CALORIMETRY
Tf is the heat-capacity–weighted average; heat lost = heat gained, audited to 1e-12.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A warmer metal block meets cooler water in an insulated container; the block gives heat to the water until both share one temperature.
One rule
Because the container is insulated, energy has nowhere to go: every joule the block loses, the water gains. Heat lost equals heat gained. Each side is mass times specific heat times temperature change, and setting them equal gives one clean equation for the one unknown final temperature.
Solve
The final temperature is a weighted average of the two starting temperatures — weighted by heat capacity, which is mass times specific heat. Water’s specific heat is enormous, several times that of any metal, so a kilogram of it barely warms while the hot block plunges. The answer sits far closer to the water’s side.
Equilibrium
Watch both thermometer columns close the gap in the physically correct directions until they meet. The shared temperature lies between the starts and is pulled toward the larger heat capacity.
Audit
The signed ledger C_block(Tf−T_block)+C_water(Tf−T_water)=0 closes across ordinary, reversed, and equal-temperature cases. Tf is between the starts, with equality only when the starts are equal.