Lecture 9 - Entropy

Lecture Recording:

Lecture Summary/Notes:

Entropy

Is an increase in RANDOMNESS, which is the driving force of the universe (not necessarily the exchange of energy, as we might suspect). A reaction is spontaneous if it contributes to the randomness of the universe. The more random you make the universe, the easier it is to go through with the reaction. Or, the more random you make the universe, the greater entropy it has.

Example) Ice melting in the heat - It increases in energy as it goes from solid state to a liquid state, but it happens spontaneously. That is because in the process of melting, it makes the universe more RANDOM. In other words, it increases in ENTROPY.

Entropy (S) can be calculated - 

qREV = The amount of heat that is reversible.

For example, if you have two buckets of water where one is higher in temperature than the other, and you pour water from one bucket to the other, heat will transfer from the hotter water to the cooler. But as the temperatures become closer and closer, the incremental increase becomes smaller and smaller, and the heat transfer becomes a state function. Conversely, the further apart the two temperatures are, the more irreversible the heat transfer is. Reversible = The amount of heat transferred is actually at equilibrium. (This never happens in reality - ideal construct, but you can still calculate what the reversible heat would be.).

Entropy (S) is always positive:

When things happen, they always go in whatever direction makes that quantity positive.

As long as S is positive, it will always go in that direction. The universe always moves in one direction. This is the reasoning for why time always moves forward.

Example) If you have two metal blocks, where one is hotter than the other:

If reversed, and the heat transfer goes the other way, the 1/273 and 1/373 would be reversed, and the quantity would come out to be negative (<0). Therefore, according to Clausius, the reaction must go the way demonstrated above because otherwise, the change in entropy would be negative (which, by intuition, we know cannot be the case - the cold brick can never make the warmer one warmer!).

In everything that happens, the entropy is ALWAYS positive. The system moves in whatever way that makes the quantity positive. This is expressed by the Clausius Inequality:

Universe = the system and its surroundings -> the sum of the entropy change of the system and its surroundings.

Second Law of Thermodynamics

The most profound statement ever said (according to Dr. Pietro):

Die Entropie der Welt strebt einem Maximum zu. 

(Entropy of the universe always strives to reach a maximum).

This is also known as the second law of thermodynamics, and is the only driving force in the universe. Everything that happens, happens to increase entropy. It is impossible for you to do anything that decreases the entropy of the universe.

Boltzmann attempted to find out what -entropy- was (Clausius did not know), and came up with the Boltzmann's equation:

S = k lnΩ

k = gas constant/Avogadro's number

Entropy equals (some proportionality constant - Boltzmann's constant) * the ln Ω. The quantity omega is the number of different ways that you can express the system (microstates).

Example) There are 3 black marbles and 3 white marbles - how many different ways can you express them in different ways? i.e. 3 white marbles beside 3 black marbles, or alternating black and white marbles?

Both have two different possibilites, or two microstates. They have the same degree of order.

How about a two black marbles beside two white marbles combination?

This pattern has 6 possibilities, or 6 microstates. This pattern is more random than the first two, hence it has a greater entropy than the first two. If the patterns were arranged completely randomly, with no forced pattern, it would have many, many more possibilites, and hence many more microstates. This is a state of maximum entropy. The universe is doing the same thing. It is getting randomer and randomer, to a point where it cannot return to its original, highly ordered state. Maximum entropy is achieved when you have the randomest state, with things cannot get... randomer! 

You can calculate that thermodynamic quantity (maximum entropy)  just by counting the different ways that you can arrange the state. In a chemical system, the states are the molecules - there are a lot of different states.

There can only be ONE purely randomized state - it has the maximum number of microstates. You can never go from a more random state to a less random state. Once the universe reaches that state, nothing else can happen -> i.e. the universe is doooomeddd :P. Once you reach that maximum entropy, the universe is doomed.

Phase Change