PHYS CHEM BIOC II
PHYS CHEM BIOC II CHEM 453
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This 8 page Class Notes was uploaded by Carmela Kilback on Wednesday September 9, 2015. The Class Notes belongs to CHEM 453 at University of Washington taught by Gabriele Varani in Fall. Since its upload, it has received 22 views. For similar materials see /class/192549/chem-453-university-of-washington in Chemistry at University of Washington.
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Date Created: 09/09/15
University of Washington Department of Chemistry Chemistry 453 Spring Quarter 2008 Lecture 3 040408 Text Reading Ch 6 p 252265 Summary of Lecture 2 0 Statistical mechanical principles allow us to calculate the properties of a complex macroscopic system from its microscopic characteristics energy levels A fundamental tool of statistical mechanics is the Boltzmann distribution that describes how energy levels are occupied for a system at thermodynamic equilibrium From the partition we can derive at least in principle thermodynamic properties such as pressure energy or entropy By being able to relate macroscopic properties to the microscopic molecular properties of a system we can use macroscopic measurements to obtain microscopic parameters such as energy differences between states In the next two lectures we will reexamine ideal gases from the point of view of their microscopic nature The ideal gas is the simplest macroscopic system and therefore an ideal introduction to more complex r39 quot quot of 39 39 By doing so we will be able to introduce molecular motion and transport and therefore apply the statistical mechanical tools we have been describing to characterize transport properties of biological molecules diffusion separation etc We will first see how molecular motion is related to temperature and mass for an ideal gas the simplest macroscopic system We will then introduce collisions between molecules as occurs in denser states of matter such as liquids In this more general case molecular motion depends not just on mass and temperature but also on size and shape We will define concept such as sedimentation the motion of a molecule in a gravitational or centrifugal force viscosity the resistance to ow when macromolecules are added to a uid its viscosity depends on the size and shape of the molecule and electrophoresis how charged molecules move in an electric eld a key technique of course to measure the size and shape of macromolecules like proteins and nucleic acids By measuring transport properties we learn about the size and shape of biological molecules in solution are they big or small compact or extended rigid or exible We will use statistical mechanical methods to relate macroscopic transport and microscopic size and shape properties ofthe molecules we study By doing so we learn how to characterize separate and purify proteins or nucleic acids for example we are able to sequence DNA by separating it on a gel under an electric eld or to measure the size of proteins or nucleic acids and how they interact with each other E Ideal Gases Statistical Mechanical Description A macroscopic description of an ideal gas was given in Chem 452 The macroscopic description of a substance consists of an equation of state which provides a relationship between the state variables P V and T An equation of state has the general mathematical form f PV T 0 where f is a mathematical function The ideal gas equation PVnRT is an example of an equation of state n is the number of moles of gas Does statistical mechanics allow us to reinterpret the ideal gas equation in terms of the molecular properties of the gas Of course it does remember that for an isolated system composed of N non interacting particle ENsz alnq 6T What is an ideal gas It is a system composed of N noninteracting particles of mass m con ned within a certain volume Vabc where a b and c are the dimensions of the container In the rst lecture I have provided the expression for the energy levels for such a system h2 n2 712 n2 Ennynz8 2b m a c We have also calculated the partition function for the ldimensional case under the assumption that the energy levels are spaced very close together corresponding to a system of large mass or a macroscopic classical system so the summation can be replaced by an integral no 2 z 2 q ZeiE kgT z J expi n h d 7 Z ma kBT o SmaZkBT h In fact because the expression for the energy level is a sum we can simply do the same calculation independently for each dimension 3 3 q qquqz azbzcz V We can now use the above expressions E Nsz a 1 q 6T skN1ni kN N T To obtain for one mole of gas E 2 RE V F Ideal gas the microscopic interpretation of temperature Here we will describe an ideal gas from in its mechanical properties because it provides insight into the microscopic meaning of temperature and on transport properties and because it will introduce the subsequent analysis of transport properties of biomolecules We will do so by relating the pressure of a gas with the collisions ofthe gas molecules against the walls of the container From the microscopic pointof view an ideal gas is a collection ofa very large number of particles molecules or atoms At a given instance in time each particle mass m in the gas has a position described by xyz and a velocity uvw where u v and W are the x y and 2 components of the velocity respectively The history of a particle s positionvelocity is called a trajectory Each point on a trajectory of a particles is described by 6 parameters xyzuvw The speed of a particle c is related to the components ofits x y and z velocity components ie u v and w respectively by the Pythagorean Theorem 6 2 u2 v2 w2 xyz39uvw A trajectory is obtained by applying the laws of classical mechanics also called Newton s laws of motion In classical mechanics there are three laws of motion I Newton s First Law ofMotz39on Law ofInertz39a Every body persists in its state ofrest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed on it I Newton s SecondLaw ofMotz39on The sum of all forces F acting on a body with mass m is related to its vector acceleration a by the equation Fma I Newton s ThirdLaw ofMotz39on To every action there is always opposed an equal reaction or the mutual actions of two bodies upon each other are always equal and directed to contrary parts Pressure is the force F exerted per unit area A of the container wall by gas molecules as they collide with the walls P In principle one should be able to obtain an expression for the pressure P by applying the laws of motion described above This mechanical view of pressure is called the Kinetic Theory of Gases The kinetic ie mechanical theory of gases is based on the following assumptions 0 A gas is composed of molecules in random motion obeying Newton s laws of motion 0 The total number of molecules in the gas N is large 0 The volume of each molecule is a negligibly small fraction of the volume V occupied by the gas 0 No appreciable forces act on the molecules except during collisions between molecules or between molecules and the container walls 0 Collisions between molecules and with the container walls conserve momentum and kinetic energy elastic collisions Consider a particle with mass m moving with velocity u ie in the x direction It collides with a wall of unit area A mm In mllixinn pmu um I mllixinn pm Momentum p of a particle is mass times velocity Just prior to a wall collision the momentum of a gas particle is pmu Just following an elastic collision the momentum is pmu the particle has the opposite direction ie it is moving in the 7x direction The change in momentum Ap is Ap mu mu 2mu Suppose in an instant of time At a particle with velocity u covers a distance d collides with the wall and subsequently covers a distance d then uAFZd or AFZdu 2 A 2 The change 1n momentum dur1ng the t1me perlod At 1s g1ven by p mu mu At 2d u d From Newton s Second Law see above this is the force exerted by the gas particle as it 2 collides with the wall f ma E m At d To get the total force F resulting from the collisions of N particles against an area A of the container wall add up all the particle forces f Ff1f2f3fNZN ZN1mulz uf 1 I This equation can be rearranged to read Where u 2 is the average of the squared velocity Now just divide both sides by A to get the pressure F Nm 2 Nm 2 u u A A d P This mechanical calculation states that the product of pressure P and volume V is proportional to the average squared velocity Temperature does not explicitly appear in this equation and it cannot temperature is not a mechanical concept it is a thermodynamic macroscopic concept Let us reconcile the macroscopic view of the pressure of an ideal gas PVriRT and the microscopic mechanical view of the gas PV Nmu2 by bringing back the statistical mechanical formulation we have give earlier Since PVriRT statistical mechanical PV Nmu2 mechanical description Thus we can interpret the temperature a thermodynamic concept of course in terms of the average velocity or kinetic energy of the molecules that compose the gas PV3NA it 2 3 2 Eka u 2 ltKgt 2 2 The pressure is related to the average kinetic energy per molecule while the temperature is a measure of the average kinetic energy of each molecule in the gas The proportionality constant that relates T and average molecular kinetic energy is Boltzmann s constant Every atom or molecule in every molecule protein nucleic acid or oxygen in any environment has an average kinetic energy proportional to its T Of course average speed depends on molecular mass in addition to T In other words the average kinetic energy of molecules or atoms depends only on the temperature of the system this is true for the translational energy of solids and liquids as well Of course not all gas molecules in a container have the same speed molecular speeds are distributed like exam grades The average speed is an important quantity for example the rate at which molecules collide an important determinants of chemical reactivity depends on it G MaxwellBoltzmann Speed Distribution Function We should now be used to the idea of averaging microscopic properties using Boltmann distribution to calculate macroscopic properties of a system For the average or mean squared speed 2 2 In the weighted average equation we are averaging over the groups of molecules with equal speeds where 71 is the number of molecules with speed 0 and nN is the fraction of molecules with velocity 0 distribution We also know that energies associated with the motions of microscopic particles are quantized The spacing between energy levels is very small for large amplitude motions such as molecular translation therefore quantization is not an important effect in molecular speed distributions Because the energy level spacing is small for the translational motion the sum M w c2 Z cfcan be replaced by C2 Jfcczdc 11 0 in general the average of any property can be calculated from the distribution as follows lt x gt Iiygrxdx The function fc represents the probability of the particle having a certain speed c and is delled MaxwellBoltzmann speed distribution function What is the form of the speed distribution function In lecture 2 we considered the general form for the Boltzmann distribution function which in quantized systems gives the population of particles in the El energy level is proportional to e E W Boltzmann distribution Let us consider that l the only energy of the system under cons1derat10n 1s kmetlc energy E Emc2 then the probability of finding a molecule between c and cdc is 7 2 gig mo 2dec dPc fcdc w 2 J39czermc made 0 Since no 1 n 32 Ixze39 dx j 47239 a 32 fc fCdC 47z3927T Czeimclmdc This is the MaxwellBoltzmann speed distribution function that can be used to calculate mechanical averages eg c 2 J fcdc 0 l o In the argument of the exponential the energy is the molecular kinetic energy Emcz 32 no 0 The constant term 1s a normalization constant to ensure IdPc l 0 o The 47239c2 term skews the distribution function toward higher speeds Clearly the probability depends on the mass of the molecules and the absolute temperature how As the temperature is increased the distribution spreads out and the peak of the distribution is shifted to higher speed that is why T has such an effect for example on reaction rates for a given mass molecules move faster encounter each other more frequently The graph shows the results of experimental measurements of the distribution of molecular speeds for nitrogen gas N2 at T0 C 1000 C and 20000C each curve also corresponds very closely to the MaxwellBoltzman distribution The y axis is the fraction of molecules with a given speed x axis 3D Velocity Distribution Functions 00018 00016 00014 00012 0001 00008 00006 00004 00002 0000000000000 LOOLOOLOOLOOLOOLOOLO Fm rol mommmowm We can use the MaxwellBoltmann distribution to calculate the mean speed ltcgt and the mean square speed lt02gt for example lt 0 gt Cfcdc 0 72m ltc2 gtJczfcdc3k T 0 m End of Lecture 3
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