What this calculator does
When two substances at different temperatures are brought together, heat flows from the warmer one to the cooler one until both reach the same final temperature: thermal equilibrium. That final temperature depends on each substance's mass, specific heat capacity and starting temperature, not just a simple average of the two starting temperatures.
The calculation rests on one physical principle: the heat lost by the warmer substance equals the heat gained by the cooler one, since no energy leaves the system. Setting m1·c1·(T1 − Tf) equal to m2·c2·(Tf − T2) and solving for Tf gives the equilibrium temperature directly.
The formula
Enter the mass, specific heat capacity and starting temperature for each of the two substances. The calculator solves Tf = (m1·c1·T1 + m2·c2·T2) / (m1·c1 + m2·c2), a mass-and-heat-capacity-weighted average of the two starting temperatures, then shows how far each substance moved to reach it and the amount of heat energy that changed hands.
| Term | Meaning |
|---|---|
| Specific heat capacity (c) | The energy needed to raise 1 kg of a substance by 1°C, in J/(kg·°C). Water is 4186; other materials are lower. |
| Thermal equilibrium | The state reached once both substances share the same final temperature and net heat flow between them stops. |
| Heat transferred (Q) | The energy that moved from the warmer substance to the cooler one while reaching equilibrium: m·c·ΔT for either side, since both sides are equal by definition. |
The inputs explained
| Field | What to enter |
|---|---|
| Mass of substance 1 (kg) | The mass of the first (typically warmer) substance. |
| Specific heat of substance 1 (J/(kg·°C)) | The specific heat capacity of the first substance. Water is 4186 J/(kg·°C); use the actual figure for other materials. |
| Initial temperature of substance 1 (°C) | The starting temperature of the first substance. |
| Mass of substance 2 (kg) | The mass of the second (typically cooler) substance. |
| Specific heat of substance 2 (J/(kg·°C)) | The specific heat capacity of the second substance. |
| Initial temperature of substance 2 (°C) | The starting temperature of the second substance. |
When to use it
Mixing hot and cold water
Pouring a known mass of hot water into a known mass of cooler water, both with the same specific heat capacity, is the simplest case: the final temperature lands closer to whichever mass is larger.
Dropping a hot object into water
A hot metal object plunged into a mass of water reaches equilibrium quickly; because metals have a much lower specific heat capacity than water, the final temperature usually sits much closer to the water's original temperature than the object's.
Checking a calorimetry experiment
A school or lab calorimetry setup calculates an unknown specific heat capacity by measuring the equilibrium temperature reached; working the equilibrium temperature forward from assumed values is a useful sanity check against the measured result.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the final temperature changes with the cooler substance's starting point
A fixed 1 kg of water starting at 80°C, mixed with 2 kg of water at a range of cooler starting temperatures.
| Substance 2 starting temperature | Final equilibrium temperature | Heat energy transferred |
|---|---|---|
| 10 °C | 33.33 °C | 195.3 kJ |
| 20 °C | 40.00 °C | 167.4 kJ |
| 40 °C | 53.33 °C | 111.6 kJ |
| 60 °C | 66.67 °C | 55.8 kJ |
| 80 °C | 80.00 °C | 0.0 kJ |
Questions
Why is the final temperature not just the average of the two starting temperatures?
A simple average only applies when both substances have equal mass and equal specific heat capacity. Otherwise the final temperature is weighted more heavily toward whichever substance has the larger mass × specific heat product, since that substance takes more energy to shift by each degree.
What if the two substances have very different specific heat capacities?
A substance with a low specific heat capacity, such as most metals, changes temperature easily for a given amount of heat, so mixing it with a high-specific-heat substance like water usually pulls the final temperature much closer to the water's original temperature.
Does this account for heat lost to the surroundings?
No. This assumes an idealised closed system where all the heat lost by the warmer substance is gained by the cooler one, with none escaping elsewhere. Real setups lose some heat to their container and surroundings, which is why lab calorimetry experiments use insulated calorimeters to minimise that loss.
How is this different from the heat capacity calculator?
The heat capacity calculator finds the energy needed to change ONE substance by a known temperature change. This calculator instead solves for the unknown final temperature reached when TWO substances at different starting temperatures exchange heat until they match.
To work out the heat energy for a single substance with a known temperature change, use the heat capacity calculator. For water specifically, the heat capacity of water calculator uses its fixed specific heat automatically.