Ch 21b - A Thermodynamics Primer/Review

We’ve spent most of Ch21b learning about the microscopic world, one that is defined by quantum mechanics. Such understanding has emerged only relatively recently within the history of chemistry, much of what we know about the transformation of chemical systems was gleaned from studies of macroscopic samples before the advent of the Schrödinger equation.

Chief among these advancements was thermodynamics. The power of this discipline lies in its generality. The field developed from observations of the natural world, it stands on its own. No molecular details of the system under study enter into classical thermodynamic analyses. The desire to bridge the macroscopic and microscopic worlds lies at the heart of statistical thermodynamics, a subject we will consider for the remainder of the quarter.

Here we’ll briefly review the fundamental “laws” of thermodynamics, in order to provide the necessary backdrop for the molecular/statistical analysis that is based on the collective behavior of extremely large numbers of microscopic quantum mechanical systems.

Page 1 23Feb2018 Ch 21b - The “Zeroth” Law of Thermodynamics

The quantitative concepts of temperature, work, internal energy, and heat play an important role in the understanding of chemical phenomena. The need to define an absolute temperature scale was not recognized until after the first and second laws of thermodynamics were established. Briefly, it states:

For three systems A, B, and C, if A is in thermal equilibrium with C and B is also in thermal equilibrium with C; then A and B are in thermal equilibrium with each other.

A C B

Page 2 23Feb2018 Ch 21b - Thermodynamic State Variables/Functions

When a system is at equilibrium under a given set of conditions, it is said to be in a definite state. State variables include things like pressure, volume and temperature (P, V, T). Those variables that depend on the size of the system are referred to as extensive (such as V, energy); those that do not are referred to as intensive (P, T, for example). Extensive variables can be converted into intensive variables by dividing be a measure of the amount of substance (the molar volume, for example).

As we’ll see next, certain quantities do not depend upon the path take by the system; these are called state functions. Some thermodynamic state functions we will be concerned with include:

U = internal energy S = entropy H = enthalpy (classically, U + PV) A = Helmholtz free enegy = U – TS G = Gibbs free energy = H – TS

Page 3 23Feb2018 Ch 21b - The First Law of Thermodynamics

This law is, essentially, a statement of the conservation of energy. Suppose a system is brought from state A to state B. The work done on the system during this change is w, and the heat absorbed by the system is q. The first law states that while w and q depend on the path taken by the system, their sum does not. This sum is a state function, and is the internal energy. Mathematically:

dU = dq + dw

where the differentials are meant to emphasize infinitesimal changes.

dU, since it is path independent, is referred to as an exact differential, while dq and dw are known as inexact differentials since their value depends upon the path taken by the system. Cyclic processes bring the system back to its initial state, and so for such processes the internal energy change is zero. B A ∫ dU = 0 Page 4 23Feb2018 Ch 21b - The Second Law of Thermodynamics

The second law is a bit more abstract, and can be stated many ways. One is: There is a quantity S, called entropy, which is a state function. In an irreversible process, the entropy of the system and its surroundings increases. For a reversible process, the entropy of the system and its surroundings remains constant. Mathematically: dS = dqrev/T where the differentials are again meant to emphasize infinitesimal changes.

Reversible processes are those in which the driving force (a difference in P, T, etc.) is infinitesimal. Any other change is called irreversible or spontaneous.

Reversible Irreversible S = S – S = dq /T or S – S > dq /T Δ A B ∫ rev A B ∫ irrev

Given the formulation above, the first and second laws can be combined to yield the well known perfect gas equation:

dU = TdS – PdV (heat + work) Page 5 23Feb2018 Ch 21b - The Third Law of Thermodynamics (Nernst Heat Theorem)

The second law relates the infinitesimal change in entropy, an exact differential, to that in the infinitesimal change in the heat exchanged (which is inexact since it depends on the path of the system) under isothermal conditions. The integral needed to calculate the change in entropy, however, has an additive constant associated with its calculation. The third law, which can be written in several forms, deals with this constant. One formulation is:

In any system in internal equilibrium undergoing an isothermal process between two states, the entropy change of the process approaches zero as the temperature of the system approaches zero. This enables us to calculate the absolute entropy of a substance via the expressions

S – S0 = ∫ dqrev/T and S0(T=0) = 0

where the integral runs from 0 to T. The restriction to states of internal equilibrium is important. Frequently, during the approach to T = 0, a system develops internal constraints that prevent the achievement of internal equilibrium (glasses cannot turn into crystalline solids, for example, at low T).

Page 6 23Feb2018