Question

\(C_{v}\) and \(C_{p}\) denote the molar specific heat capacities of a gas at constant volume and constant pressure, respectively. Then (A) \(\mathrm{C}_{\mathrm{p}}-\mathrm{C}_{\mathrm{v}}\) is larger for a diatomic ideal gas than for a monoatomic ideal gas (B) \(\mathrm{C}_{\mathrm{p}}+\mathrm{C}_{\mathrm{v}}\) is larger for a diatomic ideal gas than for a monoatomic ideal gas (C) \(\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}\) is larger for a diatomic ideal gas than for a monoatomic ideal gas (D) \(\mathrm{C}_{\mathrm{p}} \cdot \mathrm{C}_{\mathrm{v}}\) is larger for a diatomic ideal gas than for a monoatomic ideal gas

Short answer

Statements (A) and (C) are incorrect as \(C_p - C_v\) and \(C_p / C_v\) are not larger for a diatomic gas compared to a monoatomic gas. Statement (D) is correct as \(C_p \cdot C_v\) is larger for a diatomic gas. Statement (B) is also correct as \(C_p + C_v\) is larger for a diatomic gas.

Step-by-step solution

Understanding the Heat Capacities

The molar specific heat capacity at constant volume for a monoatomic gas is given by \(C_v = \frac{3}{2}R\) and for a diatomic gas, it is \(C_v = \frac{5}{2}R\), where \(R\) is the universal gas constant. The molar specific heat capacity at constant pressure is related to that at constant volume by the relation \(C_p = C_v + R\).

Comparing \(C_p - C_v\) for Monoatomic and Diatomic Gases

For a monoatomic gas, \(C_p - C_v = R\). For a diatomic gas, since \(C_v\) is \(\frac{5}{2}R\), \(C_p\) must be \(\frac{7}{2}R\), thus \(C_p - C_v = R\). There is no difference between monoatomic and diatomic gases in terms of \(C_p - C_v\).

Comparing \(C_p + C_v\) for Monoatomic and Diatomic Gases

For a monoatomic gas, \(C_p + C_v = \frac{5}{2}R + \frac{3}{2}R = 4R\). For a diatomic gas, \(C_p + C_v = \frac{7}{2}R + \frac{5}{2}R = 6R\). Thus, \(C_p + C_v\) is larger for a diatomic ideal gas.

Comparing \(C_p / C_v\) for Monoatomic and Diatomic Gases

For a monoatomic gas, \(C_p / C_v = \frac{\frac{5}{2}R}{\frac{3}{2}R} = \frac{5}{3}\). For a diatomic gas, \(C_p / C_v = \frac{\frac{7}{2}R}{\frac{5}{2}R} = \frac{7}{5}\). Therefore, the ratio \(C_p / C_v\) is larger for a monoatomic ideal gas than for a diatomic one.

Comparing \(C_p \cdot C_v\) for Monoatomic and Diatomic Gases

For a monoatomic gas, \(C_p \cdot C_v = \frac{5}{2}R \cdot \frac{3}{2}R = \frac{15}{4}R^2\). For a diatomic gas, \(C_p \cdot C_v = \frac{7}{2}R \cdot \frac{5}{2}R = \frac{35}{4}R^2\). Thus, \(C_p \cdot C_v\) is larger for a diatomic ideal gas.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation that relates the pressure, volume, temperature, and amount of an ideal gas. Expressed as \(PV = nRT\), it reveals how these four variables are interdependent. Here, \(P\) stands for pressure, \(V\) represents volume, \(n\) is the amount of substance in moles, \(R\) is the ideal gas constant, and \(T\) indicates temperature in Kelvin.

This law provides a framework for understanding how gases behave under different conditions and is crucial in calculations involving gas dynamics. In the context of molar specific heat capacities, it helps in deriving the relationship between \(C_v\) and \(C_p\), which are the heat capacities at constant volume and constant pressure respectively. Understanding the ideal gas law is essential for comprehending why certain properties of gases, such as heat capacities, differ under varying conditions.

Heat Capacity at Constant Volume

The heat capacity at constant volume, denoted as \(C_v\), is the amount of heat required to raise the temperature of a mole of gas by one degree Celsius while keeping its volume constant.

In the context of ideal gases, \(C_v\) is especially important because it is directly related to the gas's internal degrees of freedom – that is, the ways in which the gas particles can store energy. For a monoatomic gas, which has only translational degrees of freedom, \(C_v\) is \(\frac{3}{2}R\). On the other hand, diatomic gases also have rotational degrees of freedom, resulting in a higher \(C_v\) of \(\frac{5}{2}R\). These values are instrumental in calculating energy changes in thermodynamic processes.

Heat Capacity at Constant Pressure

Conversely, the heat capacity at constant pressure, \(C_p\), is the heat required to increase the temperature of a mole of gas by one degree Celsius at a constant pressure. The difference between \(C_p\) and \(C_v\) is a fundamental concept in thermodynamics, encapsulated by the equation \(C_p = C_v + R\).

This relationship emerges from the work done by the gas when it expands at constant pressure. At constant volume, no work is done, hence the difference \(R\) representing the additional energy needed to allow for expansion at constant pressure. This understanding aids in analyzing the energy involved in various thermodynamic processes, such as heating and cooling of gases.

Monoatomic and Diatomic Gases

The behavior of monoatomic and diatomic gases regarding specific heat capacities can be attributed to their atomic structure. Monoatomic gases, like helium, have atoms consisting of a single atom, which limits their energy absorption to translational motion. Thus, their \(C_v\) is lower, at \(\frac{3}{2}R\).

Diatomic gases, such as nitrogen, comprise molecules with two atoms. This allows them to absorb and distribute energy not just translationally but also rotationally and, at higher temperatures, vibrationally, resulting in a \(C_v\) of \(\frac{5}{2}R\). These additional degrees of freedom explain why their molar specific heat capacities—and consequently, the answers to parts (B) and (D) of our initial exercise—are larger as compared to monoatomic gases.