Macroscopic characterization with distribution functions Physical Chemistry Lecture 1 Distributions and Transport Processes 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 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 F of the two co ordinates x and y 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 1 Calculating average molecular properties 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 F1D v x m F3 D v 4 2 kT F 3D mv 2 m exp x 2 kT 2kT 3 2 v dv 1 mv v 2 exp 2kT 2 F 1D 1 v x dv x 0 Averages are integrals of properties weighted by the distribution function For a constant distribution the average is the single value of the temperature T 3D speed distribution functions at three different temperatures Flux A distribution changes because of transfer of a quantity e g molecules or heat from one place to another Change is quantified by a parameter called the flux J the net amount of a quantity transferred across a plane per unit area in unit time Flux may not be uniform across a macroscopic system J x constant A nonzero flux is required for a system to go from a nonequilibrium distribution to an equilibrium distribution Average speed in one dimension Average speed in three dimensions Average temperature at thermal equilibrium 3D v dv 0 v F vave 1D x 1D v x dv x mv 2 kT v x exp x dv x 2 m 2kT Examples Depends on speed v mass m and temperature T Integral must be carried out over all possible values of the independent variable f v F f 3 D ave 0 vave 3 D vF 3D v dv 0 8k T m b Tave T x F x dx T a b T FT x dx T a Gradients of distributions Nonequilibrium systems evolve towards equilibrium Spatial distributions of properties e g temperature are time dependent in nonequilibrium systems An important descriptor is the gradient of a distribution J x 1 dN A dt x A measure of how the distribution changes from one position to the next In equilibrium systems for which a property like temperature is uniform the gradient at every point is zero Not all equilibrium distributions are uniform G x0 dF dx x0 2 Determination of net flux of molecules across a plane Linear response To determine how a system evolves one must specify how the flux depends on the system s parameters If the flux is not too strongly dependent on the system s parameters it is said to provide a linear response In the linear response regime the flux is proportional to and opposite in sign to the gradient of the property The proportionality defines the transport coefficient for the process JN JT Count number of molecules in a volume on the left whose distance to the plane allows them to reach the plane in a time t Only half of the molecules on the left are moving towards the plane The number of molecules on the right who will reach the plane in the same time is the same except that the speed is negative The net flux J through the plane is the difference of these two numbers Averaging over the speed distribution gives the total flux Gives the relation between flux and gradient if all molecules were traveling along the x direction dN dx dT T dx N For electrical systems the coefficient is called the conductance For systems involving approach to thermal equilibrium the coefficient is called thermal conductivity For systems involving movement of particles the coefficient is called the diffusion coefficient The size of a transport coefficient gives a measure of how efficiently transport occurs N x Av t N x Av t x x N 1 G x Av t N 1 G x Av t 0 x 0 N L N R At x 2G x v x 2 x1D ave vx F1D v x dv xG 2 0 vave 3 D 2 G kT G 2 m vave 3 D dN 2 dx Systems may display time dependences that are unusual or unexpected under these conditions Gas phase diffusion Diffusion coefficient can be calculated from gas kinetic parameters In a three dimensional system not all molecules that are nominally traveling in the x direction reach the plane Fick s first law and the diffusion coefficient N R J Nonlinear response occurs when the flux is strongly coupled to the parameters of the system N L D k vave 3 D where k 1 3 or 1 2 from simple kinetic theory k 3 16 from accurate theory Molecules have off axis components Correct by multiplying the previously derived quantity by 2 3 the fraction that will reach the plane By comparison to Fick s law one has an equation for the diffusion coefficient of an ideal gas in terms of kinetic theory parameters Fick s first law is generally applicable to a wide variety of substances and phases The kinetic theory representation of the diffusion coefficient is only appropriate to the gas phase J 2 v3 D ave G 3 2 J DG D v3 D ave 3 v3 D ave Fick s first law 3 Calculated and experimental gas diffusion coefficients at 273 15 K and 1 01325 bar G Diffusion Coefficient m2s 1 Noble Gas S K T Calculation 1 3 Accurate Theory Experiment Neon 1 92 10 5 3 45 10 5 4 52 10 5 Argon 0 91 10 5 1 63 10 5 1 57 10 5 Krypton 0 50 10 5 0 90 10 5 0 93 10 5 Xenon 0 35 10 5 0 63 10 5 0 58 10 5 3 Time dependent changes of concentration Can measure concentration and concentration gradients as a function of time Need an expression for the time dependence of concentration Fick s second law dN dt J x J x dx A dN dN A D dx x dx x dx dN d N Gdx A D dx dx x x dx dN d N dG dG D Adx dx A D dx dx x dx x dx x dG V D dx dN dt D …