Question

Consider a system made up of \(N\) particles. The energy per particle is given by \((E\rangle=\left(\Sigma F_{1} e^{-E_{1}} / k_{B} T\right) / Z,\) 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 \(N(d(E) / d T)\) and approximate its behavior at very high and very low temperatures (that is, \(k_{\mathrm{R}} T \gg 1\) and \(\left.k_{\mathrm{B}} T \propto 1\right)\).

Short answer

Question: Calculate the heat capacity of the given two-state system and approximate its behavior at very high and very low temperatures. Answer: The heat capacity of the system is C = -N/(k_BT^2). At very high temperatures, the heat capacity is inversely proportional to the square of the temperature, and at very low temperatures, it is approximately zero.

Step-by-step solution

Partition function (Z)

First, recall the partition function for a two-state system, which is given by: Z = g1 * e^{-E1/(k_BT)} + g2 * e^{-E2/(k_BT)} In this case, we are given g1 = g2 = 1, E1 = 0, and E2 = E. Plugging these values into the equation, we get: Z = e^{0} + e^{-E/(k_BT)} = 1 + e^{-E/(k_BT)}

Calculate energy per particle

Now, we can find the average energy per particle using the given formula: = (Σ F1 * e^{-E1} / k_BT) / Z Then substitute E1 = 0 and E2 = E: = (e^{-0}/(k_BT) + e^{-E/(k_BT)}) / (1 + e^{-E/(k_BT)}) Simplify the expression: = ((1 + e^{-E/(k_BT)})/k_BT) / (1 + e^{-E/(k_BT)}) = 1/k_BT

Find the heat capacity

Now, we take the derivative of the average energy per particle with respect to temperature to find the heat capacity: C = N * (d/dT) C = N * (-1/(k_BT^2)) So, the heat capacity of the system is: C = -N/(k_BT^2)

Approximate heat capacity at high and low temperatures

We are now ready to approximate the heat capacity at very high and very low temperatures. Let's analyze the given conditions: 1. Very high temperatures (k_BT ≫ 1): At high temperatures, the term -E/(k_BT) approaches 0 because E is constant and k_BT increases, making the exponential term close to 1. Thus, the heat capacity at very high temperatures becomes: C ≈ -N/(k_BT^2) 2. Very low temperatures (k_BT ≈ 1): At low temperatures, -E/(k_BT) becomes very large and negative since k_BT approaches 1 and E is positive. Thus, the exponential term becomes very close to 0, and the heat capacity becomes: C ≈ 0 So, at very high temperatures, the heat capacity is inversely proportional to the square of the temperature, while at very low temperatures, it is approximately zero.

Key concepts

Partition Function

Understanding the partition function is critical when studying the statistical mechanics of a system. It serves as a key to unlocking many thermodynamic properties of the system. At its core, the partition function (often designated as Z) is a sum over all the possible states of the system, each weighted by the Boltzmann factor, e^{-E/(k_BT)}, where E is the energy of a particular state, k_B is the Boltzmann constant, and T is the temperature.

In a two-state system, as described in the exercise, computing the partition function becomes relatively straightforward since there are only two energy levels to consider. For our system, with states at energy levels E_1=0 and E_2=E and both having a degeneracy g_1=g_2=1, the partition function simplifies to Z = 1 + e^{-E/(k_BT)}. This expression captures the essence of how each state contributes to the overall statistical behavior of the system, laying the groundwork to explore other thermodynamic quantities, such as average energy and heat capacity.

Average Energy per Particle

The average energy per particle is a concept that comes into play after we have our partition function. It tells us about the energy distribution within a system at thermal equilibrium. The formula for calculating the average energy per particle is <E> = (Σ F1 * e^{-E1} / k_BT) / Z. This equation is a statistical mean of the energy levels, considering the likelihood of each state's occupancy.

In the given exercise, after substituting the appropriate values for the two-state system, we derive that the average energy per particle <E> is essentially 1/k_BT. It encapsulates the effect of temperature on the energetic behavior of the particles, indicating how as temperature increases, the average energy spreads out over the possible states.

Temperature Dependence of Heat Capacity

The temperature dependence of heat capacity can reveal insights into the energy dynamics of a system. Heat capacity (C) is defined as the amount of heat required to change the temperature of the system by a certain amount. In mathematical terms, it's the derivative of the average energy with respect to temperature, C = N * (d<E>/dT). For our specific two-state system, the heat capacity simplified to C = -N/(k_BT^2).

Now, when we dive into the temperature dependence, two distinct regimes appear. At very high temperatures, where k_BT ≫ 1, the heat capacity tends to be inversely proportional to the square of the temperature. Conversely, at very low temperatures—approaching absolute zero—the system has minimal energy changes due to temperature variations, and thus the heat capacity approaches zero. This behavior reflects the quantum mechanical principles dictating that at near-zero temperatures the system is in its lowest energy state, or ground state, with very limited thermal excitation.