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Entropy and the Second Law

Consider the following two experiments:

1. Behavior of gas in a container

I have two glass containers connect by a valve. In the left-hand side container I have a sample of nitrogen gas. In the right-hand side container there is nothing (i.e. a vacuum):

I then open the valve connecting the two containers. What happens?

There is a spontaneous process that involves the flow of gas molecules from the left chamber to fill in the empty right-hand side chamber
- There was no work performed either on the system or on the surroundings (w = 0)
- There was no heat input or output from the system to the surroundings (q = 0)
- The spontaneous reaction results in pressure uniformity
- Although the gases spontaneously moved from the left chamber to the right, it is highly unlikely (impossible) that the gases might spontaneously move back into the left-hand side chamber (to produce the original condition). Thus, the process appears to be spontaneous and irreversible (in the absence of some outside influence)

2. The behavior of objects of different temperature

I have two cubes of metal. One cube I stick in the oven until it gets red hot. The other I put in the freezer until it gets really cold. Then I put the two blocks next to each other:

Over time what will happen to the temperature of the two blocks?

There is a spontaneous process that involves the flow of heat energy from the hot block to the cold block (the net heat energy is unchanged, however)
- There was no work performed either on the system or on the surroundings (w = 0)
- There was no heat input or output from the system to the surroundings (q = 0)
- The spontaneous reaction results in temperature uniformity
- Although the heat energy spontaneously moved from the hot block to the cold block, it is highly unlikely (impossible) that heat energy might spontaneously move back into the left-hand side block (to produce the original condition). Thus, the process appears to be spontaneous and irreversible (in the absence of some outside influence)

In both of the above cases, we had a situation where the starting condition was highly ordered (gas on left, vacuum on right; hot block on left, cold block on right)

The final condition was one of uniformity (of pressure or temperature)

The spontaneity of the process appears to be associated with a highly ordered system going to a less-ordered, uniform state

The disorder is expressed by a thermodynamic quantity called entropy (S)

The more disordered a state, the larger its entropy (a large magnitude for S means a lot of disorder)
Entropy is a state function: DS = S_final - S_initial
- In other words, the pathway you choose to get to S_final is not important, what is important is discussing the change in entropy of a system is a comparison of the initial and final values of the entropy
- A positive value for DS indicates an increase in disorder. A negative value for DS indicates a decrease in disorder

How is the change in the enthalpy of a system (DS) defined?

(where T is some constant temperature)

Obviously an enthalpy change is in some way proportional to some kind of energy term. If a system does not do any mechanical work, then heat energy must be involved. (Note: heat, like work, is not an entity but a method of energy transfer)
If the system is at constant T, then the non-mechanical energy flow is not being used to change the temperature (it is associated with the changing order of the system)
Why is DS inversely proportional to the absolute (K) temperature?
- At high temperatures what will the system be like? It will be a highly energetic gas and it will be difficult to get much more disorder out of it (thus at high temperatures, the change in disorder (DS) will be small for a given amount of non-mechanical energy transfer to the system.
- At low temperatures the system will be more ordered (highest order is at absolute zero) and the change in disorder (DS) will be larger for a given amount of non-mechanical energy transfer to the system

One of the classic examples of entropy changes in response to non-mechanical energy transfer at a constant temperature is the melting of ice at 0°C (i.e. 273K)

At 0K liquid water is in equilibrium with solid (i.e. ice)
Non-mechanical energy can be transferred into the system without raising the temperature (this is true as long as there is some ice and water present; if things get to the point where all the ice is melted, then the temperature will raise as you heat the water; if all the liquid water is frozen, then you lower the temperature of the ice as heat is removed; otherwise, the effect of energy flow into and out of the system results in either more or less ice being present, but no temperature change)
The amount of heat transferred to the system during the fusion of ice is the heat of fusion, DH_fus (6.01kJ/mole). Thus,

DS = 6.01kJ/mole / 273K = 22 J/mol K

DS is positive, indicating that the liquid form of water has greater disorder compared to the solid form (ice) (Note: this is an equilbrium situation, not an irreversible spontaneous reaction. Thus, input energy must increase the disorder of the system)

We can also define the entropy change in relationship to expansion of a gas at constant T (i.e. isothermal expansion

DS = nR ln (V_final/V_initial)
(for an ideal gas at some constant T)

The Second Law of Thermodynamics

The law that expresses the idea that there is an inherent direction in which processes occur is called the second law of thermodynamics

We must consider the change in entropy of the system and the surroundings

Together, the system and the surroundings constitute the universe

DS universe = DS system + DS surroundings

- For an irreversible (spontaneous) process, we have an increase in entropy. Since it is irreversible, the Universe has gained entropy:

DS universe = (DS system + DS surroundings) > 0

- For a reversible process (not spontaneous, but manipulable by heat flow) there is no net DS as far as the Universe is concerned (i.e. if we input heat energy into the system, then the heat energy of the surroundings decreases)

DS universe = (DS system + DS surroundings) = 0

Thus, there is no case where DS of the universe decreases

DS universe is constantly increasing (the universe is moving towards greater and greater disorder)

Chemical reactions follow this same law

Consider an exothermic reaction that is less disordered when complete (DS decreases; the disorder decreases)

O₂(g) + 2H₂(g) ® 2H₂O(g)

Let's perform this reaction in the following way: We will keep the temperature constant. Since the reaction is exothermic, we do this by having the surroundings absorb the released heat.
- The product is two molecules of water. The reactants comprise three molecules (one of O₂ and two of H₂)
- Since temperature is constant, the overall change in entropy of the system (reactants and products) is reduced (products are fewer gas molecules due to atoms being bonded together - this reduces the entropy of the atoms because since they are bonded together they are restricted in their movements)
The released heat is taken up by the surroundings
- The surroundings are kept at same temperature.
- Since temperature is constant, the energy absorbed by surroundings is manifest as increased disorder. The surroundings become more disordered

-q_sys/T = +q_surr/T (where T is constant)

in other words

the entropy lost by the system = entropy gained by surroundings

If the entire process were reversible, DS universe = 0
If any small part is irreversible, DS > 0

Although the reaction results in a decrease in entropy, the net entropic change of the universe is either 0 or positive