### Task in CHEM 113C Physical Chemistry

(I) Consider a system of N non-interacting oscillators which are described by (i) classical CHEM 113C Physical Chemistry

(ii) quantum mechanics.

A) Derive the partition function Q for both cases, i.e., classical and quantum. Remember to take into account the the zero-point energy in the quantum mechanical description. (Note: you derived the two partition functions for a single oscillator in Homework 3.)

B) Use the partition function you obtained in the previous step and derive the expression for the internal energy of both systems. Where appropriate, use the following identities:

ddxsinh(x) = cosh(x)andcoth(x) = cosh(x)sinh(x)

C) Using the expressions for quantum and classical internal energies, determine at what conditions the two solutions will become the same (low or high temperatures)? Pay attention to the low T limit – if you knew the system’s behavior at low temperatures, would you be able to decide whether it is composed of quantum or classical oscillators? Sketch the internal energy as a function of T, derive the limit of the internal energy as T → 0 (see the hint below), and provide a detailed explanation to all the questions.

Note: you will need to inspect the behavior of hyperbolic cosine function – use the fact that for coth(x) we find the following:

limx→0+ coth(x) ≈1x(II) In this example, you will compute the partition function and the internal energy of ideal gas using only classical mechanics (i.e., you will not rely on the translational partition function). Assume that the system is composed on N identical particles in a reservoir of volume V ≡ Lx · Ly · Lz. The particles behave purely classically, they interact neither with each other nor with the surroundings, and their energy is purely kinetic. The energy is not composed of discrete levels, because we assume that the particles behave classically.

A) Compute the partition sum of the N-particle system:

Q =Z Z Z Lx,Ly,Lz0,0,0Z Z Z ∞−∞e−PNj=1 βEj (px,py,pz) dxdydzdpxdpydpzh3N,

where h is the Planck constant introduced to make the integrand dimensionless; x, y, z represent three cartesian coordinates, and px, py, pz are the momentum components in 1 three cartesian directions. The exponent contains the sum over energies of all particles (labeled j). Show that the result will be in the following form

αN VN T32 N

where α is a constant. Solve for Q and determine the expression for α.(Note that you can decompose the triple integration into a product of Gaussian integrals.)

B) Compute the internal energy of the system from the expression for Q you have obtained in the previous step. Does it agree with the expression you know from basic thermodynamics? Does it agree with the result derived using the translational partition function (see the lecture notes and slides). Provide a detailed reasoning.

(III) We will investigate an (over)simplified model for protein-ligand docking. Consider a situation in which the ligand binding is described by the Hooke’s law, i.e., the force depends linearly on the distance (d) between the protein site and the ligand with a force constant ˜k. The corresponding energy of the system thus depends on d continuously (hence we resort to classical description of the bond).

A) In the canonical ensemble of protein-ligand bound pairs, show that the probability to find a ligand at distance d is a Gaussian functionP(d) = q

−1e−β12kd˜ 2

Find the full expression for the probability and compute the partition function q. Show that q is a simple term that depends on the force constant ˜k and temperature T.

B) Using the probability distribution from the previous step, derive the expression for the average distance hdi at given temperature T. Recall that (in general) the average quantity of some function O(x) is computed as:

hOi =Z ∞0O(x)P(x)dx,where P(x) is the probability function.

You will use the following integrals:

Z ∞0x exp(−ax2)dx =12a and Z ∞0exp(−ax2)dx =12rπa

C) You found the probability to find a ligand at distance d in the first step; what is the standard deviation of the distribution and how does it explicitly depend on temperature?

D) Use the expression for the partition function q and derive the average energy of a protein-ligand bound pair in the ensemble. We assume that the energy depends merely on the distance d, i.e., our model is effective one-dimensional. Does your result agree with the equipartition theorem you know from basic thermodynamics?

(IV) When gas molecules are weakly bound to the surface of a catalyst, they can still “jump around”, i.e., they can move on the surface. Effectively, we can model this situation by a two-dimensional ideal gas in which we consider only translational motion of otherwise non-interacting particles.

A) Derive the partition function Q of an ensemble of N indistinguishable molecules of an ideal gas in two dimensions. Use only the translational degrees of freedom. Compared to the qtrans we derived in the lecture, the partition function will now depend on the area of the surface.

B) Use the partition function you have obtained in the previous step and compute the internal energy of the system. Compare your result with the internal energy of the 3D ideal gas and argue why the 2D result is expected from the equipartition theorem.

(V) Proton has a nuclear spin angular momentum and interacts with a static and homogeneous external magnetic field H~

0. The energy states of protons depend only on the orientation with respect to the external field; due to quantization of space only two orientations exist with energies ±γ|H~0|, where γ is a proportionality constant (itself composed from other elementary constants). Consider a canonical ensemble of N protons and derive an expression for the partition function and internal energy. Simplify the expression as much as possible and make use of hyperbolic functions; show all the steps of the derivation

CHEM 113C Physical Chemistry