Adapted from Wright
The 1st Law of Thermodynamics states that energy is conserved. It is neither created or destroyed; it is only transferred. This transfer of energy can result in work. For example, a fire heats water to create steam. The steam turns a turbine, performing work, to generate power or move an object.
Now imagine two objects of different temperature.
- Object 1: hot - TH
- Object 2: cold - TC
If the two objects were in contact with each other, heat spontaneously flows from the hot object to the cold object until their temperatures reach equilibrium. Energy is conserved. However, what about the case where heat spontaneously flows from the cold object to the hot object? Energy would also be conserved.
In both cases, the 1st law is not violated. However, the second case is never observed in nature.
The 2nd Law of Thermodynamics, which states that “the total entropy of a system either remains constant or increases in any spontaneous process”, addresses the impossibility of Case #2 by introducing a new property of a system, entropy, which has many definitions depending on the context. Entropy is energy that cannot be used to do work. All matter possesses entropy and at 0 Kelvin, a perfect crystal of a perfect substance is defined as zero.
A change in entropy can be seen with
- The uniform distribution of energy (such as the heat flow shown above)
- The uniform distribution of matter (think of two gases mixing… they will never de-mix)
In both cases, the process of distribution is irreversible, at least in a spontaneous way. Two gases will never un-mix spontaneously. Two objects will never spontaneously un-mix heat (they remain the same temperature).
We say, in both cases, that entropy has increased. The more spread out the energy or matter, the higher the entropy. That means, for the spontaneous, irreversible processes,
\[S_{\mathrm{final}} > S_{\mathrm{initial}}\] Entropy is an extensive property and defined as an amount of heat divided by the temperature (and given as J K–1) for a given amount of substance.
\[\Delta S = \dfrac{Q}{T}\]
Consider the heat (i.e. flow of energy) between the two objects. The hot object with a temperature of TH will experience an average temperature of
\[T_{\mathrm{H,avg}} = \dfrac{T_{\mathrm{H}} + T_{\mathrm{final}}}{2}\]
whereas the cold object with a temperature of TC will experience an average temperature of
\[T_{\mathrm{C,avg}} = \dfrac{T_{\mathrm{C}} + T_{\mathrm{final}}}{2}\] The final temperature for both objects are equal at equilibrium. Therefore,
\[T_{\mathrm{H,avg}} > T_{\mathrm{C,avg}}\]
since the hot object has a higher temperature than the cold object such that
\[T_{\mathrm{H}} > T_{\mathrm{C}}\]
The entropy for the hot object would then be
\[\Delta S_{\mathrm{H}} = \dfrac{-Q}{T_{\mathrm{H}}}\]
and the entropy for the cold object would be
\[\Delta S_{\mathrm{C}} = \dfrac{Q}{T_{\mathrm{C}}}\]
where
\[ \vert \Delta S_{\mathrm{H}} \vert < \vert \Delta S_{\mathrm{C}} \vert \]
Since both objects are the system, the entropy for the system would be given as
\[\begin{align*} S_{\mathrm{final}} &= S_{\mathrm{initial}} - \dfrac{-Q}{T_{\mathrm{H}}} + \dfrac{Q}{T_{\mathrm{C}}} \\[2ex] &= S_{\mathrm{initial}} - \Delta S_{\mathrm{H}} + \Delta S_{\mathrm{C}} \end{align*}\]
Therefore,
\[S_{\mathrm{final}} > S_{\mathrm{initial}}\]
and the entropy has increased for this spontaneous process.
If heat flowed from cold to hot,
\[S_{\mathrm{final}} < S_{\mathrm{initial}}\] which never occurs spontaneously, thereby violating the 2nd law of thermodynamics.