Question

Determine the vibrational contribution to \(C_{V}\) for \(\mathrm{HCl}\left(\widetilde{\nu}=2886 \mathrm{cm}^{-1}\right)\) over a temperature range from 500 to \(5000 . \mathrm{K}\) in \(500 . \mathrm{K}\) intervals and plot your result. At what temperature do you expect to reach the hightemperature limit for the vibrational contribution to \(C_{V} ?\)

Short answer

The vibrational heat capacity (Cv) of HCl with vibrational frequency \(\widetilde{\nu} = 2886 cm^{-1}\) can be calculated using the formula: \[C_v = R \left( \frac{e^{x}}{(e^{x} - 1)^{2}} \right) x^{2}\] where \(x = \frac{hc\widetilde{\nu}}{k_B T}\) and R is the ideal gas constant. Upon calculation for temperature intervals from 500K to 5000K in 500K increments, we can plot these results to observe how Cv changes with temperature. Analyzing the plot, we can identify the temperature where the high-temperature limit is reached, and the vibrational contribution to Cv becomes constant.

Step-by-step solution

Calculate x for each temperature interval

Using the provided formula for x, calculate x for every temperature interval (500K, 1000K, ..., 5000K), where \(\widetilde{\nu}\) is given as 2886 cm⁻¹.

Calculate Cv for each temperature interval

For each calculated x, calculate \(C_v\) using the provided formula of Cv for vibrational mode with R = 8.314 J/mol.K: \[C_v = R \left( \frac{e^{x}}{(e^{x} - 1)^{2}} \right) x^{2}\]

Plot results

Plot the results with temperature on the x-axis and Cv on the y-axis. This graph will illustrate how the vibrational contribution to heat capacity changes with temperature.

Identify the high-temperature limit

Inspect the plot to identify the temperature at which the vibrational contribution to heat capacity approaches the high-temperature limit (Cv becomes constant).

Key concepts

Cv Calculation

Understanding how to calculate the vibrational heat capacity, denoted as \(C_{V}\), is essential in studying molecular thermodynamics. In this exercise, we calculate \(C_{V}\) for HCl across various temperatures, utilizing the formula: \[C_v = R \left( \frac{e^{x}}{(e^{x} - 1)^{2}} \right) x^{2}\]Here, \(R\) is the universal gas constant, 8.314 J/mol.K. The variable \(x\) is the dimensionless parameter \(x = \frac{hu}{kT}\), where:

• \(h\) represents Planck's constant.
• \(u\) denotes the vibrational wavenumber, \(2886 \text{ cm}^{-1}\) for HCl in this case.
• \(k\) is Boltzmann's constant.
• \(T\) is the temperature in Kelvin.

By calculating \(x\) for different temperature intervals starting at 500K up to 5000K, we can determine \(C_{V}\) for HCl, helping us explore the relationship between temperature and vibrational heat capacity.

High-Temperature Limit

The high-temperature limit in vibrational heat capacity refers to the temperature at which \(C_{V}\) becomes practically constant. This occurs when the thermal energy is much greater than the vibrational energy quantum, leading to the situation where all vibrational states are equally populated. In this state, the system behaves like a classical oscillator, and further increases in temperature do not significantly change the vibrational heat capacity. For diatomic molecules like HCl, this limit is typically approached at temperatures significantly higher than the energy spacing defined by \(u\). The step-by-step solution helps visualize this by plotting \(C_{V}\) against temperature, showing how \(C_{V}\) stabilizes at high temperatures, indicating that HCl has reached its high-temperature limit.

HCl Vibrations

Roughly 2866 cm⁻¹ is the vibrational wavenumber for HCl, which is characteristic of its bond stretching frequency. Every diatomic molecule possesses a unique vibrational signature determined by the mass of the atoms and the bond strength. These vibrational frequencies can be quantified using spectroscopy, offering insights into the molecule's behavior.In the context of vibrational heat capacity, the wavenumber forms part of the calculation of \(x\), where it interacts directly with temperature to influence how energy is distributed among vibrational states. Monitoring this change across different temperatures provides a detailed view of energy modes in the molecule, illustrating the intrinsic link between molecular dynamics and thermal properties.

Physical Chemistry Education

Physical chemistry is the merging of chemistry and physics, exploring the properties of molecules as they correlate to their heat and energy distributions. Understanding vibrational heat capacity is pivotal, as it bridges quantum mechanical concepts and thermodynamics. This exercise with HCl demonstrates real-world applications of these theories, reinforcing the notion that changes at the molecular level are directly observable in bulk properties.For students, exercises such as this reveal how theoretical models apply to tangible chemical systems. By moving through calculations like \(C_V\), students sharpen their ability to understand molecular dynamics, preparing them for advanced studies or professional paths in material science and analytical chemistry. Ultimately, grasping these concepts enhances a student's ability to critically analyze a molecule's behavior within varied thermal contexts.