Question

Which species will have the largest rotational partition function at a given temperature: \(\mathrm{H}_{2}, \mathrm{HD},\) or \(\mathrm{D}_{2} ?\) Which of these species will have the largest translational partition function assuming that volume and temperature are identical? When evaluating the rotational partition functions, you can assume that the high-temperature limit is valid.

Short answer

At a given temperature, D₂ will have the largest rotational partition function since it has the smallest \(\theta_{rot}\), while H₂ will have the largest translational partition function since it has the smallest mass.

Step-by-step solution

Find the characteristic rotational temperatures for each species

First, we need to find the characteristic rotational temperature, \(\theta_{rot}\), for all three species. This can be usually found from a data table or using the formula: \(\theta_{rot} = \frac{hcB}{k}\), where \(h\) is Planck's constant, \(c\) is the speed of light, \(B\) is the rotational constant, and \(k\) is the Boltzmann constant. The rotational constants for the species are: - H₂: \(B = 60.853 \space cm^{-1}\) - HD: \(B = 44.363 \space cm^{-1}\) - D₂: \(B = 30.427 \space cm^{-1}\) Plug these values into the equation to get the rotational temperatures for each species.

Find the rotational partition functions for each species

Next, use the formula for the high-temperature limit of the rotational partition function and the \(\theta_{rot}\) values we found in Step 1: \(q_{rot} = \frac{T}{\theta_{rot}}\) We can now perform the calculation for each species to determine which has the largest rotational partition function at a given temperature.

Compare the translational partition functions for each species

Since the volume and temperature are identical for each species, we can use the ratio of their masses to compare their translational partition functions. The equation is: \(q_{trans} = (\frac{2\pi mkT}{h^2})^{3/2}V\) The masses for the species are: - H₂: \(m = 2 \space amu\) - HD: \(m = 3 \space amu\) - D₂: \(m = 4 \space amu\) Given that the other factors in the equation are the same for all species, we can simply compare their masses to determine which species will have the largest translational partition function.

Key concepts

Physical Chemistry

Physical chemistry is a branch of chemistry focused on the study of how matter behaves on a molecular and atomic level, and how chemical reactions occur. It blends principles of physics and chemistry to understand the physical properties of molecules, the forces that act upon them, and the energy changes that occur during chemical reactions. One of the key concepts in physical chemistry is the partition function, which is an essential tool for studying systems in statistical mechanics and thermodynamics. It provides valuable insights into the distribution of molecular energy states and allows us to calculate important thermodynamic properties like entropy, free energy, and heat capacity. The calculations of different partition functions, such as translational, rotational, vibrational, and electronic, help construct a comprehensive picture of the molecular energy landscape.

Translational Partition Function

The translational partition function describes the statistical distribution of particles in different translational energy states in a given volume and temperature. Translational motion refers to the movement of the entire particle from one place to another. For an ideal gas, the translational partition function, denoted as q_trans, is related to the mass m of the particles, the temperature T, the volume V, and fundamental constants like Planck's constant h and Boltzmann's constant k. The formula is expressed as:

\(q_{trans} = (\frac{2\pi mkT}{h^2})^{3/2}V\)

This function plays a crucial role in the determination of the overall energy distribution of a gas. Since each type of motion (translational, rotational, vibrational) contributes to the total energy, understanding each individual partition function is necessary for a full thermodynamic assessment of a system.

Characteristic Rotational Temperature

Every molecule has a characteristic rotational temperature, denoted as θ_rot. It is a temperature that correlates with the energy required for a molecule to occupy its first excited rotational energy level. The concept of a characteristic rotational temperature allows us to compare the rotational energy levels of different molecules. The formula to calculate it is:

\(\theta_{rot} = \frac{hcB}{k}\)

Here, h is Planck's constant, c is the speed of light, B is the rotational constant specific to each molecule, and k is Boltzmann's constant. The values of these constants are standardized, but the rotational constant B varies between molecules, as it is inversely related to the moment of inertia. Larger values of B correspond to larger rotational constants and higher characteristic rotational temperatures. This reflects how molecules with lower moments of inertia (commonly lighter molecules) require less energy to rotate, and therefore their energy levels are closer together.

High-Temperature Limit

In the context of rotational partition functions, the 'high-temperature limit' approximates the behavior of a system when the temperature is so high that a significant number of rotational states are populated. Under this condition, it is assumed that energy changes for rotation are small compared to the thermal energy (kT), allowing for a simplification of the rotational partition function, q_rot. The simplified formula used is:

\(q_{rot} = \frac{T}{\theta_{rot}}\)

This equation illustrates that as the temperature T increases, the rotational partition function also increases. It is important to note that this high-temperature limit gives an approximate value that's accurate enough for practical uses when temperature is much greater than the characteristic rotational temperature of the molecule. When using this approximation, we can more easily compare and determine the impacts of rotational energy on the thermodynamic properties of molecules at sufficiently high temperatures.