Calorimetry: Heat Lost = Heat Gained

The core calorimetry principle: in an insulated system, the heat lost by the hotter body equals the heat gained by the colder one.

What it means

Calorimetry rests on the first law of thermodynamics: energy is conserved, so in an insulated container heat cannot vanish — every joule the hot object loses must be picked up by the cold one. The negative sign captures the sign convention: q is negative for the object releasing heat and positive for the one absorbing it, so the two add to zero. Solving this equation for the final temperature is how a calorimeter turns a measured temperature change into an unknown heat or an unknown specific heat. When to use it: mixing problems (hot meets cold), coffee-cup calorimetry, and any 'find the unknown specific heat or final temperature' setup with an insulated system.

Variables

Assumptions

• The calorimeter is perfectly insulated — no heat leaks to or from the surroundings.
• Both substances reach the same final equilibrium temperature.
• Specific heats are constant over the temperature range.
• No phase changes occur during mixing.

Worked example

Given:

• 100. g hot water at 80.0°C, 200. g cold water at 20.0°C
• Same c for both, so it cancels
• Tf = final equilibrium temperature

1. Energy conservation: q_hot = −q_cold (heat lost by hot water = heat gained by cold).

2. Expand with q = mcΔT: m_hot × c × (Tf − 80.0) = −m_cold × c × (Tf − 20.0).

3. Cancel c from both sides: 100.(Tf − 80.0) = −200.(Tf − 20.0).

4. Expand: 100 Tf − 8000 = −200 Tf + 4000.

5. Collect: 100 Tf + 200 Tf = 8000 + 4000 → 300 Tf = 12,000.

6. Tf = 12,000 / 300 = 40.0°C.