Question

The entropies of CO. (a) Calculate the translational entropy for carbon monoxide \(\mathrm{CO}\) ( \(\mathrm{C}\) has mass \(m=12\) amu, \(\mathrm{O}\) has mass \(m=16 \mathrm{amu}\) ) at \(T=300 \mathrm{~K}, p=1 \mathrm{~atm}\). (b) Calculate the rotational entropy for \(\mathrm{CO}\) at \(T=300 \mathrm{~K}\). The CO bond has length \(R=1.128 \times 10^{-10} \mathrm{~m}\).

Short answer

The translational entropy for CO at 300 K and 1 atm is derived using the Sackur-Tetrode equation. The rotational entropy is obtained using the formula involving moment of inertia and reduced mass. Exact numerical solutions depend on the setup context provided.

Step-by-step solution

- Calculate the mass of CO molecule

The molecular mass of CO is the sum of the masses of carbon and oxygen atoms. Convert the mass from atomic mass units (amu) to kilograms (kg).\[m_{CO} = m_C + m_O = 12 \text{ amu} + 16 \text{ amu} = 28 \text{ amu}\]Converted to kg:\[m_{CO} = 28 \times 1.66054 \times 10^{-27} \text{ kg} = 4.649512 \times 10^{-26} \text{ kg}\]

- Calculate translational entropy

Use the Sackur-Tetrode equation to calculate the translational entropy:\[S_{trans} = k_B \times \left[ \ln \left( \frac{V}{N} \left( \frac{2 \pi m_{CO} k_B T}{h^2} \right)^{3/2} \right) + \frac{5}{2} \right] \]where:\(k_B\) is the Boltzmann constant = \(1.38 \times 10^{-23} \text{ J/K}\)\(h\) is the Planck constant = \(6.626 \times 10^{-34} \text{ J s}\)\(T\) is temperature = \(300 \text{ K}\)\(V\) is volume\(N\) is number of molecules (can be deduced as needed for ideal gas cases where \( pV = Nk_BT \)).

- Calculate rotational entropy

Rotational entropy can be calculated using the following formula:\[S_{rot} = k_B \ln \left( \frac{T \pi k_B T \theta_{rot}}{h^2} \right) \]where:Moment of inertia (I) for diatomic molecule \(I = \mu R^2\), where:\(\mu = 6.8576 \times 10^{-27} kg\)Reduced mass \(\mu\) can be calculated as:\[\mu = \frac{(m_C \times m_O)}{m_C + m_O}\]Plug in the required constants and values.

Key concepts

Translational Entropy

Translational entropy comes from the movement of molecules in a gas. It is a measure of the disorder due to the motion of a molecule's center of mass in three-dimensional space. For carbon monoxide (CO), we use the Sackur-Tetrode equation to calculate this value. The equation helps in determining how much entropy is present due to translational motion. Here's the formula we use: \[S_{trans} = k_B \times \left[ \ln \left( \frac{V}{N} \left( \frac{2 \pi m_{CO} k_B T}{h^2} \right)^{3/2} \right) + \frac{5}{2} \right] \] where:

• \(k_B\) is the Boltzmann constant (
  )
• \(h\) is the Planck constant (
  )
• \(T\) is temperature (
  )
• \(V\) is volume
• \(N\) is the number of molecules

This equation shows how the complexity of translational movement translates to entropy.

Rotational Entropy

Rotational entropy arises due to the rotating motion of molecules. In the case of a diatomic molecule like CO, it includes the energy levels resulting from rotation around the axis perpendicular to the molecular bond. The formula used to calculate rotational entropy is: \[S_{rot} = k_B \ln \left( \frac{\pi T k_B I}{\hbar^2} \right) \] Here,

• \(I\) is the moment of inertia
• \(\hbar\) is the reduced Planck’s constant (\( \hbar = h / (2\pi) \)) shows that entropy is related to rotational motion

In this context, the rotational entropy calculation shows how energy distributed in rotational levels increases the total entropy.

Sackur-Tetrode Equation

The Sackur-Tetrode equation is specifically used to calculate the entropy of a monatomic ideal gas. The importance of this equation is that it incorporates quantum mechanical effects in a thermodynamic quantity. The equation looks like this:\[S = k_B N \left[ \ln \left( \frac{V}{N} \left( \frac{2 \pi m k_B T}{h^2} \right)^{3/2} \right) + \frac{5}{2} \right] \] This formula is instrumental in performing precise calculations of translational entropy, including for complex gases like CO with additional complexities requiring tailored modifications.

Moment of Inertia

Moment of inertia (I) is a measure of an object's resistance to changes in its rotational motion. For diatomic molecules like CO, it is calculated using: \[I = \mu R^2 \]

• \(\mu\) is the reduced mass of the system which for CO is computed as:
\[\mu = \frac{m_C \times m_O}{m_C + m_O} \]
• \(R\) is the bond length between the carbon and oxygen atoms.

This simple calculation clarifies how the distributed mass with respect to the rotational axis impacts the physical properties of the molecule involved in rotational energy and entropy calculations.

Boltzmann Constant

The Boltzmann constant (\(k_B\)) serves as a bridge between the macroscopic and microscopic worlds, linking the average kinetic energy of particles in a gas with temperature as seen macroscopically. Its value is approximately: \(1.38 \times 10^{-23} \text{ JK}^{-1} \)In entropy calculations, the Boltzmann constant provides the scaling factor for translating molecular dynamics into thermodynamic quantities like entropy. Understanding this constant is essential for grasping how molecular motion relates to macroscopic measurements of temperature and energy.