From WolfWikis

LECTURE 4

Classical thermodynamics

We have seen that we can use statistics to average over the energy levels of an ensemble that consists of systems each with a great number of energy states. That brings us to the macroscopic world of classical thermodynamics. Its theory was developed before statistics, atomic theory and mechanics quantum mechanics were developed and gained acceptance.

It relies heavily upon partial differentiation of functions of multiple variables (multivariate functions). Let's first say a few things about the math, in part to refresh your memory, in part to extend your math skills into multivariate space a bit more. (See math chapter H at page 683)

Partial differentiation

The development of thermodynamics would have been unthinkable without calculus in more than one dimension (multivariate calculus) and partial differentiation is essential to the theory.

'Active' variables

When applying partial differentiation it is very important to keep in mind, which symbol is the variable and which ones are the constants.

Mathematicians usually write the variable as x or y and the constants as a, b or c but in Physical Chemistry the symbols are different. It sometimes helps to replace the symbols in your mind.

For example the van der Waals equation can be written as:

Suppose we must compute the partial differential

In this case molar volume is the variable 'x' and the pressure is the function f(x), the rest is just constants, so I could rewrite:

When calculating

I should look at the formula as:

The active variable 'x' is now the temperature T and all the rest is just constants. It is useful to train your eye to pick out the one active one from all the inactive ones. Use highlighters, underline, rewrite, do whatever helps you best.

Cross derivatives

As shown in equations H.5 and H.6 there are also higher order partial derivatives versus T and versus V. A very interesting derivative of second order and one that is used extensively in thermodynamics is the mixed second order derivative.

Of course here the 'active' variable is first T, then V. The interesting thing about it is that it does not matter whether you first take T and then V or the other way around.

Example H-2 shows an example of how mixed derivatives can be used to translate one quantity into the other. This trick is used over and over again in thermodynamics because it allows you to replace a quantity that is really hard to measure by one (or more) that are much easier to get good experimental values for.

For example:

This expression is not obvious at all. It tells you that if you study the pressure P when heating up while keeping the volume the same (which is doable) you're measuring how the entropy changes with volume under isothermal conditions. Entropy will be discussed later, suffice it to say that nobody has ever constructed a working 'entropometer'! So that is an impossible quantity to measure directly.

The decomposition of changes

A very important result of multivariate calculus is that if a quantity Q is a function of more than one variable, say A and B that we can decompose any infinitesimal change dQ into infinitesimal changes in A and B in a very simple linear way:

dQ = a.dA + b.dB

The coefficients a and b are the partial derivatives of first order versus A and B. For a derivation see the book.

This mathematical fact is something we will be using over and over.

Exact and inexact differentials. State and path functions.

Pressure is a good example of a state function (it returns to its old value if you go back to a previous state). The other (the gas gage) is a path function. (Make a detour and your bank account will tell you difference!).

The difference has its roots deep in mathematics and it comes in as soon as a function has two of more variables.

The gas law is a good example. The pressure depends on both temperature T and (molar) volume V. When changing the pressure a little bit, say by dP we can show that we can write that out in the two possible components dT and dV as:

At first I wrote arbitrary coefficients p and q, but as you can see they are really partial derivatives. This is another way that thermodynamics exploits multivariate calculus: it shows how total changes can be built up of various contributions.

The interesting thing is that if the function P is a state function (and your barometer will testify to that) that the following equality must hold:

If the function is a path function this equality does not hold.

Thermodynamics is largely based upon exploiting the above facts:

• It tries to define state functions to describe energy changes
• It tries to decompose changes into well-defined contributions
• It uses partial differentials to link known quantities to unknown ones

The zeroth and first law of thermodynamics

Thermodynamics has four major 'laws': the first, the second, the third and one that was so obvious that it was later added as the 'zeroth' law.

The zeroth law says that heat always spontaneously flows from hot to cold. That is if two objects are at a different temperature T₁>T₂ and they are in contact heat will flow from object 1 to object 2.

D'oh, you might say, but actually, this 'law' holds the key for being able to define temperatures scales and that topic can get pretty philosophical. Let's not do that here and just leave it at that.

Heat and work

Heat and work were the two most important concepts of the early development of thermodynamics. Although even the Greeks and the Romans had already played with steam and shot some cannon bullets with it, -i.e. used heat to do useful work like killing people- the real relationship of heat and work remained long a mystery.

At first the two phenomena were even given two different units. Heat was measured in calories (amount of heat needed to heat one gram of water by one Kelvin). Work has had a number of units, the most recent being the Joule or Newton.meter: the amount of work needed to move one meter against a force of one Newton.

Dissipation and the concept of energy

Sometime in the 18th century it was shown that if you moved one meter against a friction force of one Newton, things warmed up a bit. This process of turning work into heat is known as dissipation and it happens any time you hit the brake of your car. In fact dissipation is a very easy process in contrast with going the other way, from heat to work. When you release the brake you do not just regain your previous speed instead you need to give gas (and that is expensive these days).

Anyway, people dissipated some work into heat (by friction) and by measuring how much of each was involved in the process, it was found that 1 cal equals 4.2 J.

So, the two phenomena are clearly two heads of the same dragon. They are both forms of one and the same thing called: energy. (Although the point of conversion being easy in one direction and not in the other remains to be worked out).

The first law states (in one of its formulations) that energy is conserved. This is an extension of the fact that in e.g. elastic collisions (think of pool) kinetic energy is conserved etc.

Types of work

Work in general is defined as a product of a force F and a path element ds. Both are vectors and work is computed by integrating over their inproduct

w = ∫ F . ds

Moving an object against the force of friction as done in the above dissipation experiment is but one example of work.

w_friction = ∫ F_friction . ds

We could also think of electrical work. In that case we would be moving a charge e (e.g. the negative charge of an electron) against an electrical field E. The work would be:

w_electrical = ∫ eE. ds

Other examples are the stretching of a rubber band against the elastic force or moving a magnet in a magnetic field etc, etc. We come back to that.

A special case: volume work

In the case of a cylinder we can introduce the area of the piston A and forget about the vectorial nature The movement of the piston is constrained to one direction, the one in which we apply pressure (P being force F per area A). Besides: the molecules of the gas rapidly equilibrate to the applied pressure and make it into an all-sided phenomenon.

w_volume = ∫ (F/A).(A.ds) = ∫ P . dV

This particular form of work is called volume work and will play an important role in the development of our theory.

Notice however that volume work is but one form of work.

Sign conventions

Compressing a gas in a cylinder

Thermodynamics would not have come very far without cylinders to hold gases, in particular steam.

Look at fig 19.1 to see a cylinder-based version of the previous statement. Also go through the argument on page 767 to see that we can compute one type of work that is very important for gases, the volume work as follows:

w_volume = -P_extΔV.

Actually this formula only holds under isobaric conditions. If the outside pressure changes during compression we must integrate over all contributions:

(Note that the minus sign is consistent with the you pay principle. When the volume gets smaller ΔV is negative and w is positive. This is the work you have to put in).

Let us compare two ways (paths) of compressing a gas from volume 1 to volume 2.

Compression one (irreversible):

In a P-V diagram of an ideal gas P is a hyperbolic function of V see fig 19.2 but this refers to the internal pressure of the gas. It is the external one that counts when computing work and they are not necessarily the same. As long as P_extermal is constant, work is represented by a rectangle. But how can the external pressure be different from the internal one?

1. Start with cylinder 1 liter, both external and internal pressure 1 bar.
2. Peg the piston in a fixed position
3. Put cylinder in a pressure chamber with P_ext= 2 bar.

   Now you are ready to do the experiment:

4. Suddenly pull the peg

The piston will shoot down till the internal and external pressures balance out again and the volume is 1/2 L

Notice that the external pressure was maintained constant at 2 bar during the peg-pulling and that the internal and external pressures were not balanced at all time.

Compression two (reversible).

In most cases the external pressure is not constant, it consists of the pressure of the atmosphere outside plus whatever pressure your hand is exerting on the piston. Because the gas molecules adjust the internal pressure pretty fast (with the speed of sound!) to any changes your hand imposes to the outside pressure, the two pressures are pretty much the same at all times.

Reversible: P_external ≈ P_internal

Reversible: P_external = P_internal + δP

1. Start with cylinder 1 liter, both external and internal pressure 1 bar.
2. Gently push the piston until V= 1/2 l and P=2 bar

In that case both the pressure inside and out will follow the reciprocal function (a hyperbola) and w is the area under that curve. (I.e. you need to integrate).

Reversible versus irreversible

You'll say, what's the big difference. In both cases we go from 1bar/1 liter to 2bar/1/2liter.

The key is the word suddenly. By pegging we have created a situation where P_ext=2P_int artificially and now we pull.

Because it's done suddenly in path one P_external > P_internal for most of the time, because the internal pressure is struggling to catch up with the external one.

During the second compression we have P_ext=P_internal at all times

It's a bit like falling off a cliff versus gently sliding down a hill.

Path one is called an irreversible path, the second a reversible path

Work and heat are not state functions

As the work depends on the external pressure, it is not the same in the two diagrams. From the two compressions above it should be clear that work depends on how exactly (e.g. how fast, pegged or not etc.) you do things. No work is not a state, but a path function. A reversible path costs a lot less work than an irreversible one.

For the irreversible sudden path we have

w_irreversible = - P_externalΔV (the area of the rectangle)

The area under the hyperbola is not hard to find as long as the gas is ideal:

No w_irreversible and w_reversible are not the same. (See the green areas). In fact, the reversible work is always the minimum work to get from 1 to 2.

Without proof: Heat q is a path function also.

The more complete first law

The above statement is a more complete formulation of the First law of thermodynamics. The realization that work and heat are both forms of energy undergoes quite an extension by saying that it is a state function. It means that although heat and work can be produced and destroyed (and transformed into each other), energy is conserved.

This allows us to do some serious bookkeeping!

We can write the law as:

U = w + q

But the (important!) bit about the state function is better represented if we talk about small changes of the energy:

dU = δw +δq

We write a straight Latin d for U to indicate this is the change in a state function, where as the changes in work and heat are path-dependent. This is indicated by the 'crooked' δ.

We can represent changes as integrals, but only for U can we say that regardless of path we get ΔU = U2-U1 if we go from state one to state two. (I.e. it only depends on the end points, not the path).