Thermodynamic information

From the Average Molecular Energy to the Internal Energy

The partition function provides a direct connection between the microscopic properties of a system and its macroscopic thermodynamic observables. The average molecular energy, $ \langle \epsilon \rangle $, is calculated for each mode of motion, such as translation, vibration, and rotation.

The average molecular energy is given by:

\[ \langle \epsilon \rangle = -\frac{1}{q} \left( \frac{\partial q}{\partial \beta} \right)_V = k_B T^2 \left( \frac{\partial \ln q}{\partial T} \right)_V. \]

The internal energy $ U $ of a system is the total energy contained in the system and includes contributions from all molecules. For a system of $ N $ molecules, the internal energy is:

\[ U(T) = U(0) + N \langle \epsilon \rangle, \]

where $ U(0) $ represents the zero-point energy of the system. Substituting $ \langle \epsilon \rangle $:

\[ U(T) = U(0) - N \frac{1}{q} \left( \frac{\partial q}{\partial \beta} \right)_V. \]

Alternatively, the internal energy can be expressed in terms of the total partition function $ q_{tot} $:

\[ U = U(0) - \left( \frac{\partial \ln q_{tot}}{\partial \beta} \right)_V. \]

For a system of interacting molecules (an ensemble), the total partition function $ q_{tot} $ is the product of the contributions from translation, vibration, rotation, and electronic modes:

\[ q_{tot} = q_{\text{tr}} \cdot q_{\text{rot}} \cdot q_{\text{vib}} \cdot q_{\text{el}}. \]

Substituting this into the expression for internal energy, we have:

\[ U = -N \left[ \left( \frac{\partial \ln q_{\text{tr}}}{\partial \beta} \right)_V + \left( \frac{\partial \ln q_{\text{rot}}}{\partial \beta} \right)_V + \left( \frac{\partial \ln q_{\text{vib}}}{\partial \beta} \right)_V + \cdots \right]. \]

This separates the contributions to the internal energy into different degrees of freedom. Explicitly:

\[ U = U_{\text{tr}} + U_{\text{rot}} + U_{\text{vib}} + U_{\text{el}}, \]

where each term represents the internal energy contribution from the respective mode of motion.

Physical Insights: The decomposition of the internal energy allows us to understand how each molecular degree of freedom contributes to the total energy. For instance:

Translational energy: Dominant at high temperatures for all molecules.
Vibrational energy: Becomes significant at higher temperatures as higher vibrational modes are populated.
Electronic energy: Usually negligible at typical temperatures unless electronic excitation occurs.

This formulation highlights the utility of the partition function in determining thermodynamic properties.

Contribution of Translations to the Internal Energy

Translational motion contributes to the internal energy through the translational partition function, $ q_{\text{tr}} $. The translational internal energy $ U_{\text{tr}} $ is derived as:

\[ U_{\text{tr}} = -N \frac{1}{q_{\text{tr}}} \left( \frac{\partial q_{\text{tr}}}{\partial \beta} \right)_V. \]

Substituting $ q_{\text{tr}} = \frac{V}{\Lambda^3} $, we get:

\[ U_{\text{tr}} = -N \Lambda^3 \cdot \frac{\partial}{\partial \beta} \left( \Lambda^{-3} \right)_V. \]

Using $ \Lambda = \left( \frac{h^2 \beta}{2 \pi m} \right)^{1/2} $, we find:

\[ \Lambda^{-3} = \left( \frac{2 \pi m}{h^2 \beta} \right)^{3/2}. \]

Taking the derivative of $ \Lambda^{-3} $ with respect to $ \beta $:

\[ \frac{\partial}{\partial \beta} \Lambda^{-3} = -\frac{3}{2} \cdot \left( \frac{2 \pi m}{h^2} \right)^{3/2} \cdot \beta^{-5/2}. \]

Substituting this back, the negative sign cancels, giving:

\[ U_{\text{tr}} = N \cdot \frac{3}{2} \cdot \beta^{-1}. \]

Finally, substituting $ \beta = \frac{1}{k_B T} $, we have:

\[ U_{\text{tr}} = \frac{3}{2} N k_B T. \]

Using $ N k_B = nR $, the final result becomes:

\[ U_{\text{tr}} = \frac{3}{2} nRT. \]

Therefore, the translational internal energy is directly proportional to the temperature $T$, which aligns with the principle of equipartition of energy. Each translational degree of freedom contributes $\frac{1}{2}nRT$ to the internal energy. The result also demonstrates how the microscopic partition function links to a macroscopic thermodynamic property, such as internal energy. The factor $nR$ indicates the dependence on the number of moles $n$ and the gas constant $R$, making the result consistent with classical thermodynamics. For ideal gases, this is the total internal energy since there are no contributions from intermolecular interactions. This result is critical for understanding gas thermodynamics, particularly for monatomic ideal gases where $U_{tr}$ equals the total internal energy.

Contribution of Rotations to the Internal Energy

Rotational motion contributes to the internal energy through the rotational partition function, $ q_{\text{rot}} $. For a symmetric linear molecule considered as a rigid-rotor, the rotational partition function is:

\[ q_{\text{rot}} = \frac{1}{\beta hcB}, \]

where:

$ \beta = \frac{1}{k_B T} $: Inverse thermal energy,
$ h $: Planck's constant,
$ c $: Speed of light,
$ B $: Rotational constant of the molecule.

The rotational contribution to the internal energy is given by:

\[ U_{\text{rot}} = -N \frac{1}{q_{\text{rot}}} \left( \frac{\partial q_{\text{rot}}}{\partial \beta} \right)_V. \]

Substituting $ q_{\text{rot}} = \frac{1}{\beta hcB} $, we get:

\[ U_{\text{rot}} = -N \beta hcB \left( \frac{\partial}{\partial \beta} \left( \frac{1}{\beta hcB} \right) \right)_V. \]

Simplifying the derivative:

\[ \frac{\partial}{\partial \beta} \left( \frac{1}{\beta hcB} \right) = -\frac{1}{\beta^2 hcB}. \]

Substituting this result back, we have:

\[ U_{\text{rot}} = -N \beta hcB \cdot \left( -\frac{1}{\beta^2 hcB} \right) = N \beta^{-1}. \]

Using $ \beta = \frac{1}{k_B T} $, the rotational contribution becomes:

\[ U_{\text{rot}} = N k_B T = nRT, \]

where $ N k_B = nR $, with $ n $ as the number of moles and $ R $ as the ideal gas constant.

Linear vs. Nonlinear Molecules: The above result applies to linear molecules with two rotational degrees of freedom. For nonlinear molecules, which have three rotational degrees of freedom, the result generalizes to:

Linear molecules: $ U_{\text{rot}} = nRT $,
Nonlinear molecules: $ U_{\text{rot}} = \frac{3}{2} nRT $.

Physical Interpretation: The rotational contribution to the internal energy scales linearly with the temperature $ T $ and the number of moles $ n $. Each rotational degree of freedom contributes $ \frac{1}{2} nRT $, consistent with the equipartition theorem.

Contribution of Vibrations to the Internal Energy

Vibrational motion contributes to the internal energy through the vibrational partition function $ q_{\text{vib}} $, which accounts for the quantized vibrational energy levels of a molecule. The vibrational partition function is:

\[ q_{\text{vib}} = \left[ 1 - \exp(-\beta hc \tilde{\nu}) \right]^{-1}, \]

where:

$ \beta = \frac{1}{k_B T} $: Inverse thermal energy,
$ hc \tilde{\nu} $: Vibrational energy quantum, where $ \tilde{\nu} $ is the vibrational wavenumber.

Note that this expression does not include the zero-point energy $ \frac{hc \tilde{\nu}}{2} $, which is constant and does not contribute to the temperature-dependent part of the internal energy.

The vibrational contribution to the internal energy is given by:

\[ U_{\text{vib}} = -N \frac{1}{q_{\text{vib}}} \left( \frac{\partial q_{\text{vib}}}{\partial \beta} \right)_V. \]

Substituting $ q_{\text{vib}} $:

\[ U_{\text{vib}} = -N \left[ 1 - \exp(-\beta hc \tilde{\nu}) \right] \cdot \left( \frac{\partial}{\partial \beta} \left[ 1 - \exp(-\beta hc \tilde{\nu}) \right]^{-1} \right)_V. \]

Taking the derivative of $ \left[ 1 - \exp(-\beta hc \tilde{\nu}) \right]^{-1} $:

\[ \frac{\partial}{\partial \beta} \left[ 1 - \exp(-\beta hc \tilde{\nu}) \right]^{-1} = \left[ -hc \tilde{\nu} \cdot \exp(-\beta hc \tilde{\nu}) \right] \cdot \left[ 1 - \exp(-\beta hc \tilde{\nu}) \right]^{-2}. \]

Substituting back into the expression for $ U_{\text{vib}} $, we get:

\[ U_{\text{vib}} = N \cdot \frac{hc \tilde{\nu} \cdot \exp(-\beta hc \tilde{\nu})}{\left[ 1 - \exp(-\beta hc \tilde{\nu}) \right]^2} \cdot \left[ 1 - \exp(-\beta hc \tilde{\nu}) \right]. \]

Simplifying:

\[ U_{\text{vib}} = N \frac{hc \tilde{\nu} \cdot \exp(-\beta hc \tilde{\nu})}{1 - \exp(-\beta hc \tilde{\nu})}. \]

At high temperatures ($ k_B T \gg hc \tilde{\nu} $), the exponential term $ \exp(-\beta hc \tilde{\nu}) $ approaches 1. In this limit:

\[ U_{\text{vib}} \approx N \cdot \frac{hc \tilde{\nu}}{\beta hc \tilde{\nu}} = N \cdot \beta^{-1}. \]

Substituting $ \beta^{-1} = k_B T $, we find:

\[ U_{\text{vib}} \approx N k_B T = nRT, \]

where $ n $ is the number of moles, and $ R $ is the ideal gas constant.

Physical Interpretation: The vibrational contribution to the internal energy reflects the quantized nature of vibrational motion. At high temperatures, when many vibrational states are populated, the behavior approaches classical equipartition, contributing $ nRT $ to the internal energy. At low temperatures, where only the lowest vibrational states are populated, the internal energy deviates significantly from this classical result.

Total Internal Energy

The total internal energy of a system is the sum of contributions from different molecular degrees of freedom, specifically translation, rotation, vibration, and electronic motion:

\[ U_{\text{tot}} = U_{\text{tr}} + U_{\text{rot}} + U_{\text{vib}} + U_{\text{el}}. \]

The contributions from each degree of freedom are as follows:

Translational Energy:
\[ U_{\text{tr}} = \frac{3}{2} N k_B T = \frac{3}{2} nRT, \]
where $ N $ is the number of molecules, $ n $ is the number of moles, $ R $ is the gas constant, and $ T $ is the temperature.
Rotational Energy: For a linear molecule:
\[ U_{\text{rot}} = N k_B T = nRT. \]
For a nonlinear molecule:
\[ U_{\text{rot}} = \frac{3}{2} N k_B T = \frac{3}{2} nRT. \]
Vibrational Energy: At high temperatures ($ k_B T \gg hc \tilde{\nu} $), the vibrational energy contributes:
\[ U_{\text{vib}} = N k_B T = nRT. \]
Electronic Energy: The electronic contribution $ U_{\text{el}} $ is generally negligible at typical temperatures, as higher electronic states are not significantly populated. For most cases:
\[ U_{\text{el}} \approx 0. \]

Summing all the contributions for a linear molecule, the total internal energy becomes:

\[ U_{\text{tot}} = \frac{3}{2} N k_B T + N k_B T + N k_B T + 0 \]

For a nonlinear molecule:

\[ U_{\text{tot}} = \frac{3}{2} N k_B T + \frac{3}{2} N k_B T + N k_B T + 0. \]

Physical Interpretation: The total internal energy depends on the degrees of freedom associated with the molecular structure. Thus, the microscopic properties (e.g., degrees of freedom, vibrational states) are linked to macroscopic thermodynamic quantities such as the internal energy.

Heat Capacity

The heat capacity at constant volume, $ C_V $, measures the change in internal energy with temperature. It is defined as:

\[ C_V = \left( \frac{\partial U}{\partial T} \right)_V. \]

Using the relationship between $ \beta $ and $ T $ ($ \beta = \frac{1}{k_B T} $), we can also write:

\[ C_V = -k_B \beta^2 \left( \frac{\partial U}{\partial \beta} \right)_V. \]

1. Translational Contribution

The translational contribution to internal energy is:

\[ U_{\text{tr}} = \frac{3}{2} N k_B T. \]

Differentiating with respect to $ T $:

\[ C_{V,\text{tr}} = \left( \frac{\partial U_{\text{tr}}}{\partial T} \right)_V = \frac{3}{2} N k_B = \frac{3}{2} R, \]

where $ R = N_A k_B $ is the gas constant.

2. Rotational Contribution

The rotational contribution depends on whether the molecule is linear or nonlinear:

Linear Molecules:
\[ U_{\text{rot}} = N k_B T, \quad C_{V,\text{rot}} = \left( \frac{\partial U_{\text{rot}}}{\partial T} \right)_V = N k_B = R. \]
Nonlinear Molecules:
\[ U_{\text{rot}} = \frac{3}{2} N k_B T, \quad C_{V,\text{rot}} = \left( \frac{\partial U_{\text{rot}}}{\partial T} \right)_V = \frac{3}{2} N k_B = \frac{3}{2} R. \]

3. Vibrational Contribution

The vibrational contribution to the internal energy is given by:

\[ U_{\text{vib}} = \frac{N h c \tilde{\nu}}{\exp(\beta h c \tilde{\nu}) - 1}. \]

Differentiating with respect to $ T $, we find the vibrational heat capacity:

\[ C_{V,\text{vib}} = -k_B \beta^2 \left( \frac{\partial U_{\text{vib}}}{\partial \beta} \right)_V. \]

Substituting $ U_{\text{vib}} $ and its derivative, the result is:

\[ C_{V,\text{vib}} = N k_B \beta^2 (h c \tilde{\nu})^2 \cdot \frac{\exp(\beta h c \tilde{\nu})}{\left[ \exp(\beta h c \tilde{\nu}) - 1 \right]^2}. \]

Expressed in terms of the vibrational temperature $ \Theta_{\text{vib}} = \frac{h c \tilde{\nu}}{k_B} $, the heat capacity becomes:

\[ C_{V,\text{vib}} = N k_B \left( \frac{\Theta_{\text{vib}}}{T} \right)^2 \cdot \frac{\exp\left(\frac{\Theta_{\text{vib}}}{T}\right)}{\left[ \exp\left(\frac{\Theta_{\text{vib}}}{T}\right) - 1 \right]^2}. \]

4. High-Temperature Limit and Dulong-Petit Law

For solids, the vibrational heat capacity at high temperatures follows the Dulong-Petit law. In this limit ($ T \gg \Theta_{\text{vib}} $):

\[ C_{V,\text{vib}} \approx 3 N k_B = 3R. \]

This result arises because each vibrational degree of freedom contributes $ k_B T $ to the internal energy, consistent with the equipartition theorem.

The total heat capacity is the sum of the contributions from translation, rotation, and vibration. For a linear molecule at high temperatures:

\[ C_V = C_{V,\text{tr}} + C_{V,\text{rot}} + C_{V,\text{vib}} = \frac{3}{2} R + R + C_{V,\text{vib}}. \]

For nonlinear molecules:

\[ C_V = C_{V,\text{tr}} + C_{V,\text{rot}} + C_{V,\text{vib}} = \frac{3}{2} R + \frac{3}{2} R + C_{V,\text{vib}}. \]

Physical Interpretation: The heat capacity reflects the number of accessible degrees of freedom and their contribution to energy storage. Translational and rotational degrees of freedom contribute constant values ($ \frac{1}{2} R $ per degree), while vibrational contributions depend strongly on temperature, becoming significant only at higher temperatures. This section illustrates how an improtatnt macroscopic property like heat capacity arises from the microscopic characterization.