# What fraction of atoms in a sample of argon gas at 400 K have an energy of 12.5 kJ or greater

Calculate the fraction of atoms in a sample of argon gas at 400 K that have an energy of 12.5 kJ or greater.

Im not completely sure how to work this problem out. I know that i have to use the following equation but everytime i use it i come up with different answers. I tried doing a problem from my book and i end up with a total different answer. Can someone help me an explain how to do this correctly, or if im using the wrong equation.

A.  None of the answers is correct.
B.  0.023%
C.  There is not enough information provided about Argon to answer the question.
D.  10.7%
E.  2.3%
F.  99.6%

### Re: fraction of molecules

What equation are you supposed to be using?

### Re: fraction of molecules

Ideally you would need the Maxwell-Boltzmann distribution curve for argon in order to be able to answer this question.  However this involves calculating the integral for the energy distribution.  There may be a more simple equation, but I am not aware of one.

Nonetheless you can make an educated guess.  The curves show the relationship between the number of molecules and their speed or energy.The curves follow a typical Gaussian curve.  In a Gaussian curve every standard deviation gives a fraction of the occurrences that are within a deviation of the average.

For this problem the average kinetic energy of gases is independent of mass and it only depends on the Kelvin temperature thus we have

KE = 3/2 RT          where R is the ideal gas constant in energy units (8.314 J/K*mol)

plugging in 400K yields just below 5 KJ of energy.  This would represent the top point on the Gaussian curve at this temperature.  Since they are asking for 12.5 KJ, two and a half times the average kinetic energy we can make a rough estimate by calculating 2 standard deviations.  At two starndard deviations you have less than 4.56 % of the molecules at three deviations you would have less than .28% of the molecules.

Looking at a typical curve I would say that 2.5 times the energy would put you beyond three deviations.