Question

Consider a simple thermodynamic system in which particles can occupy only two states: a lower state, whose energy we define as 0 , and an upper state, energy \(E_{u}\) (a) Carry out the sun (with only two states, integration is certainly not valid) giving the average particle energy \(E\), and plot your result as a function of temperature. (b) Explain qualitatively why it should behave as it does (c) This system can be used as a model of paramagnetism. where individual atoms' magnetic moments can other be aligned or antialigned with an external magnetic field, giving a low or high energy, respectively. Describe how the average alignment or antialignment depends on temperature. Does it make sense'?

Short answer

The average energy is given by \(E = E_{u}*e^{-E_{u}/kT} / (e^{0/kT} + e^{-E_{u}/kT})\). As the temperature increases, more atoms reach the upper state, increasing average energy until it eventually levels off when all atoms have achieved the upper state. This is demonstrated on the plot. For the paramagnetism model, at low temperatures, alignment with the external field will predominate. As temperature rises and atoms can reach the higher energy, antialigned state, the net magnetic alignment decreases.

Step-by-step solution

Express Boltzmann distribution

Begin by expressing the states according to the Boltzmann distribution. For the upper state, the number of entities is given by \(n_{u} = e^{-E_{u}/kT}\) where \(E_{u}\) is the energy of the upper state, \(k\) is the Boltzmann constant and \(T\) is the temperature. For the lower state, the number of entities is \(n_{l} = e^{0/kT}\).

Calculation of Average Energy

To find the average energy per particle, take the total energy and divide by the total number of particles. The total energy is given by \(E = n_{l}*0 + n_{u}*E_{u}\). The total number of particles is \(N = n_{l} + n_{u}\). Therefore, the average energy \(E = (n_{l}*0 + n_{u}*E_{u})/(n_{l} + n_{u}) = n_{u}*E_{u}/(n_{l} + n_{u})\). Expressing \(n_{l}\) and \(n_{u}\) as \(e^{0/kT}\) and \(e^{-E_{u}/kT}\) respectively, we obtain: \(E = E_{u}*e^{-E_{u}/kT} / (e^{0/kT} + e^{-E_{u}/kT})\)

Plot

Now plot the average energy \(E\) as a function of temperature \(T\). This can be done using graph plotting tools with temperature varying on the x-axis and energy on the y-axis.

Interpret the Plot

Explain the plot. As temperature increases, more atoms reach the upper state, increasing average energy. The rate increases until it eventually levels off when all atoms have achieved the upper state. This represents a transition from a lower energy ground state to a more chaotic, high energy state with rising temperature.

Magnetic Alignment Understanding

Through the process described above, we can understand how the average alignment or antialignment depends on temperature. At low temperatures, alignment will predominate because it represents the lowest energy state. As temperature rises, there's more kinetic energy available and atoms can reach the higher energy, antialigned state. Therefore, increasing temperature reduces the net magnetic alignment. This process is consistent with how paramagnetism works.

Conclusion

Offer final thoughts on the problem. The two-state energy system helps to understand statistical thermodynamics and properties like paramagnetism. It shows how thermodynamic systems can change their state under temperature influences.

Key concepts

Boltzmann distribution

The Boltzmann distribution is a fundamental concept in statistical thermodynamics. It describes how particles distribute themselves among available energy states within a system.
The probability of a particle occupying a state with energy \(E_i\) is given by:

• \(P_i = \frac{e^{-E_i/kT}}{Z}\)

where \(e\) is the base of the natural logarithm, \(k\) is the Boltzmann constant, \(T\) is the temperature, and \(Z\) is the partition function, a sum over all states: \(Z = \sum_j e^{-E_j/kT}\).

In our two-state system, the lower state has energy 0, and the upper state has energy \(E_{u}\). The Boltzmann distribution allows us to calculate the relative occupancy of each state:

• For the lower state: \(n_{l} = e^{0/kT}\).
• For the upper state: \(n_{u} = e^{-E_u/kT}\).

This distribution helps explain why at higher temperatures, more particles occupy higher energy states.

Average energy calculation

Calculating the average energy in a two-state system involves considering the energies and probabilities of each state. The formula for average energy \(E\) is:

• \(E = \frac{n_{l} \cdot 0 + n_{u} \cdot E_u}{n_{l} + n_{u}}\)

Replacing \(n_l\) and \(n_u\) with their expressions from the Boltzmann distribution, we have:

• \(n_l = e^{0/kT} = 1\)
• \(n_u = e^{-E_u/kT}\)

Therefore, the average energy becomes:

• \(E = \frac{E_u \cdot e^{-E_u/kT}}{1 + e^{-E_u/kT}}\)

This and similar calculations help in understanding how the internal energy of a system changes with temperature.

Two-state system

A two-state system is a simplified model used to explain certain physical phenomena, such as paramagnetism, where each entity can occupy one of two possible states. In our case, these are the lower energy state with energy 0 and the upper energy state with energy \(E_{u}\).

This model allows us to observe how systems behave under different conditions. A simple two-state system is easy to analyze, making it a great introduction to more complex systems in statistical thermodynamics. It reflects how real systems can have states corresponding to different energy levels and how they respond to changing temperatures, closely mimicking many real-world physical phenomena.

Temperature dependence

Temperature plays a key role in determining how particles distribute themselves in energy states. As temperature increases, thermal energy provides particles with enough energy to jump from lower to higher energy states. In a two-state system, this manifests as more particles occupying the upper energy state with energy \(E_u\).

In our system, at low temperatures, most particles reside in the lower state, as it is energetically favorable. As the temperature rises, particles gain more energy due to increased thermal agitation, thus more particles transition to the higher energy state.
Eventually, at high enough temperatures, the population in the upper state increases, leading to a saturation point where additional temperature rise doesn't significantly change the distribution anymore.

Statistical thermodynamics

Statistical thermodynamics provides a link between the macroscopic properties of systems and their microscopic components. By considering the statistics of particles, it helps explain how macroscopic properties like temperature and energy arise from the behavior of atoms and molecules.

In our two-state system, statistical thermodynamics allows us to calculate the number of particles in each state, using the Boltzmann distribution, and from this, derive the average energy. These calculations help clarify how temperature affects energy distribution. Understanding such models offers insights into more complex thermodynamic systems, where the behavior of large numbers of particles determines the observable properties.