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

[chem_nerd09]

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

 

[Ygggdrasil]

Can you re-write the expression for p(Vx,Vy,Vz) so that the probability depends on E rather than Vx, Vy, and Vz?

 

[Blueshift5]

I would recommend starting by reviewing the fundamental concepts of classical ideal gas and the equations provided in the homework statement. This will help you understand the variables and their relationships in the problem. Then, try to break down the problem into smaller parts and identify any relevant equations that can be used to solve each part. For example, for part (a), you can use the given probability distribution function to find P(E). For part (b), you can use the definition of probability density to derive the equation. If you are still struggling, I would suggest seeking help from your instructor or classmates. It's important to understand the concepts and equations in order to solve the problem correctly.

 

[Blueshift5]

it is important to have a solid understanding of statistical thermodynamics and its applications. In this problem, we are dealing with a classical ideal gas in 3 dimensions. The first step in solving this problem would be to understand the given information and equations.

The given equation, p(Vx,Vy,Vz)=C*exp-(Vx^2+Vy^2+Vz^2), represents the probability of finding a molecule with a specific velocity in each direction. This equation is based on the assumption that the molecules are non-interacting point particles. The energy of each particle is given by E=1/2*m*|v|^2, where m is the mass of the particle and |v| represents the magnitude of its velocity.

(a) To find P(E), the overall probability that the gas particles will have a kinetic energy less than E, we can integrate the given probability density function over all possible velocities. This can be done by considering the probability in each direction (Vx, Vy, Vz) and using the formula for the probability of independent events. This will give us the overall probability as P(E)=P(Vx)*P(Vy)*P(Vz).

(b) To derive the probability density function p(E), we can use the fact that dP=p(E)dE. This means that the probability of finding a particle with an energy between E and E+dE is given by p(E)dE. We can then use the given equation for energy (E=1/2*m*|v|^2) to express the probability in terms of the velocity. This will give us p(E)=C*exp(-E/m)*dE, where dE is the infinitesimal change in energy.

In summary, to solve this problem, we need to understand the given information and equations, and then use the concepts of probability and statistical thermodynamics to find the overall probability and probability density function. It is also important to remember to always think critically and logically when solving scientific problems.