Question

Inspection of the thermodynamic tables in the back of the text reveals that many molecules have quite similar constant volume heat capacities. a. The value of \(C_{V, M}\) for \(\operatorname{Ar}(g)\) at standard temperature and pressure is \(12.48 \mathrm{J} \mathrm{mol}^{-1} \mathrm{K}^{-1},\) identical to gaseous \(\mathrm{He}(g)\) Using statistical mechanics, demonstrate why this equivalence is expected. b. The value of \(C_{V, M}\) for \(\mathrm{N}_{2}(g)\) is \(20.8 \mathrm{J} \mathrm{mol}^{-1} \mathrm{K}^{-1}\). Is this value expected given your answer to part (a)? For \(\mathrm{N}_{2}\) \(\widetilde{\nu}=2359 \mathrm{cm}^{-1}\) and \(B=2.00 \mathrm{cm}^{-1}\)

Short answer

The equivalence in heat capacities between Ar(g) and He(g) is expected since both are monoatomic gases with only translational degrees of freedom, contributing \( \frac{3}{2}R \) to the molar constant volume heat capacity. For diatomic Nitrogen (N₂), its heat capacity value of 20.8 J/(mol·K) is expected, as it has 5 significant degrees of freedom (3 translational and 2 rotational) at standard temperature. The vibrational contribution is negligible due to the high vibrational frequency.

Step-by-step solution

a. Demonstrate the equivalence of Argon and Helium heat capacities using statistical mechanics

To demonstrate the equivalence in heat capacities between Ar(g) and He(g) using statistical mechanics, we first need to understand that both the gases are monoatomic. That means, they don't have any vibrations or rotations. The molar constant volume heat capacity, \(C_{V,M}\), can be expressed as the sum of contributions from different degrees of freedom. For monoatomic gases like Ar and He, there are only translational degrees of freedom, which contribute a total of 3 degrees of freedom (one for each spatial dimension). The classical equipartition theorem states that each degree of freedom contributes \(\frac{1}{2}R\) to the molar heat capacity at constant volume, where R is the gas constant. So, for Ar(g) and He(g), the molar constant volume heat capacity can be calculated as: \(C_{V,M} = \frac{3}{2}R\) Using the value of gas constant R = 8.314 J/(mol·K), we get: \(C_{V,M} = 12.47 \mathrm{J}/(\mathrm{mol·K})\) This value is the same as the given value for Ar(g) and He(g), which demonstrates why their heat capacities are expected to be identical, as they have the same degrees of freedom.

b. Check if the heat capacity value of Nitrogen is expected

For Nitrogen(N₂), it is a diatomic gas, meaning it has rotational and vibrational degrees of freedom in addition to the translational degrees of freedom. For a diatomic gas, there are a total of 5 degrees of freedom: 3 translational, 2 rotational, and 1 vibrational. However, at standard temperature, the vibrational contribution to the heat capacity can be considered negligible, as the quantized energy levels are farther apart due to the characteristic high vibrational frequency. Therefore, we have 5 degrees of freedom that significantly contribute to the heat capacity of N₂(g). Using the classical equipartition theorem, like in part (a), we get the molar constant volume heat capacity for N₂(g): \(C_{V,M} = \frac{5}{2}R\) Using the value of gas constant R = 8.314 J/(mol·K), we get: \(C_{V,M} = 20.79 \mathrm{J}/(\mathrm{mol·K})\) This value is very close to the given value of 20.8 J/(mol·K) for N₂(g), indicating that the value is expected given our answer to part a. Moreover, at standard temperature, it is reasonable to ignore the contributions of the vibrational degree of freedom, as it is much smaller than the translational and rotational contributions.

Key concepts

Statistical Mechanics

Statistical mechanics is a powerful tool used to relate the microscopic properties of atoms and molecules to the macroscopic observable properties in thermodynamics. By treating a system as an ensemble of particles, statistical mechanics helps predict how averages come together to exhibit the characteristics we measure in experiments. This field blends principles of probability with the laws of thermodynamics.
- At its core, statistical mechanics explores the behavior of particles based on statistical rules. - By considering large numbers of particles, it adopts a probabilistic approach to avoid the complexities of individual particle motion.
In our study of heat capacity for gases, statistical mechanics provides the framework to understand how energy is distributed across various molecular degrees of freedom. Monoatomic gases like Argon and Helium utilize statistical mechanics in explaining why they possess identical heat capacities.

Equipartition Theorem

The equipartition theorem is a key principle within statistical mechanics. It provides insight into the distribution of energy across the different degrees of freedom in a system. Simply put, the theorem states that each degree of freedom contributes an equal amount of energy to the total heat capacity.
- Each degree of freedom contributes an energy of \(\frac{1}{2}kT\) per molecule, where \(k\) is the Boltzmann constant and \(T\) is the temperature.- For Avogadro's number of molecules, this translates to \(\frac{1}{2}R\) per mole, where \(R\) is the universal gas constant.
For gases, these degrees of freedom usually include translational, rotational, and sometimes vibrational motions. In monoatomic gases with only translational motion, the example demonstrated why the heat capacity is \(\frac{3}{2}R\), given three degrees of freedom all contributing equally to the system's energy.

Degrees of Freedom

Degrees of freedom in a molecular system refer to the independent ways in which a molecule can store energy. These are related to the molecule's translations, rotations, and vibrations.
- **Translational Degrees:** All gases have three translational degrees of freedom, corresponding to movement in the x, y, and z directions. - **Rotational Degrees:** Diatomic gases like N extsubscript{2} have two rotational degrees because they can rotate about two perpendicular axes perpendicular to the line connecting both atoms. - **Vibrational Degrees:** These typically become more relevant at higher temperatures as they require more energy.
Each degree of freedom contributes to the internal energy and hence the heat capacity. Monoatomic gases only consider translational degrees, while diatomic gases must account for both translational and rotational, leading to different heat capacity values.

Monoatomic Gases

Monoatomic gases are composed of single atoms and include elements like Argon (Ar) and Helium (He). These gases are straightforward examples in statistical mechanics due to their simplicity.
- They do not possess rotational or vibrational energies under normal conditions, simplifying their energy calculations to pure translational motion.- According to the equipartition theorem, the constant volume heat capacity \(C_{V,M}\) for these gases is \(\frac{3}{2}R\), exclusively from their three translational degrees.
This is why both Argon and Helium exhibit identical heat capacities at constant volume, as they exhibit no rotational or vibrational energy exchange in normal contexts. Their behavior provides a clear baseline for understanding more complex gas molecules.

Diatomic Gases

Diatomic gases consist of molecules with two atoms, like N extsubscript{2} (Nitrogen). These gases present more complexity than monoatomic gases because they introduce both rotational and sometimes vibrational energy components.
- These molecules inherently possess three translational and two rotational degrees of freedom.- Under standard temperatures, vibrational contributions are often negligible but become significant at higher temperatures.
Utilizing the equipartition theorem, the heat capacity \(C_{V,M}\) for diatomic gases at typical conditions is \(\frac{5}{2}R\), reflecting its additional degrees of freedom available. This complexity in diatomic molecules helps in understanding their enhanced heat capacities as compared to monoatomic gases, aligning well with the observed data for Nitrogen.