To calculate the molar internal energy and molar heat capacity of a system with two states, we need to consider the Boltzmann distribution and the concept of statistical thermodynamics.
The energy difference between the two states is given as kT at 10K, where k is the Boltzmann constant and T is the temperature.
- Molar Internal Energy: The molar internal energy (U) of the system can be calculated by considering the average energy of each state weighted by their respective probabilities. In this case, since we have two states, the internal energy can be calculated as:
U = (E₁ * P₁) + (E₂ * P₂)
Where: E₁ and E₂ are the energies of state 1 and state 2, respectively. P₁ and P₂ are the probabilities of state 1 and state 2, respectively.
In the Boltzmann distribution, the probability of each state is given by:
P = e^(-E / (kT))
Substituting the energy difference given in the question (E = kT) and considering that the sum of probabilities for all states is equal to 1, we can calculate P₁ and P₂:
P₁ = e^(-kT / (kT)) = e^(-1) = 1/e P₂ = 1 - P₁ = 1 - 1/e
Now we can calculate the molar internal energy:
U = (E₁ * P₁) + (E₂ * P₂) U = (0 * P₁) + (kT * P₂) U = kT * P₂ U = kT * (1 - 1/e)
- Molar Heat Capacity: The molar heat capacity (C) of the system can be calculated as the derivative of the molar internal energy with respect to temperature:
C = dU/dT
Differentiating the expression for U with respect to T:
C = d(kT * (1 - 1/e)) / dT C = k * (1/e)
Therefore, the molar internal energy is kT * (1 - 1/e), and the molar heat capacity is k/e.
Note: In this calculation, we assume that the energies of the states are multiples of kT and that the degeneracy of each state is the same.