Energy probability. Find and make careful sketch of the probability density, ρ(E), for the energ

Energy probability. Find and make careful sketch of the...

Key Concepts

Probability Density Function (PDF) in Statistical Mechanics

The probability density function provides a way to describe the likelihood of a system being found in a particular continuous state, such as having a specific energy. It is normalized so that the total probability over all states equals one, making it essential for predicting the macroscopic properties of a system from its microscopic behavior.

Boltzmann Factor

The Boltzmann factor, given by exp(–E/(kT)), quantifies the probability of the system occupying a state with energy E at temperature T. It shows that states with higher energies are less likely to be occupied at a given temperature, reflecting the exponential decrease in probability with increasing energy.

Density of States

The density of states represents the number of available microstates at each energy level and is critical when determining the overall probability distribution of energy. It must be combined with the Boltzmann factor to obtain the correct energy dependence of the probability density in a system.

Maxwell-Boltzmann Energy Distribution

For a classical non-interacting monatomic gas, the Maxwell-Boltzmann energy distribution emerges from the combination of the Boltzmann factor and the density of states. In three dimensions, the kinetic energy distribution includes a ?E dependence from the density of states and an exponential decay from the Boltzmann factor, leading to a probability density of the form ?(E) ? ?E e^(–E/(kT)).

Transcript

00:01 So first we should find the values for a and b that are asked in the problem and they are corresponding to the probabilities of p of 0 and p of 1e energy state.

00:12 Generally speaking, p of some energy is constant a times exponential function of the minus constant b times the energy.

00:25 So from the figure 10 .2 we can see that for that for p of 0, the probability is actually 0 .385.

00:39 So a is going to be 0 .385 because this is going to be 0 .385 because this is going to be 0.

01:03 The figure and the same figure we should somehow find for the energy 1e now the value of the cost to the b so p is 1a and then the sequence to cost a times b to the minus b and then 1e energy level so when we divide this width a we obtain we obtain it p 1a 1a 1 e over a is actually exponential function to b times 1a and this left hand side has a numerical value and this value is this is from the figure and also all but we already know which is equal to 0 .665 now notice that if we put both sides as as an argument of a natural logarithm function.

02:24 So we do this to both sides of the equation.

02:26 You obtain that basically minus b over 1e is equal to n of 0 .665.

02:44 Therefore, i can say that b equals, since this is going to be negative value, b equals 1e times 0 .41.

02:59 And this will be how we will look and treat this constant in the future.

03:07 So basically what we will say, but we say probability of e is 0 .385 times expression function negative and then 0 .41.

03:25 That energy over 1e as energy.

03:33 And using this and putting into a calculator the proper values, we obtain that.

03:41 For example, for p0 we know the probability.

03:44 For p1e, it would be 0 .385 times e and here we will have negative 0 .411e over 1e, which cancels out and this is equal to 0 .256.

04:06 And in the same manner, we will just have, for example, for 2e, this will be 0 .385, 385 times exponential function.

04:20 Now we will have minus 0 .41 times 2 e's and 1e and this will be just canceling and this will be 2...

Please give Ace some feedback