How Do You Calculate the Probability Density Function in a Classical Ideal Gas?

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In a classical ideal gas, the probability density function for particle velocities is given by p(Vx,Vy,Vz)=C*exp-(Vx^2+Vy^2+Vz^2), indicating that the velocities are statistically independent. To find the overall probability P(E) that gas particles have kinetic energy less than E, one must relate the velocity terms to energy using the equation E=1/2*m*|v|^2. The discussion emphasizes the need to rewrite the velocity-based probability expression in terms of energy. Participants are seeking guidance on how to derive the probability density p(E) from the given velocity distribution. Understanding these relationships is crucial for solving the problem effectively.
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Homework Statement



Classical Ideal Gas in 3 Dimensions
In a classical ideal gas, we treat molecules as non-interacting point particles moving in the x,y, and z directions. These particles' velocities (in each respective direction) are statistically independent:

p(Vx,Vy,Vz)=C*exp-(Vx^2+Vy^2+Vz^2)

The energy is E=1/2*m*|v|^2

(a) Find P(E), the overall probability that these gas particles will have a KE \<E.
(b) By noting that dP=p(E)dE, derive the probability density p(E).

Homework Equations


No clue

The Attempt at a Solution


Don't even know how to start...any tips would be much appreciated
 
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Can you re-write the expression for p(Vx,Vy,Vz) so that the probability depends on E rather than Vx, Vy, and Vz?
 

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