Question

The properties of a two-state system. Given a twostate system in which the low energy level is \(600 \mathrm{cal} \mathrm{mol}^{-1}\), the high energy level is \(1800 \mathrm{cal} \mathrm{mol}^{-1}\), and the temperature of the system is \(300 \mathrm{~K}\), (a) What is the partition function \(q\) ? (b) What is the average energy \(\langle\varepsilon\rangle\) ?

Short answer

The partition function \(q\)is calculated and the average energy \varepsilon\.

Step-by-step solution

Determine the partition function

The partition function, denoted by the symbol `q`, is given by the sum of the Boltzmann factors for all possible states. For a two-state system, the energy levels are \(E_1 = 600\text{ cal mol}^{-1}\) and \(E_2 = 1800\text{ cal mol}^{-1}\). Convert these energy levels to \( \text{J mol}^{-1}\) by multiplying with 4.184 (since \(1\text{ cal mol}^{-1} = 4.184\text{ J mol}^{-1}\)). Hence, \(E_1 = 600 \times 4.184 = 2510.4\text{ J mol}^{-1}\) and \(E_2 = 1800 \times 4.184 = 7531.2\text{ J mol}^{-1}\).

Calculate the Boltzmann factors

The Boltzmann factor for a state with energy \(E_i\) at temperature \(T\) is given by \[e^{-E_i / kT}\], where \(k = 1.987 \text{ cal K}^{-1}mol^{-1} = 8.314 \text{ J K}^{-1}mol^{-1}\) is the Boltzmann constant. For \(T = 300\text{ K}\), calculate \[ e^{-E_1 / kT} = e^{-2510.4 / (8.314 \times 300)} \] and \[ e^{-E_2 / kT} = e^{-7531.2 / (8.314 \times 300)} \].

Compute the partition function \(q\)

The partition function \(q\) for the two-state system is the sum of the Boltzmann factors: \[ q = e^{-E_1 / kT} + e^{-E_2 / kT} = e^{-2510.4 / (8.314 \times 300)} + e^{-7531.2 / (8.314 \times 300)}\]

Calculate the average energy

The average energy \langle\text{E}\rangle \is given by \[ \frac{1}{q} \times (E_1 \times e^{-E_1 / kT} + E_2 \times e^{-E_2 / kT})\]. Using the previously calculated values: \langle\text{E}\rangle = \frac{(2510.4 \times e^{-2510.4 / (8.314 \times 300)} + 7531.2 \times e^{-7531.2 / (8.314 \times 300)})}{e^{-2510.4 / (8.314 \times 300)} + e^{-7531.2 / (8.314 \times 300)}}\.

Key concepts

partition function

The partition function is a crucial concept in thermodynamics, particularly in statistical mechanics. It acts like a 'weight' for the various energy states of a system. For a two-state system, the partition function, denoted as \( q \), is the sum of the Boltzmann factors of all possible energy states.

The Boltzmann factor, which we'll cover in the next section, takes into account the energy of a state and the temperature of the system. In our example, we have two energy levels: \( E_1 = 600 \text{ cal mol}^{-1} \) and \( E_2 = 1800 \text{ cal mol}^{-1} \). These need to be converted to joules per mole using the conversion factor \( 1 \text{ cal} = 4.184 \text{ J} \).

So, \( E_1 = 600 \times 4.184 = 2510.4 \text{ J mol}^{-1} \) and \( E_2 = 1800 \times 4.184 = 7531.2 \text{ J mol}^{-1} \). Now, by computing the Boltzmann factors for these energies and summing them, we get the partition function \( q \):

\[ q = e^{-2510.4 / (8.314 \times 300)} + e^{-7531.2 / (8.314 \times 300)} \]

This gives us a single number that encapsulates the statistical properties of the two-state system.

Boltzmann factor

The Boltzmann factor is an exponential term that weighs the probability of a state having a certain energy. The formula for the Boltzmann factor is:

\[ e^{-E_i / kT} \]

where \( E_i \) is the energy of the state, \( k \) is the Boltzmann constant, and \( T \) is the absolute temperature in Kelvin.

In our specific example, we have the energy levels \( E_1 = 2510.4 \text{ J mol}^{-1} \) and \( E_2 = 7531.2 \text{ J mol}^{-1} \), and the Boltzmann constant \( k = 8.314 \text{ J K}^{-1} \text{ mol}^{-1} \).

For a temperature of 300K, let's calculate both Boltzmann factors:

• \( e^{-2510.4 / (8.314 \times 300)} \)
• \( e^{-7531.2 / (8.314 \times 300)} \)

These factors provide insight into the likelihood of the system being in either of the two states. The higher the energy, the lower the Boltzmann factor, making it less probable for the system to be in that state.

average energy

The average energy of a system, denoted as \( \langle E \rangle \), gives the expected energy considering all possible states weighted by their Boltzmann factors. For our two-state system, we use the partition function \( q \) and the energies \( E_1 \) and \( E_2 \).

The formula for the average energy is:

\[ \langle E \rangle = \frac{1}{q} \times (E_1 \times e^{-E_1 / kT} + E_2 \times e^{-E_2 / kT}) \]

This formula takes into account the energies multiplied by their respective probabilities (derived from Boltzmann factors) and normalized by the partition function.

Plugging in our values, we get:

\[ \langle E \rangle = \frac{(2510.4 \times e^{-2510.4 / (8.314 \times 300)} + 7531.2 \times e^{-7531.2 / (8.314 \times 300)})}{e^{-2510.4 / (8.314 \times 300)} + e^{-7531.2 / (8.314 \times 300)}} \]

This will give us a single number representing the average energy of our two-state system at 300K, providing a meaningful characterization of the system's thermal behavior.