Thermal Equilibrium Calculator
Enter the mass, material, and starting temperature of two objects to find the temperature they reach at thermal equilibrium and the heat transferred between them. Choose from 18 common substances or enter a custom specific heat capacity. Switch between Celsius and Fahrenheit, or metric and imperial mass units. Use the mode selector to reverse-solve for an unknown initial temperature or the thermal capacity of one body.
Formula
Worked example
A 0.5 kg block of aluminum (cp = 897 J/kg·K) at 20 °C is placed in 1.0 kg of water (cp = 4182 J/kg·K) at 80 °C. mc1 = 0.5 × 897 = 448.5 J/K; mc2 = 1.0 × 4182 = 4182 J/K. Tf = (448.5 × 20 + 4182 × 80) / (448.5 + 4182) = (8970 + 334560) / 4630.5 = 343530 / 4630.5 = 74.2 °C. Heat transferred: Q = 448.5 × (74.2 - 20) = 448.5 × 54.2 = 24309 J (about 24.3 kJ).
What is thermal equilibrium?
Thermal equilibrium is the state two objects reach when they have been in contact long enough that there is no longer any net flow of heat between them. At that point both objects share the same temperature. The process is governed by the Zeroth Law of Thermodynamics, which states that if each of two systems is in thermal equilibrium with a third, they are in thermal equilibrium with each other. In everyday life you see this whenever a cold drink warms up to room temperature, or a hot pan cools on the counter: both objects exchange heat until they settle at a common temperature.
The thermal equilibrium formula
The final temperature is found by equating the heat lost by the hotter body to the heat gained by the cooler one. For two objects with no heat lost to the surroundings: Tf = (m1 c1 T1 + m2 c2 T2) / (m1 c1 + m2 c2), where m is mass (kg), c is specific heat capacity (J/kg·K), and T is initial temperature (°C or K). The product mc is called the thermal capacity of the body: a high value means the body resists temperature change. The heat transferred is Q = m c (Tf - Ti) for either body; both should give the same magnitude if heat is conserved.
Specific heat capacity and why it matters
Specific heat capacity (c) is the energy required to raise 1 kg of a material by 1 degree Kelvin. Water has an unusually high value (4182 J/kg·K) compared with most metals, which is why it is used in cooling systems, hot water bottles, and heating radiators. Aluminum (897 J/kg·K) and copper (385 J/kg·K) are far lower, so metal cookware heats up and cools down much faster than an equal mass of water. When two objects with very different specific heats mix, the equilibrium temperature sits much closer to the starting temperature of the body with the higher thermal capacity.
Assumptions and limitations
This calculator assumes a perfectly insulated (adiabatic) system: all heat flows between the two objects and none escapes to the environment. It also assumes no phase change occurs (no melting or boiling). If the computed equilibrium temperature crosses a melting or boiling point, latent heat must be included and the result from this calculator will be inaccurate. In practice, a perfectly insulated calorimeter approaches these conditions, but any real system loses some heat. For high-accuracy work, account for the heat capacity of the container itself and correct for conduction losses.
Specific heat capacities of common materials
| Material | Specific heat (J/kg·K) | Notes |
|---|---|---|
| Water (liquid) | 4182 | Highest of common liquids; excellent thermal buffer |
| Ice (0 °C) | 2090 | Roughly half that of liquid water |
| Steam (100 °C) | 2010 | Gas phase; lower than liquid |
| Ethanol | 2440 | Common solvent; moderate thermal capacity |
| Glycerol | 2400 | Viscous liquid; useful as antifreeze component |
| Aluminum | 897 | Light metal; good thermal conductor |
| Iron / Steel | 449 | Dense; lower capacity per kg than water |
| Copper | 385 | Excellent conductor; low specific heat |
| Glass | 840 | Borosilicate; low conductivity, moderate cp |
| Granite | 790 | Rock; thermal mass material |
| Concrete | 880 | Building material; useful for passive solar |
| Wood (oak) | 1700 | Varies with moisture content |
| Gold | 129 | Dense; very low specific heat per kg |
| Lead | 128 | Very low specific heat; rapid temperature change |
| Mercury | 140 | Liquid metal; low specific heat |
| Air (25 °C) | 1005 | Gas at constant pressure (cp) |
Values at approximately 25 °C and standard pressure unless noted. Source: CRC Handbook.
Frequently asked questions
What is the thermal equilibrium formula?
The standard formula is Tf = (m1 c1 T1 + m2 c2 T2) / (m1 c1 + m2 c2). Here m is mass, c is specific heat capacity, and T is initial temperature. The denominator is the sum of the two thermal capacities. The formula follows from setting the heat lost by the hotter object equal to the heat gained by the cooler one, assuming no losses to the environment.
What is specific heat capacity and where do I find it?
Specific heat capacity (symbol c or cp) is the amount of energy needed to raise 1 kilogram of a substance by 1 kelvin. Common values: water 4182 J/(kg·K), aluminum 897 J/(kg·K), copper 385 J/(kg·K), iron 449 J/(kg·K). You can look up values in the CRC Handbook of Chemistry and Physics, NIST databases, or engineering data books. This calculator includes 18 presets so you rarely need to enter a custom value.
Does the formula work for mixing two liquids?
Yes, provided no chemical reaction occurs and no phase change happens. For two miscible liquids such as water and ethanol, the mixture specific heat is approximately the mass-weighted average of the two pure-component values. For large mixing ratios or when the liquids are immiscible, more detailed calorimetric data may be needed.
What happens when the equilibrium temperature crosses a phase boundary?
If the computed Tf is at or above the boiling point of one component, or at or below its melting point, latent heat (the energy required to change phase) must be included. For example, mixing ice at -10 °C with hot water requires first melting the ice (absorbing 334 kJ/kg at 0 °C) before the temperatures can equalize. This calculator does not model phase changes; use a calorimetry calculator with latent heat for those scenarios.
Can I use this calculator for gases?
Yes, with care. For gases, use the specific heat at constant pressure (cp) if the gas is free to expand, or specific heat at constant volume (cv) for a rigid container. Air at 25 °C has cp of about 1005 J/(kg·K). The same formula applies. However, gases have very low mass density, so a large volume of gas carries less thermal energy than a small volume of liquid.
Why does the equilibrium temperature sit closer to one starting temperature?
The final temperature is a weighted average, where the weights are the thermal capacities (mc) of each body. A body with a high thermal capacity pulls the result toward its starting temperature because it stores more energy per degree. For example, 1 kg of water (mc = 4182 J/K) overwhelms 0.1 kg of aluminum (mc = 89.7 J/K), so the final temperature ends up very close to the water's starting temperature.