Calorimetry and Mixtures
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Direct answer
When two bodies at different temperatures are mixed, heat lost by the hotter equals heat gained by the colder — the principle of mixtures, written m_h c_h (T_h − T_eq) = m_c c_c (T_eq − T_c), where the same c-form is used for each material and every phase change inserts an mL term. The anchor values from NCERT: specific heat of water 4186 J/kg/K (about 4.2 kJ), latent heat of fusion of ice 3.35 × 10^5 J/kg, and latent heat of vaporisation of steam 2.26 × 10^6 J/kg — the last is why a steam scald is far worse than a water scald, since every gram of steam dumps 2260 J while condensing before it even begins to cool. If the computed equilibrium temperature crosses 0 °C or 100 °C, a phase change is occurring and the naive average-temperature answer is wrong; working in steps (heat released to cool, then to condense or freeze) keeps the arithmetic honest.
What you must remember
- Principle of mixtures: heat lost = heat gained, always written with positive temperature drops on each side; it is energy conservation with no work done.
- Water's specific heat: 4186 J/kg/K at room temperature (NCERT Table 11.1) — the highest of common liquids, the reason water is a coolant and coastal climates are mild.
- Latent heats: fusion of ice 3.35 × 10^5 J/kg; vaporisation of water 2.26 × 10^6 J/kg; latent heat terms appear without any temperature change.
- Water equivalent: W = m c /c_w is the mass of water that would absorb the same heat as a calorimeter; add it to the cold side of the balance.
- Phase-crossing rule: if the trial answer lands below 0 °C with ice present, all the water may freeze; if above 100 °C with steam present, all may not condense — check whether the available heat can complete the transition.
- Regelation and burns: pressure melts ice (refreezing on release) and steam's latent heat dominates injury severity — two NCERT-quoted applications that appear as assertion statements.
- Units discipline: mix kilograms with J/kg/K throughout; examiners plant CGS-calorie answers (1 cal = 4.186 J) among options.
A worked steam-into-water problem
Steam at 100 °C is passed into 0.5 kg of water at 20 °C contained in a copper calorimeter of water equivalent 0.02 kg, and the final temperature is 40 °C. How much steam condensed? Heat gained by water plus calorimeter = (0.5 + 0.02) × 4186 × (40 − 20) = 0.52 × 4186 × 20 ≈ 43,534 J. Each kilogram of steam supplies latent heat 2.26 × 10^6 J condensing, plus 4186 × 60 J cooling from 100 °C to 40 °C: total 2.26 × 10^6 + 251,160 ≈ 2.51 × 10^6 J/kg. Mass condensed = 43,534/2.51 × 10^6 ≈ 0.0173 kg, about 17 grams. The instructive comparison: had 17 g of water at 100 °C been poured in instead, it would deliver only 0.017 × 4186 × 60 ≈ 4,270 J — a tenth of the steam's effect. That one-line contrast, steam versus hot water gram for gram, is the whole clinical point of the latent-heat chapter and appears in NEET as both numerical and statement questions.
How the exam frames it
The standard trap is the trial-equilibrium shortcut: averaging temperatures weighted by mass works only when no phase change intervenes, and NEET deliberately sets mixtures of ice and water where the "average" lands below 0 °C — impossible until you account for freezing. The bookkeeping rule: compute heat available from the hot side cooling to the transition temperature, compare with the latent heat needed to melt or freeze everything, and only then decide the final state (all ice, mixture at 0 °C, all water). A second planted trap is the calorimeter's own heat capacity presented as "water equivalent 20 g" — it simply adds to the water mass, but candidates drop it. Watch also for "specific heat of water = 1" in calorie problems; the arithmetic then mimics CGS textbooks, and the answer must be converted back to joules if asked.
Frequently asked questions
What is the principle of mixtures in calorimetry?
In an insulated system, the heat lost by the hotter body equals the heat gained by the colder body, m_h c_h ΔT_h = m_c c_c ΔT_c, including latent heat terms when phases change.
Why does steam at 100 degrees cause worse burns than water at 100 degrees?
Every gram of steam first releases 2260 J of latent heat while condensing and then cools like water, so it deposits roughly 540 calories more per gram than water at the same temperature.
What is water equivalent of a calorimeter?
The mass of water that would absorb the same amount of heat as the calorimeter for the same temperature rise, W = mc/c_w; it is added to the mixed water's mass in heat balances.
When does the simple mixture formula fail?
When the computed equilibrium temperature crosses a phase-change temperature (0 °C or 100 °C), because latent heat absorption or release must be accounted for, and part or all of a substance may change phase.
Why is water used as a coolant in engines and radiators?
Its specific heat capacity (4186 J/kg/K) is exceptionally high, so a given mass of water carries away a large amount of heat for a modest temperature rise.