to 80°C, as the specific heat at constant pressure is unaffected by the difference in pressure.

Step-by-step solution

Recall the heat added at constant pressure formula (Q)

The formula for heat added to the ideal gas at constant pressure is given by: \(Q = n C_p \Delta T\) where \(Q\) is the heat added, \(n\) is the number of moles of the gas, \(C_p\) is the specific heat at constant pressure, and \(\Delta T\) is the change in temperature.

Determine the change in temperature

For both cases, the change in temperature is the same. The final temperature is \(80^{\circ} \mathrm{C}\), and the initial temperature is \(50^{\circ} \mathrm{C}\). The change in temperature can be calculated as: \(\Delta T = T_{final} - T_{initial}\) \(\Delta T = 80 - 50\) \(\Delta T = 30^{\circ} \mathrm{C}\)

Analyze the effect of pressure on the specific heat

Since the pressure changes for each case but the other variables do not, we must analyze the relationship between pressure and the specific heat at constant pressure (\(C_p\)) for an ideal gas. The specific heat capacity of an ideal gas depends only on the gas's molecular properties, not pressure. Thus, for an ideal gas, \(C_p\) remains the same between the two cases, as the fixed mass and pressure fluctuations do not affect it.

Compare heat energy added in cases (a) and (b)

As the specific heat capacity and change in temperature remain the same for both cases, the heat energy added to the ideal gas will also be the same. Specifically, in each case, the energy required to heat the gas from 50 to \(80^{\circ} \mathrm{C}\) is equal. In conclusion, both cases (a) 1 atm and (b) 3 atm require the same amount of energy to heat the fixed mass of ideal gas from 50 to \(80^{\circ} \mathrm{C}\) at constant pressure. The specific heat at constant pressure is unaffected by the difference in pressure, which keeps the heat energy required equal for both cases.