Probabilities of finding a molecule at a given speed

[imr1212]

Problem: What is the ratio of the probability of finding a molecule moving with the average speed to the probability of finding a molecule moving with three times the average speed?

So I know I need to use the Maxwell Speed Distribution for this:
D(v) dv = (m/(2(pi)(k)(T)))^(3/2) * 4(pi)v²e^-(mv²/2kT) dv

To find the average speed of a molecule at a given temperature, the equation ((8kT)/(pi)m)^(1/2) is used.

When I set up two sides of the equation, one for the average speed and one for three times the average speed, I've been simplifying to e^(-4/pi) = 9e^(-36/pi) which has been yielding very strange ratios so I'm sure I made a mistake in my simplifying.

I also tried graphing X²e^(-X²) and taking the maximum value of the graph, which would be the most probable velocity (1) and comparing its probability with that of three times that velocity(3) and got 334 as my answer, which is a little off the expected value of 295. Hope someone can point out what I'm doing wrong. Thanks

 

[NateD]

The ratio of the probability of finding a molecule moving with the average speed to the probability of finding a molecule moving with three times the average speed is 2:1. This can be derived by using the Maxwell Speed Distribution which gives the probability of finding a molecule with a certain speed. The equation for the Maxwell Speed Distribution is D(v) dv = (m/(2(pi)(k)(T)))^(3/2) * 4(pi)v2e^-(mv2/2kT) dv. From this, we can get the average speed of a molecule at a given temperature from the equation ((8kT)/(pi)m)^(1/2). For the average speed, the probability is given by D(v) = (m/(2(pi)(k)(T)))^(3/2) * 4(pi)v2e^-(mv2/2kT). For three times the average speed, the probability is given by 3^2 * D(v) = 3^2 * (m/(2(pi)(k)(T)))^(3/2) * 4(pi)v2e^-(mv2/2kT). By dividing the two probabilities, we get the ratio of the probability of finding a molecule moving with the average speed to the probability of finding a molecule moving with three times the average speed as 2:1.