Law of Heat Exchange
Simple Explanation
When a hot object and a cold object are brought into contact (with no heat lost to the surroundings), heat flows from the hotter object to the cooler one until they reach a common final temperature — and by conservation of energy, the heat LOST by the hot object exactly equals the heat GAINED by the cold object.
Why Do We Need It?
This law is the basis of calorimetry — a practical technique used to determine unknown quantities like specific heat capacities or final mixing temperatures, by measuring how heat is exchanged between substances.
Formula
Law of Heat Exchange (Calorimetry)
Heat lost by hot object = Heat gained by cold object
When a hot object and a cold object are in thermal contact (with no heat lost to the surroundings), heat flows from the hot object to the cold one until they reach a common final temperature — the heat lost by one exactly equals the heat gained by the other.
- m₁c₁(T₁ − Tf)
- — Heat lost by the hotter object (mass m₁, specific heat c₁, initial temperature T₁), cooling to the final temperature Tf
- m₂c₂(Tf − T₂)
- — Heat gained by the cooler object (mass m₂, specific heat c₂, initial temperature T₂), warming to the final temperature Tf
When to use it: Use to find an unknown final temperature, mass, or specific heat capacity when two substances at different temperatures reach thermal equilibrium together.
Worked Example
Finding a final mixing temperature
0.3 kg of water at 80°C is mixed with 0.5 kg of water at 20°C (c = 4200 J/(kg·°C) for both). Find the final temperature.
Why Does This Work?
This follows directly from conservation of energy: assuming no heat escapes to the surroundings, all the heat energy leaving the hotter object must go somewhere — and that somewhere is exactly the cooler object it is in contact with, so the two heat quantities must be equal.
Real-Life Example
Tempering glass or metal by quenching
Hot metal is sometimes rapidly cooled ("quenched") in water or oil to change its properties.
The heat lost by the hot metal exactly equals the heat gained by the quenching liquid — engineers use this exact relationship (law of heat exchange) to calculate how much quenching liquid is needed, and how much its temperature will rise.
Practice
0.2 kg of water at 90°C is mixed with 0.4 kg of water at 15°C (same c for both). Find the final temperature.
HardCommon mistake
Forgetting that this law assumes NO heat is lost to the surroundings (a perfectly insulated system) — in real experiments, some heat always escapes, which is why practical calorimetry uses insulated containers to minimise this loss.
Quick Review
- Heat lost by the hotter object = heat gained by the cooler object (assuming no loss to surroundings).
- m₁c₁(T₁ − Tf) = m₂c₂(Tf − T₂) lets you solve for an unknown final temperature or other quantity.
- This is the basis of calorimetry, used to measure specific heat capacities experimentally.