Chemical Structure and Stability Dr. Toby Hudson (Room 456) t.hudson@chem.usyd.edu.au Notes: http://notes.chem.usyd.edu.au/course/hudson_t/chem2402
Statistical Thermodynamics The aim of statistical thermodynamics is to explain how the properties of individual molecules lead to bulk properties of chemical systems such as temperature, pressure, heat capacity, entropy, and chemical equilibrium. In these lectures we will develop the microscopic justification for the laws of thermodynamics.
States, Microstates, and Configurations The state of a molecule is the amount of energy in each of its degrees of freedom. e.g. the state of a diatomic molecule is specified by its electronic configuration, vibrational quantum number v, rotational quantum number J, and a translational quantum state. The total energy of that molecule is just the sum of the energies of each degree of freedom. A molecule is completely described by its molecular state.
States, Microstates, and Configurations What can we know about a system of (say, ~1023) molecules? Such a system of molecules cannot normally be completely specified. Instead we have incomplete knowledge - limited to the values of a number of bulk parameters like number of molecules, temperature, pressure, energy, volume… Some of these quantities have no counterpart in an individual molecule. They are purely properties of a system.
States, Microstates, and Configurations Consider a molecule with single set of equally-spaced energy levels . Two quanta of energy is represented thus:- Now we consider a small system of three such molecules. The molecules are identical, but distinguishable. We also assume them to be non-interacting (or “ideal”). That is, the energy levels of one molecule are not affected by the other molecules. What are the possible ways of distributing 3 quanta of energy (E = 3) among these 3 molecules (N=3)? Note that we define the problem by first fixing N and E.
States, Microstates, and Configurations There are ten different ways of distributing 3 quanta among 3 such molecules. Each such specific arrangement or distribution of energy is called a microstate. We can label our molecules because they are distinguishable. Note that there are multiple microstates for a given N and E.
States, Microstates, and Configurations The microstates can be conveniently grouped according to the occupation numbers of each energy level. A set of microstates which share the same occupation numbers is called a configuration. n3 = 1 n2 = 0 n1 = 0 n0 = 2 3 microstates n3 = 0 n2 = 1 n1 = 1 n0 = 1 6 microstates n3 = 0 n2 = 0 n1 = 3 n0 = 0 An occupation number ni tells us the number of molecules in state i. 1 microstate
Flash Quiz! The number of possible microstates in a system of N identical particles with a total of E quanta of energy is equivalent to: How many ways can N poker chips distributed amongst E players? How many ways can E keys be put in N locks? How many ways can E differently coloured balls be placed into N identical jars? How many ways can E $1 coins be deposited into N accounts?
Answer The number of possible microstates in a system of N identical particles with a total of E quanta of energy is equivalent to: How many ways can N poker chips distributed amongst E players? How many ways can E keys be put in N locks? How many ways can E differently coloured balls be placed into N identical jars? How many ways can E $1 coins be deposited into N accounts? We are distributing E amongst N, not N amongst E. Particles are allowed to have more than one quanta of energy, but a lock cannot have more than one key. The N particles are distinguishable, and the E energy quanta are not. Here it is the other way around, the N jars are not distinguishable, but the E balls are.
States, Microstates, and Configurations Every microstate and configuration must satisfy the constraints of total particle number, N, and total energy, E. Total Particle Number is just the sum of the number of molecules in each state i. In this example, For configuration I N = 2 + 0 + 0 + 1 = 3 For configuration II N = 1 + 1 + 1 + 0 = 3 For configuration III N = 0 + 3 + 0 + 0 = 3
States, Microstates, and Configurations Total Energy is just the sum of the number of molecules in each state i times the energy of that state In this example, For configuration I E = 0 + 0 + 0 + 3 = 3 For configuration II E = 0 + 1 + 2 + 0 = 3 For configuration III E = 0 + 3 + 0 + 0 = 3 Review: A molecular state is a quantum state of an individual molecule. A microstate is a possible distribution of the total energy among the molecules. A configuration is a possible distribution of the molecules among the molecular states.
How Many Microstates? We will not be able to deal with even moderately large systems by continuing to enumerate microstates. Instead, we can calculate the number of microstates, W(n0, n1, n2,…) that occur for a given allowed configuration (set of occupation numbers) as follows Where N is the total number of molecules in the system, and ni is the number of molecules in state i. N! (factorial N) = N x (N-1) x (N-2) x (N-3) x… 3 x 2 x 1, and 0! = 1 by definition. In this example, I n0=2, n1=0, n2=0, n3=1 WI = 3!/(2! 0! 0! 1!) = 6/(2111) = 3 II n0=1, n1=1, n2=1, n3=0 WII = 3!/(1! 1! 1! 0!) = 6/(111) = 6 III n0=0, n1=3, n2=0, n3=0 WIII = 3!/(0! 3! 0! 0!) = 6/(1611) = 1 …and the total number of microstates is Wtot = configurations W (= WI + WII + WIII , in the example) = 3 + 6 + 1 = 10.
Homework Problem Draw out all 15 possible microstates for a system of three molecules with equally-spaced (1 quantum) energy levels and four quanta of energy (i.e. N=3, E=4). Group the microstates by their occupation numbers into their configurations. Check that the number of microstates for each configurations equals that predicted by the formula for W(n0,n1,n2,...) .
Summary Next Lecture You should now Be able to define all the following terms degree of freedom; state; microstate; occupation number, configuration Know the important degrees of freedom:- electronic, vibrational, rotational, translational Discriminate between the properties of individual molecules and those that only exist for entire chemical systems. Do the Homework Problem. Next Lecture Statistics and Configurations in large systems. What is Equilibrium?