Lecture1 - Macroscopic characterization with distribution...

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1 Physical Chemistry Lecture 1 Distributions and Transport Processes Macroscopic characterization with distribution functions A macroscopic system contains a LARGE number of particles, not all of which have the same set of microscopic properties To give a macroscopic system’s state requires a distribution function , F , that describes the “amounts” of properties, either macroscopic or microscopic, as a function of independent variables The equilibrium state is described by a unique distribution function for each property A Gaussian distribution function, , of the two co-ordinates x and y Particles in a Box Example distribution function for particles in a box, showing two regions Left side has more particles than the right This particular distribution can be considered bimodal Real distribution functions are more complex functions of position Equilibrium particle distribution under no outside constraints Particle density is uniform (i.e. a constant, independent of position) Two equilibrium distributions Two other simple equilibrium distributions Thermal equilibrium No external constraints Temperature is independent of position Mechanical equilibrium No external constraints Pressure is independent of position Not all equilibrium distributions are constants, independent of the variable
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2 Boltzmann’s distribution: the speed distribution at equilibrium An equilibrium distribution that depends on the variable Boltzmann distribution of speeds Only kinetic energy Compromise between minimal energy and maximal entropy Normalized Depends on speed ( v ), mass ( m ), and temperature ( T ) kT mv v kT m v F D 2 exp 2 4 ) ( 2 2 2 / 3 3 1 ) ( 1 ) ( 1 0 3 x x D D dv v F dv v F kT mv kT m v F x x D 2 exp 2 ) ( 2 1 3D speed distribution functions at three different temperatures Calculating average molecular properties Averages are integrals of properties weighted by the distribution function Integral must be carried out over all possible values of the independent variable Examples Average speed in one dimension Average speed in three dimensions Average temperature at thermal equilibrium For a constant distribution, the average is the single value of the temperature, T 0 2 exp 2 ) ( 2 1 1 ,  x x x x x D x D ave dv kT mv v m kT dv v F v v 0 3 , 3 ) ( ) ( dv v F v f f
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Lecture1 - Macroscopic characterization with distribution...

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