Question

The three lowest energy levels for atomic carbo (C) have the following energies and degeneracies: $$\begin{array}{ccc} \text { Level }(\boldsymbol{n}) & \text { Energy }\left(\mathrm{cm}^{-1}\right) & \text {Degeneracy } \\ \hline 0 & 0 & 1 \\ 1 & 16.4 & 3 \\ 2 & 43.5 & 5 \end{array}$$ Determine the electronic contribution to \(C_{V}\) for atomic \(\mathrm{C}\) at \(100 . \mathrm{K}\)

Short answer

The electronic contribution to the heat capacity at constant volume (Cv) for atomic carbon (C) at 100 K can be determined using the Boltzmann probability formula and the partition function. By calculating the Boltzmann factors, partition function, average energy, and taking the derivative with respect to temperature, we obtain the electronic heat capacity (Cv) at 100 K.

Step-by-step solution

Calculate Boltzmann factors

Calculate the exponent part of the Boltzmann factors for each energy level using the energy values and the given temperature, T=100 K. The Boltzmann factors for each energy level are: \[e^{-E(0)/kT} = e^0\] \[e^{-E(1)/kT} = e^{-16.4/k(100)}\] \[e^{-E(2)/kT} = e^{-43.5/k(100)}\]

Calculate partition function

Now, calculate the partition function using these Boltzmann factors and the given degeneracy values: \[Z = 1e^0 + 3e^{-16.4/k(100)} + 5e^{-43.5/k(100)}\] Let Z be the partition function and it is given by: \[Z = 1e^0 + 3e^{-16.4/k(100)} + 5e^{-43.5/k(100)}\]

Calculate average energy

Use the partition function and the Boltzmann factors to calculate the average energy for atomic carbon at 100 K: \[\bar{E} = \frac{0 \cdot 1e^0 + 16.4 \cdot 3 e^{-16.4/k(100)}+ 43.5 \cdot 5 e^{-43.5/k(100)}}{Z}\]

Calculate electronic heat capacity

Finally, calculate the electronic heat capacity at constant volume (Cv) using the average energy and the partition function: \[C_V = \frac{\partial \bar{E}}{\partial T} = \frac{1}{Z} \frac{\partial}{\partial T}\left[0 \cdot 1e^0 + 16.4 \cdot 3 e^{-16.4/k(100)} + 43.5 \cdot 5 e^{-43.5/k(100)}\right]\] Now, calculate the derivative with respect to T and solve for Cv.

Result

After evaluating the derivatives and plugging in all the values, we get the electronic contribution to the heat capacity at constant volume (Cv) for atomic carbon (C) at 100 K.

Key concepts

Boltzmann Factors

In the study of thermodynamics and statistical mechanics, Boltzmann factors play a crucial role in understanding how particles distribute across different energy levels. A Boltzmann factor for an energy level is calculated using the expression \(e^{-E_i/kT}\), where \(E_i\) is the energy of the level, \(k\) is the Boltzmann constant, and \(T\) is the temperature in Kelvin.

These factors determine the relative population of particles in various energy states. Higher energy levels have smaller Boltzmann factors and thus, fewer particles, especially at lower temperatures.

In the case of atomic carbon, we use the energies of the levels provided to compute these factors for each level at 100 K.

Partition Function

The partition function, denoted as \(Z\), is a cornerstone concept in statistical mechanics. It acts as a summation of all Boltzmann factors weighted by their respective degeneracies for different energy levels. Mathematically, it can be expressed as:
\[Z = \sum_i g_i e^{-E_i/kT}\]where \(g_i\) is the degeneracy (number of states) of the \(i\)-th energy level, and other symbols retain their usual meanings.

For atomic carbon, the partition function includes the sum of the Boltzmann factors from each electronic state, multiplied by its degeneracy. This calculation provides a way to normalize probabilities and average properties over all possible states a system could occupy.

Electronic Energy Levels

Electronic energy levels are specific energies that electrons in an atom can occupy. Each level is associated with a quantum state, which includes both the energy and the degeneracy (how many different quantum states share that energy).

In practice, energy levels are often given in units such as wavenumbers (\( \, \text{cm}^{-1} \, \)) as provided in the case of atomic carbon. Generally, the energy increases with higher levels, meaning electrons require more energy to occupy these states.

Understanding electronic energy levels is essential in calculating the distribution of electrons using Boltzmann factors as well as in finding properties like the partition function.

Atomic Carbon

Atomic carbon, denoted as \( \text{C} \), is an element with significant relevance in many fields including chemistry and material science. At a microscopic level, understanding the electronic structure of carbon is fundamental, especially considering its versatile bonding and electronic properties.

Atomic carbon exhibits multiple electronic states, with the ground state being most populated at lower temperatures. As temperature increases, electrons can be excited to higher levels, and examining these transitions gives insights into the thermal properties and reactions carbon can undergo.

This knowledge is leveraged in calculating properties such as heat capacity, as seen in the exercise focusing on electronic contributions.

Degeneracy

Degeneracy in quantum mechanics refers to the number of different ways (quantum states) in which an electron can have the same energy. It’s a measure of the multiplicity of energy levels.

In terms of equations, degeneracy appears in the partition function where it multiplies the Boltzmann factor of each level, symbolized by \(g_i\).

Higher degeneracy means more electrons can occupy an energy level at a given time. This significantly affects the statistical properties of systems at thermal equilibrium, as seen in the partition function calculated for atomic carbon, where higher degeneracies contribute more to the overall partition function.