Question

Heat capacity for \(\mathrm{Cl}_{2} .\) What is \(C_{V}\) at \(800 \mathrm{~K}\) for \(\mathrm{Cl}_{2}\) treated as an ideal diatomic gas in the high-temperature limit?

Short answer

At 800 K, the heat capacity at constant volume for \[\mathrm{Cl}_2\] as an ideal diatomic gas is approximately 29.1 J/mol K.

Step-by-step solution

Understand the problem

Determine the heat capacity at constant volume (\(C_V\)) for chlorine gas (\reviews evaluating the limiting behavior of a high-temperature ideal diatomic gas.

Recall the formula for heat capacity of a diatomic gas

For a diatomic gas, in the high-temperature limit, the total degrees of freedom (including translational, rotational, and vibrational) contribute to the heat capacity. The formula is given by: \[ C_V = \frac{f}{2} R \]where \( f \) is the degrees of freedom, and \( R \) is the gas constant.

Identify the degrees of freedom for \(\mathrm{Cl}_2\)

A diatomic gas has 3 translational, 2 rotational, and 2 vibrational degrees of freedom. The total degrees of freedom at high temperature are: \[ f = 3 + 2 + 2 = 7 \]

Substitute the values into the formula

Plug in \( f = 7 \) and \( R = 8.314 \; \mathrm{J/mol \, K} \) into the formula: \[ C_V = \frac{7}{2} R = \frac{7}{2} \times 8.314 \approx 29.1 \; \mathrm{J/mol \, K} \]

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental principle used in thermodynamics to describe the behavior of ideal gases. It is represented by the equation:
\( PV = nRT \).

In this equation:

• \( P \): Pressure of the gas
• \( V \): Volume of the gas
• \( n \): Amount of substance (in moles)
• \( R \): Ideal gas constant \( (8.314 \, \text{J/mol K}) \)
• \( T \): Absolute temperature in Kelvin

The ideal gas law helps us understand how gases expand and contract depending on temperature, pressure, and volume changes.

Degrees of Freedom

In thermodynamics, degrees of freedom refer to the number of independent ways in which a system can store energy. For a diatomic gas like chlorine gas (\( \text{Cl}_2 \)), we consider translational, rotational, and vibrational movements.

• Translational: The gas molecules move in three-dimensional space (x, y, z directions). Thus, there are three translational degrees of freedom.
• Rotational: For a diatomic molecule, rotation can occur around two axes perpendicular to the bond axis. Hence, there are two rotational degrees.
• Vibrational: Diatomic molecules have one vibrational mode. Each mode counts as two degrees (one for kinetic and one for potential energy). Thus, there are two vibrational degrees of freedom.

This summation gives a total of seven degrees of freedom for diatomic gases at high temperatures (
\(3 + 2 + 2 = 7\)).

High-Temperature Limit

At high temperatures, the vibrational modes of diatomic molecules are fully excited along with the translational and rotational modes. This is why we consider all seven degrees of freedom (3 translational, 2 rotational, and 2 vibrational) for calculating heat capacity.

Normally, at room temperature, vibrational modes contribute very little because their energy levels are not accessible. However, as temperature increases, these modes get populated and contribute significantly to heat capacity calculations.

Heat Capacity Calculation

Heat capacity is a property that describes how much energy a substance can absorb per unit temperature increase. For a diatomic gas at constant volume (\(C_V\)), it can be calculated using degrees of freedom.

The formula is:
\[ C_V = \frac{f}{2} R \]
where:

• \( f \): Degrees of freedom (which is 7 for diatomic gases at high temperatures)
• \( R \): Ideal gas constant (\( 8.314 \, \text{J/mol·K} \))

So, for chlorine gas (\( \text{Cl}_2 \)) at 800 K:
\[ C_V = \frac{7}{2} \times 8.314 \, \text{J/mol·K} \ C_V \ \approx 29.1 \, \text{J/mol·K} \]

Diatomic Molecules

Diatomic molecules are molecules composed of only two atoms. Examples include chlorine gas (\( \text{Cl}_2 \)), oxygen gas (\( \text{O}_2 \)), and hydrogen gas (\( \text{H}_2 \)).

For diatomic gases, we consider:

• Movements: The atoms can move translationally and rotate.
• Bond Vibration: The bond between the two atoms can stretch and compress (vibrational modes).

Understanding these properties is essential for computing thermal properties such as the heat capacity in different thermodynamic conditions.