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HyperPhysics - Entropy as Time's Arrow

PY105 Notes - Entropy and the second law
Conversational, intuitive conceptual introduction to the concept of entropy.

PY105 Notes - Entropy and the second law
Conversational, intuitive conceptual introduction to the concept of entropy.

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Special points of emphasis

Heat and Temperature

The Second Law of Thermodynamics and Heat Engines

The Second Law of Thermodynamics dictates that heat flows from the warmer to the cooler object when two objects are in thermal contact.

When heat flows from the warm object, it loses an amount of entropy equal to Q/Th, and the cold object gains an amount of entropy equal to Q/Tc.

Therefore, heat flowing from a hot object to a cold one increases the disorder of the universe. The entropy gained by the cold object is greater than the entropy lost by the warm object. Such a change is spontaneous.

The First Law of Thermodynamics

The Second Law of Thermodynamics and Heat Engines

The Second Law of Thermodynamics and Heat Engines

An isothermal expansion occurs when a piston is slowly expanded at constant temperature. To maintain constant temperature, thermal equilibrium must be maintained with a heat sink in contact with the piston throughout the expansion. Entropy increases in the system during the isothermal expansion as heat flows in. Because the hot sink is the same temperature as the piston, it experiences an equal decrease of entropy in losing heat to the piston. The overall entropy of the universe is unchanged in this nonspontaneous, reversible heat flow. (Do you remember from the 1st law why heat must flow in to the piston during the isothermal expansion?)

A good, traditional conceptual problem is to compare the isothermal expansion discussed above to another kind of thermodynamic transformation called an adiabatic free expansion. The initial state of an adiabatic free expansion consists of a canister in which a membrane divides an area containing a gas from the second portion of the canister which is empty. When the membrane is pierced, the gas spontaneously diffuses to fill the container.

The initial and final states of the system in an adiabatic free expansion are identitical to the initial and final states of an isothermal expansion. What is interesting about an adiabatic free expansion, though, is that while the initial and final states of the system are the same as with the isothermal expansion, the initial and final states of the surroundings are not the same as with the isothermal expansion. While the isothermal expansion is reversible, the adiabatic free expansion is irreversible.

The internal energy change is zero (no work is performed and no heat flows in during the free expansion) but, like the isothermal expansion, the entropy of the canister has increased, but unlike the isothermal expansion, a compensating decrease in the entropy of the surroundings did not occur.

While entropy did not increase in the universe for the isothermal expansion, it did increase in the universe for the adiabatic free expansion. Think about it for a minute, and you will intuitively realize that the while an isothermal expansion is reversible, an adiabatic free expansion is not, though the entropy changes within the system are equivalent for the two transformations.

The Ideal Gas and Kinetic Theory

The Second Law of Thermodynamics and Heat Engines

Let us develop our understanding of entropy by touching on the advanced topic of statistical mechanics. Don't worry. Statistical mechanics has its quantitative side, which is beyond the scope of the MCAT, and its conceptual side, which provides some helpful perspective for understanding more basic ideas.

Statistical mechanics approaches particle motion within a thermodynamic system as a distribution of states. Some particles move slowly. Some move fast. Most are in between. The characteristic curve describing the distribution of molecular speed in a sample of gas is called the Maxwell/Boltzman distribution.

One thing about the Maxwell/Boltzman distribution is that he higher the temperature, the wider the distribution. What this means is that there are more possible combinations of motion available to a thermodynamic system at a higher temperature. In other words, entropy increases with higher temperature. Think about that.

The more possible states available to the molecules, the greater the total chaos, the more probable the disordered macrostate because there are so many combinations that produce disorder.

In these terms, the Second Law is telling us that molecules will not assume fewer possible states, will not become restricted in possibilities, without there being a balancing increase in disorder to the surroundings.

It is much more likely for a deck of cards sorted into suits to fall to the floor and spontaneously become chaotic than for a deck which is mixed to fall to the floor and spontaneously sort into suits. One state, the higher entropy, is much more probable than the other. You could sort the cards yourself and impose order on the system, but time is passing and soon it would be time for dinner.

Events occur irreversibly because the direction is going toward greater chaos. If the future were as likely as the past, though, events could go backwards as well as forwards. You aren't getting any younger.

The Second Law of Thermodynamics and Heat Engines

The Second Law of Thermodynamics and Heat Engines

Chemical Thermodynamics and the Equilibrium State

We're on the way to Chemical Thermodynamics with all of this, so let's have a little preview of the concept of free energy.

During a chemical reaction, the balance of the change in entropy in the surroundings due to the flow of heat (Δ H/T) in or out of the system is balanced against the change in the entropy of the system itself, (Δ S). This balance determines the free energy change (Δ G). Free energy is about the balance of two kinds of entropy change.

We previously mentioned the conception of the temperature as a potential function for the escaping tendency of heat. Think of the free energy change as the measure of the available work that can be realized from the escaping tendency of any poised configuration, any system outside of equilibrium.

Remember that spontaneity is ultimately about increasing total disorder, entropy. Chemical reactions move toward the equilibrium state. This is another way of saying that the entropy of the universe is always increasing. If there is a difference in free energy between two possible states of the system, we are saying that change is set to spontaneously occur which increases the disorder of the universe.

In terms of the equations, which are for thinking about (not for plugging and chugging) if there is a difference in free energy comparing Δ H/T going one way with Δ S going the other way, then chemical change is occurring spontaneously in one direction or the other. If a system possesses free energy in a given state, that state is less probable compared to the equilibrium state, and heat flows in the direction will be more likely.

Equilibrium for the system is a way for the universe to maximize its own disorder. Probability dictates. It is about the particles finding a macrostate with many microstates, the more likely one. So the free energy is expended until the equilibrium state is reached, and, now, there, at equilibrium, all heat flows are reversible (Δ H/T = Δ S) or (Δ G = 0). Heat can flow microscopically into the system from the surroundings, but it is just as likely to flow right back out.

Events in the equilibrium state are analogous to events in the Carnot cycle, where heat flow and work can occur without increasing the entropy of the universe and are completely reversible.

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