The three lowest energy levels of a certain molecule are E1=0, E2= ε, E3=10 ε. Show that at suff

The three lowest energy levels of a certain molecule are...

The three lowest energy levels of a certain molecule are $E_1=0, E_2=$ $\varepsilon, E_3=10 \varepsilon$. Show that at sufficiently low temperatures (how low?) only levels $E_1, E_2$ are populated. Find the average energy $E$ of the molecule at temperature $T$. Find the contributions of these levels to the specific heat per mole, $C_v$, and sketch $C_v$ as a function of $T$.

Key Concepts

Boltzmann Distribution

The Boltzmann distribution explains how particles in a system occupy discrete energy states at thermal equilibrium, with the probability of a state of energy E being proportional to exp(–E/(kT)), where k is the Boltzmann constant and T is the temperature. This concept is fundamental in determining the relative populations of energy levels and understanding the temperature dependence of statistical properties.

Partition Function

The partition function is a central quantity in statistical mechanics that sums over all possible energy states, each weighted by the Boltzmann factor. It acts as a normalization constant for probabilities and is used to derive macroscopic thermodynamic properties such as the average energy, free energy, and entropy of a system.

Low Temperature Approximation

At sufficiently low temperatures, only the energy states with the smallest energies are significantly populated because the Boltzmann factors for higher energy states become negligibly small. This approximation simplifies analysis by effectively reducing the system to include only a subset of available energy levels, typically those for which E is comparable to or less than kT.

Specific Heat and Its Temperature Dependence

Specific heat measures how a system's internal energy changes with temperature. In the context of discrete energy levels, the specific heat can be derived by differentiating the temperature-dependent average energy. The distribution of energy levels and thermal activation between them leads to characteristic peaks or features in the specific heat as a function of temperature, reflecting the activation of higher energy states as temperature increases.

Transcript

0:00 Hi there.

00:01 So for this problem, we need to show that the energy of the system is given by this expression here.

00:13 Now, n distinguishable atoms are distributed over two energies levels, which is the energy 1, which is equal to 0, and the energy 2, which is equal to the energy.

00:34 So with that said, since the atoms are distinguishable, we need to use balsamount estuistic.

00:43 So for that, we state that the number n1 is equal to a times the exponential of minus the energy 1.

00:54 Let's put it just like this.

00:55 Well, let's put it in the same way as they did.

00:59 The energy 1 divided by balsman constant times the temperature.

01:05 And then the energy 2, sorry, the number 2, which is a times the exponential of the energy divided by balsman constant times the temperature.

01:24 Remember that in here, the energy 1 is 0.

01:31 So we know that the exponential of 0 is just 1.

01:34 So this expression reduces just to simply a.

01:39 That we we denote n as the total number of atoms total number of atoms so the sum of these two values n1 plus n2 this is equal to the total number n and then we will have that that total number n is equal to a times one plus the esponential of the energy divided by bolzman constant times the temperateur.

02:19 And then if we solve for a, we will find that a then is equal to the total number n.

02:28 And this divided by 1 plus the s penumptial of minus the energy divided by bolzman constant times the temperature.

02:37 So the total energy is then, the total energy that we call just e is just simply the number n, 1 times the energy 1 plus n2 times the energy 2.

02:54 Now in this case, we know that the energy 1 is 0, so this whole term cancel.

02:59 So with this just simply reduces to n2 times the energy 2.

03:05 So we will have that this is a times the exponential of minus the energy divided by bolzman constant times the temperature.

03:13 And this times the energy in here.

03:21 So with that said, we can substitute the expression that we obtain for a.

03:29 So that will be then that this is the total number times the energy, and this times the exponential of minus the energy divided by bolzman constant times the temperature, and this divided by 1 plus the exponential of minus the energy divided by boltzmann constant times the temperature.

03:50 And this is the expression that we wanted to obtain for this problem.

03:58 Now for part b, we are asked about to show that the specific heat is given by the following expression.

04:12 Let me put that equation in here.

04:23 So that expression is the following.

04:34 Okay.

04:36 So we will have it in here.

04:38 Let me just move it down here...

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