Question

Heat capacities of liquids. (a) \(C_{V}\) for liquid argon (at \(T=100 \mathrm{~K}\) ) is \(18.7 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\). How much of this heat capacity can you rationalize on the basis of your knowledge of gases? (b) \(C_{V}\) for liquid water at \(T=10^{\circ} \mathrm{C}\) is about \(75 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\). Assuming water has three vibrations, how much of this heat capacity can you rationalize on the basis of gases? What is responsible for the rest?

Short answer

For liquid argon, a significant portion of the heat capacity (18.7 vs 12.471 J K^{-1} mol^{-1}) can be explained using gas-phase properties. For liquid water, about half (75 vs 37.413 J K^{-1} mol^{-1}) can be explained, with the remainder due to liquid phase interactions.

Step-by-step solution

Understanding Heat Capacities for Gases

The heat capacity at constant volume (\(C_{V}\)) for a monatomic ideal gas can be determined using the equation: \[ C_{V} = \frac{3}{2} R \] where \(R\) is the universal gas constant \(8.314 \text{ J K}^{-1} \text{ mol}^{-1}\).

Calculate Heat Capacity for Liquid Argon

Given: Heat capacity for liquid argon \(C_{V} = 18.7 \text{ J K}^{-1} \text{ mol}^{-1}\). For a monatomic gas like argon, we use the formula from Step 1: \[ C_{V} = \frac{3}{2} R = \frac{3}{2} \times 8.314 \text{ J K}^{-1} \text{ mol}^{-1} = 12.471 \text{ J K}^{-1} \text{ mol}^{-1} \]

Compare the Heat Capacity of Liquid Argon

Comparing \( 18.7 \text{ J K}^{-1} \text{ mol}^{-1} \) (given) and \( 12.471 \text{ J K}^{-1} \text{ mol}^{-1} \) (calculated), a significant portion of the heat capacity of liquid argon can be rationalized based on the ideal gas value.

Understanding Heat Capacities for Polyatomic Molecules

The heat capacity for a polyatomic gas (like water), taking into account translational, rotational, and vibrational modes, is given by: \[ C_{V} = \left( \frac{3}{2} \right) R + R + \left( 3 \times R \right) = \left( \frac{3}{2} + 1 + 3 \right) R \]Where the translational contribution is \(\left( \frac{3}{2} R \right)\), rotational contribution is \( R \), and vibrational contribution is \( 3R \)

Calculate Heat Capacity for Liquid Water

Given: Heat capacity for liquid water \(C_{V} = 75 \text{ J K}^{-1} \text{ mol}^{-1}\). Using the formula from Step 4, we have:\[ C_{V} = \left( \frac{3}{2} + 1 + 3 \right) R = \left( \frac{3}{2} + 1 + 3 \right) \times 8.314 \text{ J K}^{-1} \text{ mol}^{-1} = 37.413 \text{ J K}^{-1} \text{ mol}^{-1} \]

Compare the Heat Capacity of Liquid Water

Comparing \( 75 \text{ J K}^{-1} \text{ mol}^{-1} \) (given) and \( 37.413 \text{ J K}^{-1} \text{ mol}^{-1} \) (calculated), around half of the heat capacity can be rationalized based on gas phase modes. The rest is attributed to intermolecular interactions in the liquid phase.

Key concepts

Heat Capacity

Heat capacity is the amount of heat energy required to raise the temperature of a substance by one degree Celsius. It is usually represented by the symbol, \(C\), and measured in joules per Kelvin per mole (\(\mathrm{JK}^{-1} \, \mathrm{mol}^{-1}\)). Heat capacity can be measured at constant volume (\(C_{V}\)) or constant pressure (\(C_{P}\)). For gases, the heat capacity is influenced by translational, rotational, and vibrational motions of the molecules. Each type of motion contributes differently to the heat capacity. For solids and liquids, intermolecular forces also play a significant role.

Monatomic Gases

Monatomic gases consist of single atoms, such as argon (Ar). They have a simple structure, with only translational modes of motion contributing to their heat capacity. The heat capacity at constant volume (\(C_{V}\)) for a monatomic ideal gas can be calculated using the formula:\[ C_{V} = \frac{3}{2}R \] where \(R\) is the universal gas constant \(8.314 \, \mathrm{JK}^{-1} \, \mathrm{mol}^{-1}\). For example, in the given exercise, the calculated heat capacity for liquid argon is:\[ C_{V} = \frac{3}{2} \times 8.314 = 12.471 \, \mathrm{JK}^{-1} \, \mathrm{mol}^{-1} \] However, comparing this with the provided heat capacity of 18.7 \, \mathrm{JK}^{-1} \, \mathrm{mol}^{-1}, it indicates that while a significant portion of the heat capacity for liquid argon can be explained by its monatomic nature, additional interactions in the liquid phase also contribute.

Polyatomic Molecules

Polyatomic molecules, like water (H₂O), have more complex structures compared to monatomic gases. They possess translational, rotational, and vibrational modes of motion, each contributing to the total heat capacity. The heat capacity for a polyatomic ideal gas can be expressed as: \[ C_{V} = \left( \frac{3}{2} \right) R + R + \left( 3 \times R \right) = \left( \frac{3}{2} + 1 + 3 \right) R \] Here, the contributions are:

• Translational: \(\left( \frac{3}{2}R \right)\)
• Rotational: \(R\)
• Vibrational: \(3R\) (assuming three vibrational modes)

For liquid water, given its heat capacity of 75 \(\mathrm{JK}^{-1} \, \mathrm{mol}^{-1}\), we compare it with the calculated value: \[ C_{V} = \left( \frac{3}{2} + 1 + 3 \right) \times 8.314 = 37.413 \, \mathrm{JK}^{-1} \, \mathrm{mol}^{-1} \] This shows that approximately half of the heat capacity for liquid water can be rationalized by gas phase modes, while the rest arises from additional factors.

Intermolecular Interactions

In liquids, intermolecular interactions play a crucial role in determining heat capacities. These interactions are more significant than in gases due to the closer proximity of molecules. These forces are responsible for the extra heat capacity observed in liquids that cannot be explained solely on the basis of ideal gas behavior. For instance, the heat capacity of liquid water is much higher than that calculated from gas phase modes because:

• Hydrogen bonding
• Van der Waals forces
• Dipole-dipole interactions

These forces require additional energy to overcome during heating, contributing to a higher heat capacity. Similarly, the difference between the calculated and given heat capacities for liquid argon can also be attributed to the additional energy required to handle intermolecular forces present in the liquid state.