Why does heat flow from hot to cold and never the other way on its own? Why can't the Carnot limit be beaten? One quantity answers both: entropy.
Why it works
Define the entropy change of something that takes in heat $Q$ at a steady kelvin temperature $T$ as $\Delta S = Q/T$. Now let heat $Q$ flow from a hot reservoir to a cold one. The hot one loses $Q/T_H$, the cold one gains $Q/T_C$, and since $T_C < T_H$ the gain is bigger: the total goes up. Heat flowing from cold to hot would make the total go down, and that is exactly what never happens by itself. At the level of molecules, entropy counts the ways to arrange the energy; spread-out energy has far more arrangements, so it is overwhelmingly likely.
Key idea
$\Delta S = \dfrac{Q}{T}$ for heat $Q$ at kelvin temperature $T$. Second law: the total entropy of an isolated system never decreases. It rises in every real process and stays the same only in an ideal, reversible one.
Worked examples
Worked example: Heat flowing downhill
$1200\,\text{J}$ flows from a reservoir at $600\,\text{K}$ to one at $300\,\text{K}$. Find the total entropy change.
- Hot: $\Delta S_H = -\dfrac{1200}{600} = -2.0\,\text{J/K}$. Cold: $\Delta S_C = +\dfrac{1200}{300} = +4.0\,\text{J/K}$.
- Total: $+2.0\,\text{J/K}$. Positive, so it happens on its own.
Answer: $+2.0\,\text{J/K}$
Worked example: Melting ice
Find the entropy change of $0.10\,\text{kg}$ of ice melting at $0\,^\circ\text{C}$.
- Heat: $Q = mL_f = (0.10)(3.34 \times 10^{5}) = 33{,}400\,\text{J}$, taken in at $273\,\text{K}$.
- $\Delta S = \dfrac{33{,}400}{273} = 122\,\text{J/K}$.
Answer: $122\,\text{J/K}$
Worked example: Inside a refrigerator
A refrigerator takes $600\,\text{J}$ from food at $270\,\text{K}$ and, using $200\,\text{J}$ of work, gives $800\,\text{J}$ to a kitchen at $300\,\text{K}$. Find the total entropy change.
- Food: $-\dfrac{600}{270} = -2.22\,\text{J/K}$. Kitchen: $+\dfrac{800}{300} = +2.67\,\text{J/K}$.
- Total: $+0.44\,\text{J/K}$. The food's entropy fell, but the total rose.
Answer: $+0.44\,\text{J/K}$
Common mistake
Thinking entropy can never go down anywhere. It can go down in one part of a system, such as the food in a refrigerator or water freezing into ice, as long as the entropy of the surroundings rises by more. The second law is about the total for an isolated system. Check: list every object that exchanges heat, and add up their $Q/T$ values.
For Carnot's ideal engine the total is exactly zero: $Q_H/T_H = Q_C/T_C$, which gives its efficiency $1 - T_C/T_H$. Any real engine adds entropy, so it gives off more heat and does less work.