1.6 - Entropy and the Second Law of Thermodynamics

The Second Law of Thermodynamics

The second law of thermodynamics says that the total entropy of an isolated system can never decrease. It only stays constant when every process is reversible.

For an isolated system: -

\Delta S_{\text{total}} \geq 0

\Delta S = 0

only for reversible processes -

\Delta S > 0

for all real, irreversible processes

The consequence is that natural processes are irreversible and push the total entropy up.

What Is Entropy?

You can describe entropy qualitatively in two ways: 1. The tendency of energy to spread out. 2. The share of a system’s energy that isn’t available to do work.

At the microscopic level, entropy measures disorder, or the number of possible microstates.

Energy Dispersal

Localized energy naturally spreads out. A few examples: - Hot coffee cools as its thermal energy spreads into the surroundings. - Perfume diffuses through a room as its molecules spread out. - A gas expands to fill its container as its particles spread out.

Energy naturally flows from concentrated states toward dispersed ones.

Energy That Can't Do Work

The second description of entropy is about usefulness. Energy flowing from a hot object to a cold one can run an engine and do work along the way. Once the energy has spread out and everything is at the same temperature, the total energy is still there (the first law holds), but there's no temperature difference left to drive anything. That energy is no longer available to do work. Higher entropy means more of a system's energy is in this spread-out, unusable form.

Properties of Entropy

It’s a State Function

Entropy is a state function, so it only depends on the current state of a system, not on how the system got there. That means: - It’s path-independent. - It’s set by the current conditions (P, V, T, and so on). -

\Delta S

only depends on the initial and final states.

Compare this to heat and work, since

Q

and

W

are path-dependent and are not state functions.

Example: on a PV diagram (Topic 1.4), take a gas from state A to state B along two different paths. The areas under the paths are different, so

W

is different, and so is

Q

. But

\Delta U

and

\Delta S

are the same for both paths, because they depend only on states A and B.

Maximum Entropy at Equilibrium

Entropy hits its maximum when a system reaches thermodynamic equilibrium. At that point: - Equilibrium is the state of maximum probability. - Energy is spread out as much as it can be. - No further spontaneous change is possible. - It’s the most stable configuration.

Change in Entropy

How a system’s entropy changes is set by how it interacts with its surroundings.

Isolated Systems

Isolated systems spontaneously move toward thermodynamic equilibrium. The process goes: - The system starts in a non-equilibrium state with lower entropy. - Spontaneous processes happen and raise the entropy. - The system evolves toward equilibrium, which is maximum entropy. - It reaches equilibrium and stays there.

Some examples: - Hot and cold objects settle to the same temperature. - A concentrated gas ends up uniformly distributed. - An ordered arrangement becomes disordered.

Entropy in Different Systems

Isolated System

No energy or matter is exchanged with the surroundings.

$\Delta S_{\text{system}} \geq 0$ always.

Closed System

Energy can be exchanged, but not matter.

The system’s entropy can decrease as long as the surroundings’ entropy increases by more.

Example: water freezing in a freezer loses entropy, because energy is transferred out of it by cooling. The freezer dumps that energy, plus the energy from its motor, into the kitchen, so the kitchen's entropy goes up by more.

$\Delta S_{\text{system}} + \Delta S_{\text{surroundings}} \geq 0$

Open System

Both energy and matter can be exchanged.

Same rule applies: the total entropy of the system plus its surroundings never decreases.

Practice Problems

1. A hot cup of coffee cools down to room temperature. Describe the entropy changes for the coffee and for the room.

2. Why can’t you “unscramble” an egg?

3. Which has higher entropy: (a) ice at 0°C, or (b) water at 0°C?

4. A gas is compressed isothermally, and the system’s entropy decreases. Does this break the second law? Explain.

5. Use the idea of entropy to explain why perpetual motion machines can’t work.

Solutions

1. For the coffee (the system): - It loses energy. - Its temperature drops. - Its entropy decreases, because energy is transferred out of it and its molecules have less energy to share.

For the room (the surroundings): - It gains energy from the coffee. - Its entropy increases.

For the total (coffee plus room): -

\Delta S_{\text{total}} > 0

, so the second law is satisfied. - The room’s entropy increase is bigger than the coffee’s entropy decrease. - The net result is an increase in total entropy.

2. Scrambling an egg is an irreversible process that sharply increases entropy: - The organized structure, with the yolk separate from the white, becomes a disordered mixture. - Chemical bonds break and proteins denature. - The system goes from low entropy (ordered) to high entropy (disordered).

Unscrambling it would require a spontaneous drop in entropy, which breaks the second law. Even though it’s theoretically possible with outside work, the egg’s chemical changes are irreversible, since the proteins can’t spontaneously refold and separate on their own.

3. Water at 0°C has higher entropy.

Liquid water: - Its molecules move around freely, which is disorder. - There are more possible arrangements. - More energy states are accessible.

Ice at 0°C: - Its molecules are locked into a crystal lattice, which is order. - There are fewer possible arrangements. - Motion is restricted.

They’re at the same temperature, but the liquid phase is more disordered, so it has higher entropy.

4. No, it doesn’t break the second law.

The gas isn’t an isolated system, since it interacts with its surroundings during the compression. - Compression takes work input from the surroundings. - The system loses entropy as it becomes more ordered. - During isothermal compression, heat is released to the surroundings. - The surroundings gain at least as much entropy as the system loses (exactly as much only for an ideal, reversible compression, and more for any real one). -

\Delta S_{\text{total}} = \Delta S_{\text{system}} + \Delta S_{\text{surroundings}} \geq 0

The second law applies to isolated systems, or to the total of system plus surroundings. The system on its own can lose entropy as long as the surroundings make up for it.

5. Perpetual motion machines break the laws of thermodynamics. There are two types:

Type 1 (breaks the first law): - It would create energy from nothing, which breaks conservation of energy.

Type 2 (breaks the second law): - It would turn heat completely into work with nothing else changing. - That would require entropy to decrease, pulling ordered energy out of disordered heat. - It’s impossible, since natural processes always increase entropy.

Every energy conversion: - Increases total entropy. - Makes some energy unavailable for work. - Comes with efficiency below 100%, always. - Eventually runs down as entropy climbs to its maximum.

Common Mistakes

Thinking entropy always increases for any system. That’s only true for isolated systems. Open and closed systems can lose entropy locally if the surroundings make up for it.

Confusing entropy with energy. Entropy measures the disorder or spread of energy, not the amount of energy. A system can lose energy and gain entropy at the same time.

Thinking equilibrium means static. Equilibrium is dynamic, since processes still happen, there’s just no net change. It’s maximum entropy, not zero activity.

Assuming entropy depends on the path. Entropy is a state function, so it’s path-independent. Only the initial and final states matter for $\Delta S$ .

Thinking low entropy is always good. Low entropy means organized and ordered, but spontaneous processes naturally raise entropy, meaning disorder.

Confusing reversible with possible. A reversible process is an idealization, meaning infinitely slow and always in equilibrium. Real processes are irreversible.

Thinking entropy breaks energy conservation. The second law is separate from the first law. Energy is still conserved, but entropy increases, so the energy just becomes less useful.

Assuming heat death means no energy. The universe approaching maximum entropy means energy still exists, it’s just spread out uniformly, with no temperature differences left to drive any processes.

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