Entropy is in some sense a measure of disorder.

The symbol for entropy is S, and the units are J/K.

A container of ideal gas has an entropy value, just as it has a pressure, a volume, and a temperature. Unlike P, V, and T, which are quite easy to measure, the entropy of a system is difficult to calculate.

On the other, a change in entropy is easy to determine.

Change in Entropy

Entropy changes whenever there is a transfer of heat. The change in entropy is the heat added divided by the temperature at which the transfer took place. Expressed as an integral:

ΔS = Sf - Si = dQ/T

If the heat transfer takes place at a single temperature, the change in entropy is simply:

isothermal process: ΔS = Q/T

If the heat transfer takes place over a range of temperatures then, as long as ΔT is small compared to the absolute temperature T, the change in entropy is approximately:

ΔS = Q/Tavg

For an ideal gas, it can be shown that the change in entropy is given by:

ΔS = nR ln(Vf / Vi) + nCV ln(Tf / Ti)

Reversible and Irreversible Processes

If you spill a glass of milk, what the glass and the milk droplets do is governed by the laws of physics. If you videotaped the spill and then played the film backwards, it would be obvious to you that the film was running backwards. Even though Newton's Laws and The Laws of Conservation of Energy and Conservation of Momentum would be obeyed when you played the film backwards, the probability that all the milk and the glass would spontaneously come together to form a full glass of milk is incredibly small.

This is an example of an irreversible process. The entropy postulate connects the concept of entropy with such processes:

Entropy Postulate: If an irreversible process occurs in a closed system, the entropy S of the system always increases.

Some processes are reversible. In these there is no change in entropy in a closed system.

The Second Law of Thermodynamics

The Second Law of Thermodynamics states that:

The entropy of a closed system is constant for reversible processes and increases for irreversible processes. It never decreases.

ΔS ≥ 0

This is why the glass of spilled milk never spontaneously transforms itself back into an upright full glass of milk - that would decrease the entropy.

Entropy is often called time's arrow. Time moves in the direction of increasing entropy.

Sample Problem

You have two styrofoam containers of water. Each holds 1 kg of water. In one the water temperature is 17C, while in the other it is 37C. The colder water is then poured into the warmer water, and the system is allowed to come to equilibrium.

Is this process reversible or irreversible?

  1. Reversible
  2. Irreversible

The process is irreversible - any process involving a transfer of heat from a higher-temperature region to a lower-temperature region is irreversible.

Assuming no heat is exchanged with the surroundings or the environment, what is the change in entropy in the mixing process?

First, determine how much heat is involved. The final temperature will be 27C. The heat transferred from the hot water to the cold water is therefore:

Q = mcΔT

m = 1, c = 4186, and ΔT = 10.

Q = 41860 J

Calculate the change in entropy for the hot and cold water using the equation:

ΔS = Q/Tavg

The T's are in kelvin. For the cold water, Tavg = 22C = 295 K. For the hot water Tavg = 32C = 305 K.

Q is positive for the cold water, because heat was added, and negative for the hot water. Therefore, for the mixing process:

ΔScold + ΔShot = 41860/295 - 41860/305 = +4.65 J/K.