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. c) all of the above. b) a diatomic atomic gas. d) none of the above.

Short answer

Question: Is the molar specific heat at constant pressure, \(C_p\), larger than the molar specific heat at constant volume, \(C_V\) for: a) a monoatomic ideal gas b) a diatomic ideal gas c) all of the above Answer: c) all of the above

Step-by-step solution

Recall the specific heat capacity relationships for monoatomic and diatomic ideal gases

For a monoatomic ideal gas, the molar specific heat capacity at constant volume is given by \(C_{V} = \frac{3}{2}R\), where R is the gas constant. The molar specific heat capacity at constant pressure is given by \(C_{p} = \frac{5}{2}R\). For a diatomic ideal gas, the molar specific heat capacity at constant volume is given by \(C_{V} = \frac{5}{2}R\), and the molar specific heat capacity at constant pressure is given by \(C_{p} = \frac{7}{2}R\).

Compare \(C_p\) and \(_C\) for a monoatomic ideal gas

Compare the values of \(C_p\) and \(C_V\) for a monoatomic ideal gas: \(C_{p} = \frac{5}{2}R > C_{V} = \frac{3}{2}R\) Since \(C_p\) is greater than \(C_V\), the statement is true for a monoatomic ideal gas.

Compare \(C_p\) and \(C_V\) for a diatomic ideal gas

Compare the values of \(C_p\) and \(C_V\) for a diatomic ideal gas: \(C_{p} = \frac{7}{2}R > C_{V} = \frac{5}{2}R\) Since \(C_p\) is greater than \(C_V\), the statement is also true for a diatomic ideal gas.

Choose the correct option

Based on Steps 2 and 3, we can conclude that the molar specific heat at constant pressure (\(C_p\)) is larger than molar specific heat at constant volume (\(C_V\)) for both: a) a monoatomic ideal gas and b) a diatomic atomic gas. Thus the correct answer is: c) all of the above.

Key concepts

Ideal Gas Law

Understanding the ideal gas law is critical to grasping the principles of molar specific heat capacity for gases. The ideal gas law is a fundamental equation in chemistry and physics given by the formula: \( PV = nRT \), where

• \(P\) denotes the pressure of the gas,
• \(V\) is the volume it occupies,
• \(n\) represents the number of moles,
• \(R\) is the ideal gas constant, and
• \(T\) is the absolute temperature in Kelvin.

This law describes the behavior of an ideal gas, which is a hypothetical gas that perfectly follows this equation under all conditions. In reality, no gas is truly ideal, but many gases behave like ideal gases under certain temperatures and pressures, making this law a good approximation for many applications.
In connection to specific heat capacities, the ideal gas law helps explain why these values vary with the conditions of volume and pressure. Since the specific heat depends on the amount of energy required to change the temperature of a substance, the relationships between pressure, volume, and temperature are paramount.

Specific Heat at Constant Volume

The specific heat at constant volume, \(C_V\), for an ideal gas, is a measure of the amount of energy needed to raise the temperature of one mole of the gas by one degree Kelvin without allowing the volume to change. For a monoatomic ideal gas, where the atoms have only translational motion, the molar specific heat capacity at constant volume is given by: \(C_{V} = \frac{3}{2}R\). This formula indicates that \(C_V\) is directly proportional to the ideal gas constant \(R\) and is inherent to the kinetic theory of gases, which assumes that the particles are in constant, random motion and that there are no intermolecular forces acting upon them when the gas volume is constant.
Understanding \(C_V\) is essential when examining thermodynamic processes that occur at a fixed volume as it enables accurate predictions of energy exchange.

Specific Heat at Constant Pressure

In contrast, the specific heat at constant pressure, \(C_p\), pertains to the energy required to raise the temperature of one mole of a gas by one degree Kelvin while maintaining a constant pressure. The presence of constant pressure allows the gas to expand upon heating, which requires additional energy.
For a monoatomic gas, this quantity is expressed as: \(C_{p} = \frac{5}{2}R\), highlighting that it is greater than \(C_V\). This is because at constant pressure, work is done by the gas as it expands. Therefore, \(C_p\) also includes the energy needed to do this work against the external pressure, in addition to the energy required to increase the temperature.
For a diatomic gas, the specific heat capacities are higher due to the presence of additional degrees of freedom, like rotational and vibrational movements. This results in: \(C_{p} = \frac{7}{2}R\) and \(C_{V} = \frac{5}{2}R\), indicating that \(C_p\) is still larger than \(C_V\) because of the work done during expansion at constant pressure. Consequently, the energy required at constant pressure is more than at constant volume for both monoatomic and diatomic ideal gases.