Velocity Distribution and Average Energy of Gases

We can use the Boltzmann distribution to give the velocity distribution for a gas at equilibrium. If we are only interested in the y-direction (in other words, we want to know the probability of finding different values of vy, independent of the values of vx or vz) the Boltzmann distribution gives  

This is called the one-dimensional velocity distribution, since only the y direction is included. It can also be converted to a probability distribution P vy)dvy, which should be interpreted as the chance of finding any one molecule with a velocity between vy and 

The term in brackets in Equation 4.31 is the normalization constant, chosen so that 

Equation 4.31 is a Gaussian in vy with standard deviation a = fks T/m. The distribution is peaked at vy = 0 (remember vy can be positive or negative). The average value = 0, since as many molecules are going left as right. The mean-squared velocity in the y direction v2 (square before averaging) is evaluated in the same way as 

M2 was for the coin toss distribution in Section 4.2 this quantity is just equal to o2 for a Gaussian. We thus have 

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