Question

Determine the standard molar entropy of \(\mathrm{OClO}\), a nonlinear triatomic molecule where \(B_{A}=1.06 \mathrm{cm}^{-1}\) \\[ \begin{array}{l} B_{B}=0.31 \mathrm{cm}^{-1}, B_{C}=0.29 \mathrm{cm}^{-1}, \widetilde{\nu}_{1}=938 \mathrm{cm}^{-1} \\ \tilde{\nu}_{2}=450 . \mathrm{cm}^{-1}, \widetilde{\nu}_{3}=1100 . \mathrm{cm}^{-1}, \text {and } P=1.00 \mathrm{atm} \end{array} \\]

Short answer

The standard molar entropy of OClO can be found by calculating the individual entropy contributions from translational, rotational, and vibrational motion and summing them together. Using the given rotational constants and vibrational frequencies, along with the Boltzmann constant and Planck constant, the total standard molar entropy of OClO can be determined using \( S_{Total} = S_{trans} + S_{rot} + S_{vib} \).

Step-by-step solution

Calculate the Boltzmann constant and Plank constant

First, we need to find the Boltzmann constant (kB) and the Plank constant (h). Boltzmann constant, \(k_B\) = 1.38 x 10^{-23} J K^{-1} Planck constant, \(h\) = 6.626 x 10^{-34} Js We will be using these values later as our calculations' basis to find the entropy.

Calculate translational entropy

To calculate the translational entropy (Strans), we use the following formula: \[S_{trans} = R\left(\frac{5}{2} + \ln [V(2\pi m kT/h^2)^\frac{3}{2} / P]\right) \] We are given standard pressure (P) = 1.00 atm, and we need to convert it to J/m³: \[ P = 1.00\,\mathrm{atm} \times 101325\,\mathrm{Pa}/\mathrm{atm} = 101325\,\mathrm{Pa} = 101325\,\mathrm{J}/\mathrm{m³}\] For the volume (V), we can use the molar volume at standard conditions (T = 1K): \[ V = \frac{RT}{P} \] Where T = 298K, R = 8.314 J K^{-1} mol^{-1}

Calculate rotational entropy

We are given the rotational constants \(B_A\), \(B_B\), and \(B_C\), and we will use them to calculate the rotational entropy (Srot) using the following formula for a nonlinear molecule: \[ S_{rot} = R\left(\ln(\sigma T^{\frac{3}{2}}(B_A B_B B_C)^{-\frac{1}{2}})\right) \] Here, \(\sigma\) refers to the symmetry number of the molecule. Since OClO has no symmetry, its symmetry number is 1.

Calculate vibrational entropy

We are given the vibrational frequencies \(\tilde{\nu}_{1}\), \(\tilde{\nu}_{2}\), and \(\tilde{\nu}_{3}\). To calculate the vibrational entropy (Svib), we use the following formula: \[ S_{vib} = R \sum_{i=1}^{3} \left[\frac{\tilde{\nu}_i}{T}\frac{1}{e^{\frac{\tilde{\nu}_i}{T}}-1} - \ln(1-e^{-\frac{\tilde{\nu}_i}{T}})\right] \] Make sure to convert the vibrational frequencies from \(\mathrm{cm}^{-1}\) to J/mol.

Calculate the total standard molar entropy

Now that we have calculated the entropy contributions from translational, rotational, and vibrational motion, we can add them together to find the total entropy (STotal): \[ S_{Total} = S_{trans} + S_{rot} + S_{vib} \] By calculating all the individual entropies and summing them up, we will obtain the standard molar entropy of OClO in J K^{-1} mol^{-1}.

Key concepts

Translational Entropy

Translational entropy is associated with the motion of molecules along a three-dimensional space. It is an important contribution when calculating the total entropy of a system, especially in gases where molecules have significant freedom to move. Translational entropy, denoted as \(S_{trans}\), can be calculated using the equation:\[S_{trans} = R\left(\frac{5}{2} + \ln \left[ \frac{V(2\pi m k T/h^2)^{\frac{3}{2}}}{P} \right] \right) \]In this formula:

• \(R\) is the universal gas constant.
• \(V\) is the molar volume, which can be calculated using the temperature and pressure.
• \(m\) is the mass of the molecule.
• \(k\) is the Boltzmann constant.
• \(h\) is the Planck constant.

The entropy increases with a higher volume and temperature, emphasizing how space for molecular motion and energy availability affect translational entropy.

Rotational Entropy

Rotational entropy is linked to the molecule's ability to rotate around its axes. For nonlinear molecules like OClO, which consists of three different rotational axes, this is a critical factor influencing their entropy. The formula used to calculate rotational entropy \(S_{rot}\) for a nonlinear molecule is:\[ S_{rot} = R\left(\ln(\sigma T^{\frac{3}{2}}(B_A B_B B_C)^{-\frac{1}{2}})\right) \]Important components include:

• \(\sigma\) is the symmetry number, reflecting the molecule's mirror-image symmetry. For OClO, \(\sigma\) is 1.
• \(B_A, B_B, B_C\) are the rotational constants for each axis.

These constants help quantify how rotation around different axes contributes to the total entropy. The symmetry number accounts for whether identical rotations produce indistinguishable configurations, impacting entropy calculations.

Vibrational Entropy

Vibrational entropy arises from the quantized vibrational motions within a molecule. Each vibrational mode contributes to the molecule's energy distribution at a particular temperature. The vibrational entropy \(S_{vib}\) is determined using the frequencies of these modes:\[ S_{vib} = R \sum_{i=1}^{3} \left[ \frac{\tilde{u}_i}{T} \frac{1}{e^{\frac{\tilde{u}_i}{T}}-1} - \ln(1-e^{-\frac{\tilde{u}_i}{T}}) \right] \]Key highlights of the formula are:

• \(\tilde{u}_i\) represents each vibrational frequency, initially given in \(\mathrm{cm}^{-1}\) and needing conversion to energy units.
• The exponential terms arise from the quantum mechanical treatment of vibrations.

This aspect of entropy shows how the periodic stretching and compressing of molecular bonds contributes to the total entropy of a system.

Boltzmann Constant

Named after the Austrian physicist Ludwig Boltzmann, the Boltzmann constant \(k\) serves as a bridge between macroscopic and microscopic physics. It forms part of the equation \( S = k \ln \Omega \), where \(\Omega\) is the number of microstates corresponding to the macrostate of a system.The Boltzmann constant value is \(1.38 \times 10^{-23} \mathrm{J}\,\mathrm{K}^{-1}\). This constant is crucial for translating microscopic properties of individual particles to macroscopic thermochemical properties measurable in laboratories. It plays a fundamental role in entropy calculations of systems and is essential in formulas that involve temperature and energy scale comparisons, linking thermodynamic laws with statistical mechanics.

Planck Constant

The Planck constant \(h\) is a fundamental parameter in quantum mechanics, introduced by Max Planck. It connects the energy of photons to their frequency and is part of the fundamental equation \(E = h u\), where \(E\) is energy and \(u\) is frequency. Its value is \(6.626 \times 10^{-34} \mathrm{Js}\).Within the context of entropy, particularly in calculating translational and vibrational contributions, the Planck constant governs the quantization of energy levels. It helps determine how discrete the energy states of a system can be, making it indispensable for the precise computation of entropy in quantum systems.