Question

A fixed mass of an ideal gas is heated from 50 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

Explain your answer. Answer: More energy will be required to heat the fixed mass of an ideal gas from 50°C to 80°C at constant pressure (b). This is because at constant pressure, some of the absorbed heat energy is used not only to increase the internal energy of the gas but also to do work on the surroundings by expanding the gas, whereas at constant volume, there is no expansion work done, and all the absorbed heat energy contributes only to the increase of internal energy of the gas.

Step-by-step solution

Recall the formulas for heat energy absorbed under constant volume and constant pressure

Under constant volume (isochoric process), the heat energy absorbed (\(Q_V\)) by the ideal gas is given by the formula: \(Q_V = nC_V\Delta T\) Under constant pressure (isobaric process), the heat energy absorbed (\(Q_P\)) by the ideal gas is given by the formula: \(Q_P = nC_P\Delta T\) Here, \(n\) represents the number of moles of the gas, \(C_V\) and \(C_P\) are the molar heat capacities of the gas at constant volume and constant pressure, respectively, and \(\Delta T\) is the change in temperature.

Relate the molar heat capacities

Using the Mayer's relation, we can relate the molar heat capacities: \(C_P = C_V + R\) where \(R\) is the universal gas constant. Since \(R\) is always positive, we can conclude that \(C_P > C_V\).

Compare heat energy absorbed under constant volume and constant pressure

Now we can compare the heat energy absorbed in both cases using the formulas from Step 1: \(Q_V = nC_V\Delta T\) \(Q_P = nC_P\Delta T\) From our conclusion in Step 2 that \(C_P > C_V\), we can infer that: \(Q_P > Q_V\)

Explain the findings

Therefore, the energy required to heat the fixed mass of an ideal gas from 50°C to 80°C at constant pressure will be greater than at constant volume. The reason for this is that at constant pressure, when the gas is heated, some of the absorbed heat energy is used not only to increase the internal energy of the gas but also to do the work on the surroundings by expanding the gas. In contrast, at constant volume, there is no expansion work done as the volume is kept constant, and all the absorbed heat energy contributes only to the increase of internal energy of the gas.

Key concepts

Isochoric Process

The isochoric process, also known as a constant volume process, is a fundamental concept in thermodynamics where the volume (\( V \)) of a system remains unchanged. During this process, the work done on or by the system is zero because volume does not change, and hence no expansion or compression of the gas occurs. Instead, any heat (\( Q_V \)) added to the system exclusively affects the internal energy (\( U \)) of the gas.

Ideal gases undergoing an isochoric process comply with the equation \( Q_V = nC_V\triangle T \), where \( n \) is the number of moles of gas, \( C_V \) represents the molar heat capacity at constant volume, and \( \triangle T \) denotes the temperature change. It is important to note that all the heat energy added to the system exclusively raises the temperature of the gas, as there is no work done on the surroundings.

The absence of work leads to a straightforward relation between heat added and temperature change, making calculations regarding isochoric processes simpler compared to processes where volume changes.

Isobaric Process

In contrast to the isochoric process, the isobaric process involves changes under a constant pressure (\( P \)). When an ideal gas is subject to an isobaric process, it absorbs heat energy (\( Q_P \)) and can do work on its surroundings since the volume can change as a response to the heat input.

The equation governing an isobaric process is \( Q_P = nC_P\triangle T \), where \( C_P \) stands for the molar heat capacity at constant pressure. Unlike the isochoric process, part of the heat energy absorbed (\( Q_P \)) in an isobaric process is used for doing work as the gas expands. This contributes to less increase in internal energy per unit of heat absorbed compared to the isochoric process when considering the same temperature change.

This means that, under an isobaric process, we need to supply more heat to achieve the same temperature rise, as some of this heat is 'consumed' by the expansion work against the external pressure. This nuanced balance between heat, work, and temperature change is crucial for understanding phenomena like the heating and cooling of gases in various engineering systems and natural processes.

Molar Heat Capacities

Molar heat capacities are a measure of how much heat energy is required to raise the temperature of one mole of a substance by one degree Celsius (or one Kelvin). There are two specific types of molar heat capacities relevant to gases — at constant volume (\( C_V \)) and at constant pressure (\( C_P \)).

\( C_V \) is the amount of heat required to raise the temperature of a gas when its volume is held constant. It is a measure of the internal energy change of a gas when it is heated without being allowed to expand. On the other hand, \( C_P \) is the heat required to raise the temperature of the gas when its pressure is held constant, which will involve volume changes and, therefore, work done by the gas.

Following Mayer's relation, \( C_P \) is always greater than \( C_V \) by an amount equal to the universal gas constant (\( R \)): \( C_P = C_V + R \). The difference accounts for the extra energy needed at constant pressure to perform work by expanding against the ambient pressure.

Understanding these concepts can help explain why heating a gas at constant pressure requires more energy than heating the same gas at constant volume, as some of the energy goes into doing work in the former while all of it increases the internal energy in the latter.