Question

Consider a system made up of \(N\) particles. The average energy per particle is given by \(\langle E\rangle=\left(\sum E_{i} e^{-E_{i} / k_{B} T}\right) / Z\) where \(Z\) is the partition function defined in equation \(36.29 .\) If this is a two-state system with \(E_{1}=0\) and \(E_{2}=E\) and \(g_{1}=\) \(g_{2}=1,\) calculate the heat capacity of the system, defined as \(N(d\langle E\rangle / d T)\) and approximate its behavior at very high and very low temperatures (that is, \(k_{\mathrm{B}} T \gg 1\) and \(k_{\mathrm{B}} T \ll 1\) ).

Short answer

Answer: At very high temperatures, the heat capacity is approximately 0, while at very low temperatures, the heat capacity is proportional to \(N E^2 / k_{B} T^2\).

Step-by-step solution

Calculate the partition function Z

Since we have a two-state system with \(E_{1}=0\) and \(E_{2}=E\), and \(g_{1}=g_{2}=1\), the partition function Z is given by the sum of the Boltzmann factors for these two states: $$Z = g_{1}e^{-E_{1}/k_BT} + g_{2}e^{-E_{2}/k_BT} = e^0 + e^{-E/k_BT} = 1 + e^{-E/k_BT}$$

Calculate the average energy per particle

Using the formula for average energy per particle, let's plug in the values for \(E_{1}\), \(E_{2}\), and Z: $$\langle E\rangle = \frac{\sum E_{i} e^{-E_{i} / k_{B} T}}{Z} = \frac{0\cdot e^0 + E\cdot e^{-E/k_BT}}{1 + e^{-E/k_BT}} = \frac{E\cdot e^{-E/k_BT}}{1 + e^{-E/k_BT}}$$

Differentiate the average energy per particle with respect to temperature T

Now let's differentiate \(\langle E\rangle\) with respect to T, using quotient rule: $$\frac{d\langle E\rangle}{dT} = \frac{-E^{2}/k_{B} e^{-E/k_BT}(1 + e^{-E/k_BT}) - E\cdot e^{-E/k_BT}(-E/k_BT)e^{-E/k_BT}}{(1 + e^{-E/k_BT})^2}$$ After some simplification: $$\frac{d\langle E\rangle}{dT} = \frac{E^{2}/k_{B} e^{-E/k_BT}}{k_{B} T^2 (1 + e^{-E/k_BT})^2}$$

Calculate the heat capacity

The heat capacity is defined as \(N(d\langle E\rangle / d T)\). Plug in the result from Step 3: $$C = N\frac{E^2 / k_{B} e^{-E/k_BT}}{k_{B} T^2 (1 + e^{-E/k_BT})^2} = \frac{N E^2 e^{-E/k_BT}}{k_{B} T^2 (1 + e^{-E/k_BT})^2}$$

Approximate the heat capacity behavior at very high and very low temperatures

At very high temperatures (\(k_BT \gg 1\)), \(e^{-E/k_BT} \approx 0\), so the heat capacity becomes: $$C \approx \frac{N E^2 \cdot 0}{k_{B} T^2 (1 + 0)^2} = 0$$ At very low temperatures (\(k_BT \ll 1\)), \(e^{-E/k_BT} \approx \infty\), so the heat capacity simplifies to: $$C \approx \frac{N E^2 \cdot \infty}{k_{B} T^2 (\infty)^2} = \frac{N E^2}{k_{B} T^2}$$ So, at very high temperatures, the heat capacity is approximately 0, while at very low temperatures, the heat capacity is proportional to \(N E^2 / k_{B} T^2\).

Key concepts

Two-State System

In a two-state system, we consider a scenario where each particle within the system can exist in one of only two possible energy states. This is a simplified model that helps in understanding basic statistical mechanics concepts.
The two energy levels in such a system are often denoted as \(E_1 = 0\) and \(E_2 = E\), with degeneracies \(g_1 = g_2 = 1\). Here, \(E_1\) being zero is a reference that signifies the ground state, and \(E_2\) is the excited state.

• The partition function \(Z\), which is crucial in statistical mechanics, is given by the sum of the Boltzmann factors of these two states: \(Z = 1 + e^{-E/k_BT}\). This function plays a key role in determining the statistical properties of the system.
• The average energy \(\langle E\rangle\), describes the mean energy per particle when the system is at thermal equilibrium. It is crucial because it can be used to determine other thermodynamic quantities, like heat capacity.

Heat Capacity

Heat capacity, denoted as \(C\), measures the amount of heat required to raise the temperature of a system by a given amount. In the context of our two-state system, it reflects how much energy is needed to increase the temperature of the entire system of \(N\) particles.
The heat capacity can be mathematically expressed as \(C = N \frac{d\langle E\rangle}{dT}\), where \(\langle E\rangle\) is the average energy per particle.
For our two-state system, heat capacity is computed using the change in average energy with temperature. Utilizing the quotient rule for differentiation, the formula simplifies to: \[ C = \frac{N E^2 e^{-E/k_BT}}{k_{B} T^2 (1 + e^{-E/k_BT})^2} \] This result tells us how the energy distribution evolves with temperature changes, emphasizing important concepts like energy dispersion and thermal response of the system.

High and Low Temperature Behavior

Understanding the high and low temperature behavior of a two-state system provides insights into how heat capacity varies with temperature.
At **high temperatures** \((k_B T \gg 1)\):

• The Boltzmann factor \(e^{-E/k_BT}\) becomes very small, effectively approaching zero.
• This simplification leads to the heat capacity \(C \approx 0\), as thermal energy is sufficiently high to populate excited states uniformly.

At **low temperatures** \((k_B T \ll 1)\):

• Conversely, the factor \(e^{-E/k_BT}\) becomes very large, indicating that most particles are in the ground state as the system has insufficient thermal energy to excite particles.
• The heat capacity then resembles \(\frac{N E^2}{k_{B} T^2}\), which shows a dependence inversely proportional to \(T^2\). This reflects increased sensitivity to temperature changes as only a few particles transition to the excited state as temperature varies.

Such temperature-dependent behaviors highlight how systems react under thermal variations and are key in disciplines like thermodynamics and material science.