Classical Statistical Mechanics in the Canonical Ensemble Canonical

Classical Statistical Mechanics in the Canonical Ensemble

Canonical Ensemble in Classical Statistical Mechanics • As we’ve seen, classical phase space for a system with f degrees of freedom is f generalized coordinates & f generalized momenta (qi, pi). • The classical mechanics problem is done in the Hamiltonian formulation with a Hamiltonian energy function H(q, p). • There may also be a few constants of motion such as energy, particle number, volume, . . .

The Canonical Distribution in Classical Statistical Mechanics The Partition Function has the form: Z ≡ ∫∫∫d 3 r 1 d 3 r 2…d 3 r. N d 3 p 1 d 3 p 2…d 3 p. N e(-E/k. T) A 6 N Dimensional Integral! • This assumes that we have already solved the classical mechanics problem for each particle in the system so that we know the total energy E for the N particles as a function of all positions ri & momenta pi. E E(r 1, r 2, r 3, …r. N, p 1, p 2, p 3, …p. N)

CLASSICAL Statistical Mechanics: • Let A ≡ any measurable, macroscopic quantity. The thermodynamic average of A ≡ <A>. This is what is measured. Use probability theory to calculate <A> : P(E) ≡ e[-E/(k T)]/Z B <A>≡ ∫∫∫(A)d 3 r 1 d 3 r 2…d 3 r. N d 3 p 1 d 3 p 2…d 3 p. NP(E) Another 6 N Dimensional Integral!

The Classical Ideal Gas • So, in Classical Statistical Mechanics, the Canonical Probability Distribution is: P(E) = [e-E/(k. T)]/Z Z ≡ ∫∫∫d 3 r 1 d 3 r 2…d 3 r. N d 3 p 1 d 3 p 2…d 3 p. N e(-E/k. T) • This is the tool we will use in what follows. • As we. ve seen, from the partition function Z all thermodynamic properties can be calculated: pressure, energy, entropy….

• Consider an Ideal Gas from the point of view of microscopic physics. • The ideal gas is the simplest macroscopic system. • Therefore it is useful to introduce the use of the Canonical Ensemble in Classical Statistical Mechanics. • The ideal gas Equation of State is PV= n. RT n is the number of moles of gas). • Lets reinterpret the ideal gas equation of state in terms of the molecular properties of the gas.

• We will do Classical Statistical Mechanics, but very briefly, lets consider the simple Quantum Mechanics of an ideal gas & then take the classical limit. • From the microscopic perspective, an ideal gas is a system of N non interacting particles of mass m confined in a volume V = abc. (a, b, c are the box’s sides) • Since there is no interaction, each molecule can be considered a “Particle in a Box” as in elementary quantum mechanics. • The energy levels for such a system have the form: nx, ny, nx = integers

• The energy levels for each molecule in the Ideal Gas have the form: (nx, ny, nx = integers) (1) • The Ideal Gas molecules are non-interacting, so the gas Partition Function has the form: Z = (q)N (2) where q One Particle Partition Function • Using (2) in the Canonical Ensemble formalism gives the expressions on the right for the mean energy E, the equation of state P & the entropy S:

• The Partition Function for the 1 - dimensional particle in a “box” under the assumption that the energy levels are so closely spaced that the sum becomes an integral over phase space can be written: (3) • For the 3 – dimensional particle in a “box”, the 3 dimensions are independent so that the Partition Function can be written as the product of 3 terms like equation (3). That is: (4)

• Now, using the Canonical Ensemble expressions from before: • Mean Energy & the Equation of State can be obtained (per mole): • To obtain, for one mole of gas:

• The Entropy can also be obtained: E E (5) • As first discussed by Gibbs, The Entropy in Eq. (5) is NOT CORRECT! Specifically, its dependence on particle number N is wrong! “Gibbs’ Paradox” in the first part of Ch. 7!

• Here The we will an idealinterpretation gas from inofitstemperature mechanical ideal describe gas: microscopic properties • We aim to provide insight into the microscopic meaning of temperature and on transport properties • It will also introduce the subsequent analysis of transport properties of biomolecules • We will do so by relating the pressure of a gas with the collisions of the gas molecules against the walls of the container

• An ideal gas is a collection of a 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 (x, y, z) and a velocity (u, v, w), where u, v, and w are the x, y, and z components of the velocity, respectively. The history of a particle’s position/velocity is called a trajectory. Each point on a trajectory of a particles is described by 6 parameters (x, y, z, u, v, w). The speed of a particle c is related to the components of its x, y, and z velocity components (i. e. u, v, and w, respectively):

The ideal gas: microscopic interpretation of temperature

• A trajectory is obtained by applying the laws of classical mechanics, also called Newton’s laws of motion • Pressure is is the force F exerted per unit area A of the container wall by gas molecules as they collide with the walls • In principle, one should be able to obtain an expression for the pressure P by applying Newtons’ laws of motion • This mechanical view of pressure is called the Kinetic Theory of Gases

• The kinetic (i. e. mechanical) theory of gases is based on the following assumptions: A gas is composed of molecules in random motion obeying Newton’s laws of motion The volume of each molecule is a negligibly small fraction of the volume V occupied by the gas No appreciable forces act on the molecules except during collisions between molecules or between molecules and the container walls 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 (i. e. in the +x direction). It collides with a wall of unit area A

• Momentum p of a particle is mass times velocity. Just prior to a wall collision the momentum of a gas particle is p=mu. Just following an elastic collision the momentum is p=-mu (the particle has the opposite direction, i. e. it is moving in the –x direction). The change in momentum Dp is: • Suppose in an instant of time Dt a particle with velocity u covers a distance d, collides with the wall and subsequently covers a distance d; then u. Dt=2 d or Dt=2 d/u; the change in momentum during the time period Dt is given by

• From Newton’s Second Law (see above) this is the force exerted by the gas particle as it collides with the wall: • 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 fi:

• This equation can be rearranged to read: Now just divide both sides by A to get the pressure:

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 PV= n. RT and the microscopic (mechanical) view of the gas; let us being back the statistical mechanical formulation we have give earlier: PV= n. RT statistical description mechanical description

PV=n. RT statistical description mechanical description • We can interpret the temperature, a thermodynamic concept, in terms of the average velocity or kinetic energy of the molecules that compose the gas

• Pressure is related to the average kinetic energy per molecule • 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 • Average speed depends on molecular mass in addition to T

• 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 • The average speed is an important quantity; for example, the rate at which molecules collide (an important determinants of chemical reactivity) depends on it • Of course not all gas molecules in a container have the same speed: molecular speeds are distributed (like exam grades) • What is the distribution like?

• Of course not all gas molecules in a container have the same speed: molecular speeds are distributed (like exam grades); what is the distribution like? • We should now be used to the idea of averaging microscopic properties using Boltzmann distribution to calculate macroscopic properties of a system • For the average (or mean) squared speed:

• In the weighted average equation we are averaging over the groups of molecules with equal speeds, where ni is the number of molecules with speed ci, and fi=ni/N is the fraction of molecules with velocity ci (distribution) • We also know that energies associated with the motions of microscopic particles are quantized. However, the spacing between energy levels is very small for large amplitude motions such as molecular translation; thus, quantization is not an important effect in molecular speed distributions • Because the energy level spacing is small for the translational motion, the sum

• In general, the average of any property can be calculated from the distribution as follows:

• The function f(c) represents the probability of the particle having a certain speed c and is called Maxwell. Boltzmann 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 Ei energy level is proportional to

• Let us consider that the only energy of the system under consideration is kinetic energy • since: • then:

• This is the Maxwell-Boltzmann speed distribution function that can be used to calculate mechanical averages: In the argument of the exponential, the energy is the molecular kinetic energy ensures that • The term skews the distribution function toward higher speeds

• How does the probability depends on the mass of the molecules and the absolute temperature?

• How does the probability depends on the mass of the molecules and the absolute temperature? • The graph shows the results of experimental measurements of the distribution of molecular speeds for nitrogen gas N 2 at T=0 o. C, 1000 o. C, and 2000 o. C; each curve also corresponds very closely to the Maxwell-Boltzman distribution. The y axis is the fraction of molecules with a given speed (x axis)

• 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)

• We can use the Maxwell-Boltmann distribution to calculate the mean speed <c> and the mean square speed <c 2>: