Thermal Equilibrium Calculator

What is thermal equilibrium?

When two materials at different temperatures are brought together, heat flows from the hotter to the colder. The flow continues until the two temperatures are equal. That common temperature is the equilibrium temperature. If the system is insulated, the heat given off by the hot material equals the heat taken in by the cold one:

m₁ × c₁ × (T − T₁) = m₂ × c₂ × (T₂ − T)

Solving for the final temperature:

T = (m₁ × c₁ × T₁ + m₂ × c₂ × T₂) / (m₁ × c₁ + m₂ × c₂)

The equilibrium temperature is an average weighted by heat capacity (m × c), and it always lies between the two initial temperatures.

Example: mixing water

2 kg of water at 20 °C is mixed with 1 kg at 80 °C. The materials are the same, so the specific heats cancel: T = (2 × 20 + 1 × 80) / 3 = 40 °C. The heat exchanged is 2 × 4186 × 20 = 167 440 J.

Example: hot metal into water

200 g of aluminium at 100 °C is dropped into 500 g of water at 20 °C. T = (0.5 × 4186 × 20 + 0.2 × 900 × 100) / (2093 + 180) = 26.3 °C. The metal cools by 74 degrees while the water warms by only 6, because the heat capacity of the water is much larger.

Finding specific heat by calorimetry

This is how the specific heat of an unknown material is measured: heat the sample, drop it into water of known mass and temperature, and read the equilibrium temperature. The second tab of the calculator does this calculation.

Limits

The calculation is for an insulated system and materials that do not change phase. If ice melts or water evaporates, the heats of fusion and vaporisation have to be included separately. Heat taken up by the container and heat lost to the surroundings are neglected.