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1、Distribution of gas molecules in free pathThe mean frequency of intermolecular collisions and mean free path introduced earlier can represent the main characteristics of intermolecular collisions,but do not reflect the random nature of intermolecular collisions.In fact,if a molecule has just been co
2、llided at x=0,the time of subsequent second collision is completely random.So the distance it travels between collisions is also random.In order to describe this random property,we must find the probability that it will be collided in the range of x x+dx,that is,the probability that the free path of
3、 the molecule is in the range of x x+dx.One way to derive the distribution of molecular free path is to prepare a molecular beam composed of N0 molecules.The molecules in the molecular beam happen to be at the same place(x=0)just once and then move in the x direction.At the time of t=0(i.e.the momen
4、t when the gas molecule escapes the hole),the N0 molecules that have collided exactly once at x=0(i.e.,in the hole)and are moving in the same direction(x-axis direction).The molecular beam is constantly subject to the collision of background gas molecules during the course of the movement,causing th
5、e number of molecules to decrease gradually.As long as you know that the molecular beam in the xx+dx range of reduced molecular number dN,you can get free path distribution.If at time t,there are N molecules left at x,after dt time,the molecular beam moves to x+dx and is collided again|dN|molecules
6、off(that is,the number of molecules with free path xx+dx is-dN),the number of molecules dN reduced in the distance xx+dx is proportional to the number N of molecules at x,and also proportional to the size of dx.ddNKN xddNK xN After integral,we get 00lndxNK xN 0dexp()dNKKxxN0dexp()dNKNKxx The upper f
7、ormula represents the collided N0 molecules emitted from x=0,the number of molecules that travel to x,and N declines exponentially.Differentiate the right side of the above formula,we can get (-dN)indicates the average number of molecules in N0 molecules with a free path of xx+dx,and()is the probabi
8、lity of the free path of the molecule in the range of xx+dx.Probability distribution of molecular free path0exp()NNKx0d/N NMean free path01exp()dx KKxxK0exp()NxN0d1exp()dNxxN0d1()dexp()dNxP xxxN 0dexp()dNKNKxx)exp(0KxNNThe probability of the residual movement of the molecular beam to x is also the p
9、robability of a free path from x to infinity rangeProbability of a molecular collision P(x)between xx+dxProbability distribution of molecular free path The area of the slash area indicates the residual probability.The area of the horizontal line indicates the probability that the free range is in the xx+dx range.0exp()NxN0d1exp()dNxxNThank you
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