Maxwell-Boltzmann Energy distribution

In summary: This avoids confusion about whether the exponent applies only to the E, or to the entire denominator.In summary, the conversation discusses finding the average energy of a system with n energy states using the equations P(E) = e-BE/Z and <E>=∑(nE* (e-BE)n) /Z. The individual steps and calculations for finding the solution are also mentioned.
  • #1
WrongMan
149
15

Homework Statement


find the average energy of a system with n energy states (0, 1E, 2E, 3E...nE)

Homework Equations


P(E) = e-BE/Z - where B=1/KbT and Z= ∑e(-BE)n
<E>=∑(nE* (e-BE)n) /Z

The Attempt at a Solution


i feel like I've gone down the correct path - that is finding result of the sums.
Z - if S=R0+R1+...+Rn i do S-RS and get S=R0-Rn+1/1-R ... so Z= e0-e-BE(n+1)/1-eBE
now it gets tricky, i tried to evaluate the sum ∑(nE* (e-BE)n) in a similar way,
so S2=0R0+1R1+2R2...
and RS2=0R+1R2+2R3...
so S2-RS2=S-1 (remember S=Z; "-1" cause R0 is missng)
so S2=(1-e-BE(n+1)/(1-e-BE)2-1/(1-e-BE).

substituting into average fomula and simplifying my answer is: (1/1-e-BE) - (1/1-eBE(n+1))
is this correct?
 
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  • #2
WrongMan said:
now it gets tricky, i tried to evaluate the sum ∑(nE* (e-BE)n) in a similar way,
so S2=0R0+1R1+2R2...
and RS2=0R+1R2+2R3...
so S2-RS2=S-1 (remember S=Z; "-1" cause R0 is missng)
You're missing a term on the right hand side of the last equation. (Don't forget to consider the last terms in S2 and RS2).
 
  • #3
TSny said:
You're missing a term on the right hand side of the last equation. (Don't forget to consider the last terms in S2 and RS2).
oh right RS2 ends with nRn+1 so S2 ends with "-nRn+1" (the missing term)
so my new answer is: (1/1-e-BE) + (-1-ne-BE(n+1)/1-e-BE(n+1))
 
  • #4
OK, that looks right for S2 / Z. But that's not quite the answer for <E>.

When you have expressions like 1/1-e-BE , you should include parentheses around the entire denominator: 1/(1-e-BE).
 

FAQ: Maxwell-Boltzmann Energy distribution

What is the Maxwell-Boltzmann Energy distribution?

The Maxwell-Boltzmann Energy distribution is a probability distribution that describes the distribution of velocities of particles in a gas at a given temperature. It is named after James Clerk Maxwell and Ludwig Boltzmann, who independently developed the theory in the late 19th century.

What factors affect the shape of the Maxwell-Boltzmann Energy distribution?

The shape of the Maxwell-Boltzmann Energy distribution is affected by the temperature of the gas, the mass of the particles, and the number of particles present. As temperature increases, the distribution becomes wider and more spread out, with a higher peak at higher velocities. As mass increases, the distribution becomes narrower with a lower peak at lower velocities. As the number of particles increases, the distribution becomes taller and narrower.

How is the Maxwell-Boltzmann Energy distribution related to the kinetic theory of gases?

The Maxwell-Boltzmann Energy distribution is a direct consequence of the kinetic theory of gases, which states that the average kinetic energy of gas particles is directly proportional to the temperature of the gas. The distribution describes the range of kinetic energies that gas particles can have at a given temperature.

What is the most probable speed in the Maxwell-Boltzmann Energy distribution?

The most probable speed in the Maxwell-Boltzmann Energy distribution is the speed at which the peak of the curve occurs. This is also known as the mode of the distribution. It is directly related to the temperature of the gas and can be calculated using the Maxwell-Boltzmann distribution equation.

How does the Maxwell-Boltzmann Energy distribution change with increasing temperature?

As temperature increases, the Maxwell-Boltzmann Energy distribution shifts to the right, with a wider range of velocities and a higher peak at higher velocities. This is because at higher temperatures, the gas particles have more kinetic energy and therefore a wider range of velocities. The average velocity also increases with temperature, as does the most probable speed.

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