Macroscopic characterization with distribution functions. Physical Chemistry. Particles in a Box. Two equilibrium distributions

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1 Macroscopic characterization with distribution functions Physical Chemistry Lecture istributions and ransport Processes A macroscopic system contains a LARE number of particles, not all of which h the same set of microscopic properties o gie 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 ariables he equilibrium state is described by a unique distribution function for each property A aussian distribution function, F, of the two co-ordinates and y Particles in a Bo Eample distribution function for particles in a bo, showing two regions Left side has more particles than the right his particular distribution can be considered bimodal Real distribution functions are more comple functions of position Equilibrium particle distribution under no outside constraints Particle density is uniform (i.e. a constant, independent of position) wo equilibrium distributions wo other simple equilibrium distributions hermal equilibrium o eternal constraints emperature is independent of position Mechanical equilibrium o eternal constraints Pressure is independent of position ot all equilibrium distributions are constants, independent of the ariable

2 Boltzmann s distribution: the speed distribution at equilibrium An equilibrium distribution that depends on the ariable Boltzmann distribution of speeds Only kinetic energy Compromise between minimal energy and maimal entropy ormalized epends on speed ( ), mass (m), and temperature ( ) F ( ) / F m m 4 ep k k F d F ( ) d m m ep k k speed distribution functions at three different temperatures Calculating rage molecular properties Aerages are integrals of properties weighted by the distribution function Integral must be carried out oer all possible alues of the independent ariable Eamples Aerage speed in one dimension Aerage speed in three dimensions Aerage temperature at thermal equilibrium For a constant distribution, the rage is the single alue of the temperature,, f ) F f, F ( ) d k m m ep d k, F d b ( ) F ( ) d a b F ( ) d a ( d 8k m Flu radients of distributions 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 flu, J, the net amount of a quantity transferred across a plane per unit area in unit time. Flu may not be uniform across a macroscopic system: J() constant A nonzero flu is required for a system to go from a nonequilibrium distribution to an equilibrium distribution. J ( ) A onequilibrium systems eole towards equilibrium Spatial distributions of properties, e.g. temperature, are time-dependent in nonequilibrium systems An important descriptor is the gradient of a distribution A measure of how the distribution changes from one position to the net In equilibrium systems for which a property (like temperature) is uniform, the gradient at eery point is zero. ot all equilibrium distributions are uniform. df ( ) d

3 Linear response etermination of net flu of molecules across a plane o determine how a system eoles, one must specify how the flu depends on the system s parameters If the flu is not too strongly dependent on the system s parameters, it is said to proide a linear response. In the linear-response regime, the flu is proportional to and opposite in sign to the gradient of the property. he proportionality defines the transport coefficient,, for the process. For electrical systems, the coefficient is called the conductance. For systems inoling approach to thermal equilibrium, the coefficient is called thermal conductiity. For systems inoling moement of particles, the coefficient is called the diffusion coefficient. he size of a transport coefficient gies a measure of how efficiently transport occurs. onlinear response occurs when the flu is strongly coupled to the parameters of the system. Systems may display time dependences that are unusual or unepected under these conditions. J J d d d Count number of molecules in a olume 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 moing towards the plane he number of molecules on the right who will reach the plane in the same time is the same, ecept that the speed is negatie he net flu, J, through the plane is the difference of these two numbers. Aeraging oer the speed distribution gies the total flu ies the relation between flu and gradient if all molecules were trling along the direction. L At At R At At L R J At,, F ( ) d, d k m Fick s first law and the diffusion coefficient In a three-dimensional system, not all molecules that are nominally trling in the direction reach the plane. Molecules h off-ais components. Correct by multiplying the preiously deried quantity by /, 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 ariety of substances and phases. he kinetic-theory representation of the diffusion coefficient is only appropriate to the gas phase. J J,, Fick s first law, as-phase diffusion iffusion coefficient can be calculated from gas-kinetic parameters k, where k / or / from simple kinetic theory k /6 from accurate theory Calculated and eperimental gas diffusion coefficients at 7.5 K and.5 bar oble as eon Argon Krypton Xenon S-K- Calculation (/) iffusion Coefficient (m s - ) Accurate heory Eperiment

4 ime-dependent changes of concentration Solution of Fick s second law Can measure concentration and concentration gradients as a function of time eed an epression for the time-dependence of concentration Fick s second law J J d A A d d d d A d d d d da d d d d ( V ) d d d d ( Ad) d he solution of Fick s second law depends on the boundary conditions. For diffusion from a plane that is populated with molecules, the solution is a aussian that broadens as a function of time. In the limit of infinite time, the result is a constant density of molecules across the sample space. (, t) ep A t 4t Another solution of Fick s equation ypical diffusion coefficients An infinitely long tube Initial condition All molecules to the left of the tube Like sugar water diffusing into pure water in a pipe wo ways to measure diffusion Monitor time dependence of concentration at one position Monitor position dependence of concentration at one time Concentration profiles as a function of time: <t <t <t < Sizes of quantities are important to remember iffusion coefficients of gases are larger than diffusion coefficients of liquids Heaier molecules often h smaller diffusion coefficients, if the two molecules are in the same phase as (C) (m s - ) Liquid (5C) (m s - ) H.5-4 H O.4-9 O.9-5 CH OH C 6 H CO. -5 Hg.7-9 C H C H 5 OH. -9 Xe.5-5 C H 7 OH.6-9 4

5 iffusion coefficients of proteins Proteins tend to h smaller diffusion coefficients than small molecules in liquids Eample diffusion coefficients at C Ferredoin -lobulin Insulin Protein Alcohol dehydrogenase Cytochrome C Hemoglobin A (m s - ) Lactalbumin -Lactalbumin Lipoprotein, H Myoglobin Pepsin Protein Pyruate kinase (m s - ) Summary Calculations with distribution functions gie rage alues of quantities such as speed onequilibrium distributions eole to equilibrium distributions Inhomogenous distribution of molecules eoles to a homogeneous distribution iffusion Fick s laws goern the return to the equilibrium mass distribution iffusion coefficients are epressible in terms of kinetic-theory parameters Allows estimation of the diffusion coefficient of a gas that is nearly ideal Can determine parameters like molecular diameter from diffusion measurement iffusion occurs in other phases iffusion in liquids Self-diffusion Interdiffusion of two different liquids iffusion of molecules in solids (often ery, ery slow) One eample is diffusion of H in metals Sizes of diffusion coefficients Fast diffusion for gases ( -5 m s - ) iffusion of liquids ( -9 m s - ) 5