Maxwell-Boltzmann distribution help

In summary: The fraction of molecules with speeds in the range 240 to 250 m s-1 would be the area under the linear curve between those two speeds divided by the total area under the curve. This can be calculated by finding the area of a trapezoid. In summary, to find the fraction of F2 molecules at 500 K with speeds in the range 240 to 250 m s-1 using the Maxwell-Boltzmann distribution, you can assume that the distribution curve is linear over this small range and calculate the area of a trapezoid. This area would represent the fraction of the total area under the curve.
  • #1
Lily Wright
9
0

Homework Statement


Use the Maxwell-Boltzmann distribution of speeds to find the fraction of F2 molecules at 500 K which have speeds in the range 240 to 250 m s -1
(HINT: determine the fraction of the total area under the distribution curve represented by the part between these two speeds. Assume that the distribution curve is linear over this small range)

Homework Equations


2}e^{-\frac{mv^{2}}{2KT}}dv.gif


The Attempt at a Solution


So I'm guessing I want to integrate this equation between 240 and 250 m s-1
So it simplifies to ∫v2e-mv2/2KTdv
I'm not very good at integrating so I genuinely have no idea what happens after that
 
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  • #2
Lily Wright said:

Homework Statement


Use the Maxwell-Boltzmann distribution of speeds to find the fraction of F2 molecules at 500 K which have speeds in the range 240 to 250 m s -1
(HINT: determine the fraction of the total area under the distribution curve represented by the part between these two speeds. Assume that the distribution curve is linear over this small range)

Homework Equations


2}e^{-\frac{mv^{2}}{2KT}}dv.gif


The Attempt at a Solution


So I'm guessing I want to integrate this equation between 240 and 250 m s-1
So it simplifies to ∫v2e-mv2/2KTdv
I'm not very good at integrating so I genuinely have no idea what happens after that
You don't have to integrate if you use the given hint: "Assume that the distribution curve is linear over this small range"
 
  • #3
Ah ok so would I just work it out for each velocity?
 
  • #4
Yes.
 

1. What is the Maxwell-Boltzmann distribution?

The Maxwell-Boltzmann distribution is a probability distribution that describes the distribution of speeds of particles in a gas at a given temperature. It is named after James Clerk Maxwell and Ludwig Boltzmann, who developed the theory of kinetic theory of gases.

2. How is the Maxwell-Boltzmann distribution derived?

The Maxwell-Boltzmann distribution is derived from the kinetic theory of gases, which assumes that gas particles are in constant random motion and that their average kinetic energy is directly proportional to the temperature. This leads to a probability distribution that describes the speeds of the particles in the gas.

3. What is the relationship between temperature and the Maxwell-Boltzmann distribution?

The Maxwell-Boltzmann distribution shows that as the temperature increases, the distribution of particle speeds shifts to higher values and becomes wider. This means that as the temperature increases, there is a greater range of particle speeds present in the gas.

4. How does the Maxwell-Boltzmann distribution relate to the ideal gas law?

The ideal gas law, which describes the relationship between pressure, volume, and temperature of a gas, can be derived from the Maxwell-Boltzmann distribution. The average kinetic energy of gas particles is directly related to temperature, and this is incorporated into the ideal gas law through the constant "k", known as the Boltzmann constant.

5. What are some practical applications of the Maxwell-Boltzmann distribution?

The Maxwell-Boltzmann distribution is used in various fields of science, such as thermodynamics, statistical mechanics, and astrophysics. It is also applied in engineering and technology, such as in the design of gas turbines and in the study of gas dynamics in supersonic jets. It is also used in the analysis of particle collisions in particle accelerators.

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