Maxwell speed distribution

In summary, the conversation discusses the Maxwell speed distribution in relation to an ideal gas system with n particles constrained to a unit box. The phase space for this system is described as ([0,1]^3 x R^3)^n, and the probability distribution for the first particle's velocity is examined on a surface of constant energy E=n/2. The question is whether this distribution is equivalent to the Maxwell speed distribution, which is the distribution for the velocity of a particle found in a system chosen uniformly over all systems with the same energy E. The usual derivation of the Maxwell distribution assumes contact with an external reservoir, but in the limit of infinite particles, an exact energy may also produce the correct result.
  • #1
yall
1
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I want to know if the Maxwell speed distribution is the following.

An ideal gas system of n particles, say constrained to the unit box, has the phase space ([0,1]^3 x R^3)^n. That is, [0,1]^3 for the position of a particle, R^3 for the velocity, and all to the n since there are n particles. Now in this space we can take the surface of constant energy say E=n/2, so that the average energy of a single particle is 1. This surface has finite surface area, so we can put a uniform probability distribution on it, and ask what the distribution of the first particle's velocity is.

Is said distribution the Maxwell speed distribution, in the limit as n->infinity?

In other words, is the Maxwell speed distribution just the distribution for the velocity of a particle found in a system chosen uniformly over all systems of the same energy E?

Thanks in advance!
 
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  • #2
The usual derivation assumes that the particles are in contact with some external reservoir and that the total energy can vary a bit. In the limit of infinite particles, I would expect that an exact energy gives the correct result, too.
 

What is the Maxwell speed distribution?

The Maxwell speed distribution is a probability distribution that describes the speed of particles in a gas at a given temperature. It is named after Scottish physicist James Clerk Maxwell who first derived it in the 19th century.

What is the significance of the Maxwell speed distribution?

The Maxwell speed distribution is significant because it helps us understand the behavior of particles in a gas and their kinetic energy. It also allows us to make predictions about the properties of a gas and how it will behave under different conditions.

What factors affect the shape of the Maxwell speed distribution curve?

The shape of the Maxwell speed distribution curve is affected by the temperature of the gas, the mass of the particles, and the type of gas. Higher temperatures result in a wider and flatter curve, while heavier particles and gases with higher molar masses result in a narrower and taller curve.

How does the Maxwell speed distribution relate to the ideal gas law?

The Maxwell speed distribution is directly related to the ideal gas law, which describes the relationship between the pressure, volume, temperature, and number of moles of a gas. The distribution allows us to calculate the average kinetic energy of gas particles, which is a key component of the ideal gas law.

What is the most probable speed according to the Maxwell speed distribution?

The most probable speed according to the Maxwell speed distribution is the speed at which the highest number of particles in a gas are moving. It is also known as the peak of the distribution curve and is directly related to the temperature of the gas.

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