Dust particles are trapped on a surface. There are two energy levels particles can be in: absorbed on a surface with zero energy, and wiggling with energy E=3/2kT. However, since the surface area is limited, not all particles can be in any state. There are 1000 particles and 1000 "holes" for them, 20 % of which are the wiggling ones. What is the ratio of expected energy of the dust particles and kT?
Details and assumptions
The probability that a particle has energy E is proportional to exp(−E/(kT)).
The particles are identical.
To find the ratio of the expected energy of the dust particles to kT, we need to calculate the average energy contribution from both energy levels, taking into account the probabilities.
Let's break down the problem step by step:
Step 1: Determine the number of particles in each energy state.
We know that there are 1000 particles and 1000 "holes," and 20% of those holes correspond to particles in the wiggling state. Therefore, the number of particles in the wiggling state is 20% of 1000, which is 0.20 x 1000 = 200 particles. The remaining particles (800) are absorbed on the surface with zero energy.
Step 2: Calculate the probabilities of each energy state.
According to the given information, the probability that a particle has energy E is proportional to exp(-E/(kT)). Since we have two energy levels, let's calculate the probabilities for both.
For the wiggling state:
The energy of the wiggling state is given as E = (3/2)kT. Therefore, the probability of a particle being in the wiggling state is proportional to exp(-(3/2)kT/(kT)) = exp(-3/2).
For the zero energy state:
Since the energy of the zero energy state is 0, the probability of a particle being in this state is proportional to exp(-0/(kT)) = exp(0) = 1.
Step 3: Calculate the average energy contribution from each state.
The average energy contribution from each state can be calculated by multiplying the energy of the state by its respective probability and summing them up.
For the wiggling state:
Average energy contribution from wiggling particles = (3/2)kT x exp(-3/2).
For the zero energy state:
Average energy contribution from absorbed particles = 0 x 1 (since probability = 1 for the zero energy state).
Step 4: Calculate the total average energy contribution.
The total average energy contribution is obtained by summing up the contributions from both states.
Total average energy contribution = Average energy contribution from wiggling particles + Average energy contribution from absorbed particles.
Step 5: Calculate the ratio of the total average energy contribution to kT.
The ratio can be obtained by dividing the total average energy contribution by kT.
Ratio = Total average energy contribution / kT.
Now, you can use the information given and the steps outlined above to calculate the ratio of the expected energy of the dust particles to kT.