Thermal Equilibrium: the Final Temperature of a Mixture

Finds the temperature two bodies settle at when they touch. The heat lost by the hotter one equals the heat gained by the cooler one. Specific heats are in J/(g·K): 4.2 for water, 0.45 for iron.

Explanation

Put a hot body against a cold one and they end up at the same temperature. That temperature follows from a single statement: heat lost equals heat gained.

T=m1c1T1+m2c2T2m1c1+m2c2T = \dfrac{m_1 c_1 T_1 + m_2 c_2 T_2}{m_1 c_1 + m_2 c_2}

Here mm is mass and cc is specific heat. The product mcmc is the heat capacity, the energy needed to raise the body by one kelvin. So the formula is nothing but an average weighted by heat capacity.

Example

Drop 50 g of iron at 100 °C into 100 g of water at 20 °C. Water has a specific heat of 4.2 J/(g·K), iron only 0.45.

T=100×4.2×20+50×0.45×100100×4.2+50×0.45=8400+2250442.5=24.1 °CT = \dfrac{100 \times 4.2 \times 20 + 50 \times 0.45 \times 100}{100 \times 4.2 + 50 \times 0.45} = \dfrac{8400 + 2250}{442.5} = 24.1\ °C

Iron at boiling point, and the water warms by only four degrees.

Why water refuses to warm up

The specific heat of water, 4.2, is more than nine times that of iron. Gram for gram and degree for degree, water swallows nine times the energy.

In the example the water's heat capacity is 420 against the iron's 22.5, a factor of nineteen. The final temperature is dragged almost all the way to the water's side.

This stubbornness of water is what makes Earth's climate liveable. The oceans absorb the swings of day, night and season, which is why coasts are mild. Inland deserts, with no water to buffer them, can swing 30 degrees between noon and midnight.

Where it is used