Entropy and T-ds Relationships

The Essentials

Part of the Second Law of Thermodynamics is that the universe will tend to move toward a state of disorder more readily than toward a state of order. For example, imagine a sand castle on a windy beach. The sand castle is a state of high order. If the system is left for a long time, the wind will naturally move the sand from a castle (order) to a shapeless pile of sand (disorder). The measure of the movement from order to disorder is known as entropy.

Entropy and the Clausius Inequality

Entropy, unlike matter and energy, can only be created, not destroyed. Thus, the total sum of entropy in the universe will only ever increase. The entropy of a system that does not include the entire universe, however, can increase or decrease. Because the entropy of the universe can only increase, a system that creates entropy is said to be irreversible, while a system with no entropy generation is said to be reversible. The Clausius Inequality states that the change in entropy is greater than or equal to the integral of the differential heat transfer divided by the temperature of the system. $$\Delta S\ge \int_1^2\frac{\delta Q}{T}$$ This inequality can be broken down into two parts: the entropy generated by an irreversible process and the change in entropy of the system. $$\Delta S=\int_1^2\frac{\delta Q}{T}+S_{gen}, S_{gen}>0$$ For a reversible process, there is no entropy generated, so the change in entropy of the system can be given as a the integral term of them above equation. From this equation, an equation for the entropy of a system can be derived. $$\Delta S_{sys}=S_{in}-S_{out}+S_{gen}=S_2-S_1$$

Example

This section should have a good example to help demonstrate each step of the topic. Try to choose an example that goes through a lot of the situations a student might run into, not the easiest one you can find (I’m sure we’ve all seen that before and it’s the worst). If you feel like this topic has a need for more than one example then go ahead.

Practice

This section should have at least 3 examples for the student to go through. It’s best that these aren’t pulled from a textbook and instead we make our own so 1) we can guarantee that they come out with nice numbers and 2) so it’s a unique bit of practicing rather than something they may have already done.

Solutions: Here is where you’d put the solutions to the practice problems above

More Resources

This section is where you could put links to other resources you feel might be helpful OpenStax: Comparison Tests