Entropy & the Second Law
The Second Law of Thermodynamics
Entropy measures how spread out or disordered a system's energy is. Whenever heat flows into an object, its entropy goes up — and the second law says the total entropy of an isolated system never decreases, only stays the same or grows. That's why heat always flows from hot to cold on its own, and why engines can never turn all their heat into useful work.
The formula
ΔS = Q / T
- ΔS — entropy change (J/K): how much the system's entropy increases
- Q — heat (J): the heat energy flowing into the system
- T — temperature (K): the absolute temperature at which the heat is added
Worked example
600 joules of heat flows into a large tank of water held at a steady 300 kelvin. How much does the water's entropy increase?
- Q = 600 J
- T = 300 K
- ΔS = Q / T
- ΔS = 600 J / 300 K
ΔS = 2 J/K
Test yourself
The same 600 joules of heat now flows into a cooler tank at 200 kelvin instead. What is the entropy change?
- 1200 J/K
- Correct answer: 3 J/K
- 0.33 J/K
Right! ΔS = 600 J ÷ 200 K = 3 J/K. A cooler object gains more entropy for the same amount of heat.
Why does heat flow from a hot cup of coffee into the cooler air of the room, and never the other way around?
- Correct answer: Because that direction increases total entropy
- Because that direction decreases total entropy
- Because both directions keep entropy exactly equal
Exactly — the same heat produces a bigger entropy gain in the cooler air than the entropy it removes from the hot coffee, so total entropy rises.
Where you see this
A hot coffee cools to room temperature and never spontaneously reheats; a dropped mug shatters and never reassembles; ice melts in your drink and never refreezes there. Every one of these one-way everyday events is the second law showing its face — each happens the way it does because that direction raises total entropy.
Common mistakes
Treating entropy as a vague synonym for messiness hides the actual arithmetic: the change is heat divided by temperature, ΔS = Q / T, so 600 J flowing into a 300 K tank raises entropy by 2 J/K. Then the counterintuitive half: the same 600 J into a cooler 200 K tank gives 3 J/K — a cooler object gains MORE entropy from the same heat, which is precisely why heat flows from hot to cold and never back. The law governs total entropy of isolated systems; a refrigerator cools its interior only by raising entropy elsewhere.
How it connects
The second law is the never behind this module's two earlier nevers: heat flowing only from hot to cold (calorimetry's opening fact) and no engine reaching full conversion (the Carnot limit) are both entropy bookkeeping. It closes the thermodynamics unit as physics' only fundamental law with a built-in direction of time — the universe's total entropy climbs, irreversibly.