Physics
Calorimetry calculator: heat, thermal equilibrium, expansion
Pick the relation and clear the quantity you are after: the calculator inverts the formula for you. Mind the distinction the whole chapter rests on — sensible heat changes the temperature, latent heat changes the state and leaves the thermometer where it was.
Sensible heat and latent heat: the distinction that settles the problem
Almost every mistake in calorimetry comes from treating heat as though it were temperature. Heat is energy in transit, measured in joules; temperature is a state, measured in degrees. The link between them is Q = m·c·ΔT, and it already carries two warnings: the same heat produces different temperature steps in different masses, and different steps in equal masses of different substances. That is exactly what specific heat is — the energy one degree costs per kilogram — and water's is among the highest you will meet.
Then there is the case where heat goes in and the temperature does not move. During a change of state the energy goes into pulling molecules apart rather than shaking them harder: ice at zero becomes water at zero, and the thermometer sits still until the last fragment has melted. The latent heat of fusion of water is 334 kJ/kg, as much as heating that same water by eighty degrees; the latent heat of vaporisation is nearly seven times larger, which is why sweating cools so effectively.
The rest of the chapter applies the same balances. In a calorimeter the heat given up by the hot body is the heat taken in by the cold one, and the final temperature always falls between the two starting ones, pulled towards whichever has the larger thermal capacity. In expansion, temperature acts on the dimensions rather than on the internal energy, with coefficients of a few millionths per degree that nonetheless amount to centimetres on a bridge or a rail. And in conduction the heat does not accumulate but passes through: Fourier's law says how much per second, and it is the arithmetic behind every insulation decision.
Common mistakes
- Confusing heat with temperature. Two bodies at the same temperature can hold very different amounts of heat if their masses or specific heats differ.
- Using Q = m·c·ΔT during a change of state. There the temperature does not change: the right formula is Q = m·L, and serious problems have to be split into stages, one per phase.
- Forgetting that the two ΔT in a calorimeter balance have opposite signs. The hot body cools, the cold one warms, and the heats exchanged sum to zero.
- Entering temperatures in kelvin where a difference is wanted and in Celsius where an absolute value is. On this page only differences appear, so degrees Celsius are fine throughout.
- Mixing up the linear and volume coefficients of expansion. For a solid the second is about three times the first, and using the wrong one is off by a factor of three.
Frequently asked questions
What is the difference between heat and temperature?
Heat is energy passing from one body to another, measured in joules; temperature is a state of the body, measured in degrees. A lukewarm bath holds far more heat than a scalding cup of coffee even though it is colder: mass and specific heat come into it too.
Why does the temperature not rise while something melts?
Because the energy supplied goes into breaking the bonds of the solid lattice, not into faster thermal motion. As long as ice and water sit together the thermometer stays at 0 °C, and all the heat going in is latent heat.
Why does water heat up so slowly?
Because its specific heat is unusually high: 4186 J per kilo per degree, against about 450 for iron. It takes nearly ten times the energy to warm a kilo of water as a kilo of iron, which is why maritime climates are milder than continental ones.
Where does the equilibrium temperature of two bodies fall?
Always between the two starting temperatures, and closer to the one with the larger thermal capacity — that is, the larger mass times specific heat. A red-hot spoon in a glass of water barely warms the water, because its thermal capacity is tiny by comparison.
Why is the volume coefficient three times the linear one?
Because the body expands in all three directions at once. Expanding (1 + λΔT)³ and dropping the higher-order terms, which are minute, leaves 1 + 3λΔT: hence α ≈ 3λ.
How this calculation works
Sensible heat: Q = m·c·ΔT, with c the specific heat in J/(kg·K); the thermal capacity of the body is C = m·c and the water equivalent is C/4186. Latent heat: Q = m·L, exchanged at constant temperature during a change of state. Thermal equilibrium in an ideal calorimeter: m₁c₁(T_e − T₁) + m₂c₂(T_e − T₂) = 0, hence T_e = (m₁c₁T₁ + m₂c₂T₂)/(m₁c₁ + m₂c₂); the result always lies between T₁ and T₂. Heat delivered at constant power: Q = P·t. Linear expansion: ΔL = λ·L₀·ΔT, with L = L₀(1 + λΔT). Volume expansion: ΔV = α·V₀·ΔT, and for a solid α ≈ 3λ because the body expands in all three directions. Steady-state conduction, Fourier's law: P = k·A·ΔT/d, with thermal resistance R = d/(k·A) and P = ΔT/R. Temperatures enter only as differences, so degrees Celsius and kelvin are interchangeable.