Maxwell's speed distribution law

In summary, the conversation is about the derivation of Maxwell's speed distribution for gases and the use of independence in the combined probability. The speaker is uncomfortable with the assumption of exponential dependence and is unsure about the use of P as a partition function. The argument is based on the independence of V_x, V_y, and V_z.
  • #1
Kolahal Bhattacharya
135
1
In a non-traditional type of derivation of Maxwell's speed distribution for gases,I happen to face the following problem:
They say since P(v_x),P(v_y),P(v_z) are independent,so the combined probability wil be P=P(v_x)P(v_y)P(v_z).
This much is OK.Then they say the only function having the property f(a+b+c)=f(a)f(b)f(c)
is an exponential function.So, consider the P(v_x) as to have exponential dependence P(v_x)=K exp[-L*(v_x)^2].This makes me uncomfortable.Did we have P=P(v_x+P_y+P_z)?I am a bit new to statistical ideas,so really cannot be sure when we said the joint probability is P,it means P=P(v_x+P_y+P_z).
Please help.
 
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  • #2
Do you what a partition function is?
 
  • #3
OK, I do not know.Is P acting as a partition function here?How do we know that?And after all,what does it do?
 
  • #4
I think the argument holds because

[tex]V^2 = V_x^2 + V_y^2 + V_z^2[/tex]
therefore
[tex]P(V^2) = P( V_x^2 + V_y^2 + V_z^2 )[/tex]
and the next step follows from the independence of V_x, V_y and V_z.
 
  • #5
Thank you all very much
 

1. What is Maxwell's speed distribution law?

Maxwell's speed distribution law, also known as the Maxwell-Boltzmann distribution, is a mathematical formula that describes the distribution of speeds of particles in a gas at a given temperature. It is a fundamental concept in thermodynamics and statistical mechanics.

2. What does Maxwell's speed distribution law tell us?

Maxwell's speed distribution law tells us the probability of finding particles in a gas with a particular speed. It also provides insights into the average and most probable speed of particles in a gas at a given temperature.

3. How is Maxwell's speed distribution law derived?

Maxwell's speed distribution law is derived from the kinetic theory of gases, which assumes that gas particles are in constant random motion and interact only through collisions. By applying statistical mechanics and probability theory, the formula for the speed distribution is derived.

4. What factors affect the shape of Maxwell's speed distribution curve?

The shape of Maxwell's speed distribution curve is affected by the temperature of the gas, the mass of the gas particles, and the type of gas. Higher temperatures result in a broader and flatter curve, while heavier particles and gases with stronger intermolecular forces lead to a narrower and taller curve.

5. What are the practical applications of Maxwell's speed distribution law?

Maxwell's speed distribution law has various practical applications, such as in understanding gas diffusion, calculating the rate of evaporation, and predicting the behavior of gases in chemical reactions. It is also used in the design of gas compressors, turbines, and other industrial processes that involve gases.

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