Question

You have two samples of the same gas in the same size container, with the same pressure. The gas in the first container has a Kelvin temperature four times that of the gas in the other container. . . 1) The ratio of the number of moles of gas in the first container compared to that in the second is?. 2) The ratio of the average velocity of particles in the first container compared to that in the second is?. . Please explain.

Answer 1

We assume that the gases in the container both behave as ideal gas. The number of moles (n) is calculated through the ideal gas law,
                                       n = PV / RT
From the two gases,
                                n1 / n2 = (PV / R(4T2)) / (PV / RT2)
The value of n1:n2 is therefore equal to 1:4. 
The equation for the average velocity of the gas particles is sqrt (3RT/M). Substituting,
                                 u1 / u2 = sqrt (3R(4T2)/M) / sqrt (3RT2/M)
Simplifying gives an answer of 2:1. 

Answer 2

The formula for the ideal gas law is PV = nRT. Manipulating the variables we get n = PV/RT.

so if the first container has a Kelvin temperature four times than the other container, n = PV/R(4T), then the ratio of the number of moles of gas in the first container compared to that in the second is 1/4.

following the kinetic energy of a molecule whcih is represented by squared speed = sqrt(3RT/M), the ratio of the average velocity of particles in the first container compared to that in the second is
squared speed = sqrt(3R(4T)/M)
squared speed = sqrt(12RT/M)
squared speed = 2sqrt(3RT/M)
2 times that of the other container