Question

Molar specific heat at constant pressure, \(C_{p}\), is larger than molar specific heat at constant volume, \(C_{V}\), for a) a monoatomic ideal gas. b) a diatomic atomic gas. c) all of the above. d) none of the above.

Short answer

a) monoatomic ideal gas b) diatomic atomic gas c) all of the above Answer: c) all of the above.

Step-by-step solution

Recall the relationships between molar specific heats

For an ideal gas, the relationship between molar specific heat at constant pressure (\(C_{p}\)) and molar specific heat at constant volume (\(C_{V}\)) is given by: \(C_{p} = C_{V} + R\), where R is the gas constant. From this relationship, we can see that the molar specific heat at constant pressure is always higher than the molar specific heat at constant volume for any ideal gas.

Analyze the case of a monoatomic ideal gas

For a monoatomic ideal gas, the atoms have only translational degrees of freedom, and the molar specific heat at constant volume, \(C_{V}\), is given by: \(C_{V} = \frac{3}{2} R\) Now, we can calculate the molar specific heat at constant pressure (\(C_{p}\)) using the relationship between \(C_{p}\) and \(C_{V}\): \(C_{p} = C_{V} + R = \frac{3}{2} R + R = \frac{5}{2} R\) Since \(\frac{5}{2}R > \frac{3}{2}R\), the molar specific heat at constant pressure is larger than the molar specific heat at constant volume for a monoatomic ideal gas.

Analyze the case of a diatomic atomic gas

For a diatomic atomic gas, the atoms have translational, rotational and (in some cases) vibrational degrees of freedom. The molar specific heat at constant volume, \(C_{V}\), is given by: \(C_{V} = \frac{5}{2} R\) Now, we can calculate the molar specific heat at constant pressure (\(C_{p}\)) using the relationship between \(C_{p}\) and \(C_{V}\): \(C_{p} = C_{V} + R = \frac{5}{2} R + R = \frac{7}{2} R\) Since \(\frac{7}{2}R > \frac{5}{2}R\), the molar specific heat at constant pressure is larger than the molar specific heat at constant volume for a diatomic atomic gas.

Analyze the given options

From Steps 2 and 3, we have seen that \(C_{p} > C_{V}\) holds true for both a monoatomic ideal gas and a diatomic atomic gas. Therefore, the correct answer to this exercise is: c) all of the above.

Key concepts

Thermodynamics

Thermodynamics is a branch of physics concerned with heat and temperature and their relation to energy and work. It defines macroscopic variables, such as internal energy, entropy, and pressure, that partly describe a body of matter or radiation.

In the context of our exercise, thermodynamics provides the framework to understand the behavior of gases—particularly the concept of specific heat. Specific heat is a property that describes how much heat energy is needed to raise the temperature of a substance. It is often divided into two categories: molar specific heat at constant volume (\(C_V\)) and molar specific heat at constant pressure (\(C_p\)).

The distinction between these two specific heats is crucial because they relate to how a gas will respond to heat under different conditions. Constant volume implies that the gas does not expand, and thus all the heat energy goes into raising the temperature. In contrast, at constant pressure, some of the heat energy does the work of expanding the gas, in addition to increasing the temperature. This results in a larger value for molar specific heat at constant pressure than at constant volume for any ideal gas, as the exercise demonstrates.

Ideal Gas Law

The ideal gas law is a fundamental equation that describes the state of an ideal gas. It combines the relationships of pressure, volume, temperature, and the number of moles of gas in one straightforward formula: \(PV = nRT\), where \(P\) stands for pressure, \(V\) for volume, \(n\) for the number of moles, \(R\) for the ideal gas constant, and \(T\) for temperature in Kelvin.

This equation is central to the exercise because it allows us to relate the molar specific heats with the gas constant \(R\). The difference between the molar specific heats, \(C_p\) and \(C_V\), is equal to the gas constant \(R\), showing how the ideal gas law interconnects different gas properties.

When dealing with ideal gases in thermodynamics, using the ideal gas law helps predict how changing one property of the gas will affect the others. For instance, when the temperature of a gas increases at constant volume, its pressure must also increase. The ideal gas law simplifies the complex behaviors of gases to fundamental, predictable formulas.

Degrees of Freedom

Degrees of freedom in thermodynamics refer to the number of ways an individual particle can store energy. This concept is critical when it comes to determining the specific heats of gases. The energy can be in the form of translational, rotational, or vibrational motion.

A monoatomic ideal gas, comprised of single atoms, has three translational degrees of freedom, corresponding to its movement in three-dimensional space (x, y, and z directions). Meanwhile, diatomic or polyatomic gases have additional degrees of freedom due to their ability to rotate and, in the case of polyatomic molecules, to vibrate.

Translation, Rotation, and Vibration

For simplicity, imagine that a single atom of a monoatomic gas can move left-right, up-down, or forward-backward—constituting its three degrees of freedom. In contrast, a diatomic molecule can also rotate about its center of mass and, if the temperature is high enough, can vibrate as its atoms move relative to each other. This grants diatomic gases more degrees of freedom.

The degrees of freedom are directly related to the molar specific heats: the more degrees of freedom a gas molecule has, the more energy it can absorb without a significant change in temperature, leading to a higher value of \(C_V\) and subsequently \(C_p\). Thus, understanding degrees of freedom is vital for explaining why \(C_p > C_V\) for all ideal gases, as each degree of freedom contributes a certain amount of energy to the specific heat value.