Thermal Equilibrium Calculator

Background.

Drop a hot metal block into a cup of cool water and the two drift toward a shared temperature. The block cools, the water warms, and once they match, heat stops flowing between them. That common temperature is thermal equilibrium, and if no heat escapes to the surroundings it can be predicted before the block ever touches the water.

The prediction is energy bookkeeping. Every joule the hot body gives up, the cold body absorbs. How far each one's temperature moves depends on its heat capacity, mass times specific heat, which is the number of joules needed to shift its temperature by one kelvin. The final temperature is therefore an average of the two starting temperatures weighted by heat capacity, not by mass alone.

That weighting explains results that can look odd at first. Water's specific heat, about 4,186 J/(kg·K), is several times that of common metals, so when equal masses of metal and water meet, the metal's temperature swings much further than the water's. Two bodies with equal heat capacities, on the other hand, meet exactly halfway, however different their masses.

This is the method of mixtures from school calorimetry, run forwards. It is an idealized model: the pair is sealed off from everything else, specific heats stay constant, and neither substance melts, boils or reacts. A real cup absorbs some heat itself and the mixture loses more to the air. In the usual experiment, a hot object dropped into room-temperature water, both effects pull the measured final temperature below the prediction.

What is thermal equilibrium calculator?

Thermal equilibrium is the condition in which bodies in thermal contact have reached the same temperature, so no net heat flows between them. The zeroth law of thermodynamics adds that two bodies each in equilibrium with a third are in equilibrium with each other, which is what makes a thermometer reading meaningful. The quantity here is the final common temperature of two bodies that exchange heat only with one another, each with a constant specific heat.

How to use this calculator.

1. 01For each body enter the mass in kilograms; divide grams by 1,000.
2. 02Enter each specific heat in J/(kg·K). A value quoted in J/(g·°C) is multiplied by 1,000, so water's 4.186 becomes 4,186. Values per °C go in unchanged, because a Celsius degree and a kelvin are the same size.
3. 03Enter both starting temperatures in kelvin by adding 273.15 to the Celsius reading. Zero and negative values are refused, as they would be physically impossible.
4. 04Read the equilibrium temperature in kelvin and subtract 273.15 for Celsius. It always lies between the two starting temperatures; if it sits suspiciously close to one of them, check that both specific heats use the same units.

The formula.

The starting point is conservation of energy: heat lost by the hot body equals heat gained by the cold one, m₁c₁(T₁ − T_f) = m₂c₂(T_f − T₂). Solving for the final temperature gives T_f = (m₁c₁T₁ + m₂c₂T₂) ÷ (m₁c₁ + m₂c₂). Here m is mass in kilograms, c is specific heat in J/(kg·K), T is initial temperature in kelvin, and each product m·c is a heat capacity in joules per kelvin. Written that way the result is plainly a weighted average: it always falls between T₁ and T₂, closer to whichever body has the larger heat capacity. It does not matter which body you label 1 or which one starts hotter. The heat exchanged is m₁c₁ times the hot body's temperature drop, and the same amount appears in the cold body's rise.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, University Physics Volume 2, section 1.4, Calorimetry.
2. [2]IUPAC Gold Book, thermal equilibrium.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.