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MHT-CET Physics · Thermal Properties of Matter

Calorimetry: Heat Lost Equals Heat Gained

When hot and cold bodies share heat in isolation, the heat one loses equals the heat the other gains, Q = mcΔT for a temperature change and Q = mL for a change of state; a body's heat capacity mc is the heat that raises it by one degree.

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Why this matters

4 PYQs, none HARD: the definition of thermal capacity, a metal's heat capacity from the water it warms, the temperature rise of a bullet that stops in a wall, and ice mixed with warm water. One card.

Concept 1 of 1: Mixing, Melting and Heat From Motion

Write the heat each body exchanges and set losses equal to gains. For a temperature change Q = mcΔT; the product mc is the body's heat capacity (thermal capacity), the heat for a one-degree rise. For melting, Q = mL with L = 80 cal/g for ice. When ice meets warm water, first check whether the warm water can melt all the ice: if not, the mixture stays at 0 °C. Mechanical energy turned into heat enters the same balance, divided by J = 4.2 J/cal when the heat is in calories.

Definition

  • Q=mcΔTQ = mc\Delta T; heat capacity C=mcC = mc (heat per degree); water c=1c = 1 cal/g °C =4200= 4200 J/kg K.
  • Change of state: Q=mLQ = mL; ice L=80L = 80 cal/g.
  • Heat lost = heat gained. Check first whether all the ice can melt.
  • Bullet stopping with fraction f of its KE as heat: msΔT=f mv22Jms\Delta T = \dfrac{f\,mv^2}{2J} (50% ⇒ ΔT=v24Js\Delta T = \dfrac{v^2}{4Js}).

Heat balance

m1c1(T1−T)=m2c2(T−T2),Q=mLm_1c_1(T_1 - T) = m_2c_2(T - T_2), \qquad Q = mL

Worked example

A metal at 850 K is dropped into 1 kg of water at 300 K; they settle at 350 K. Heat capacity of the metal?
Practice this conceptself-check · 1 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 4th May Shift 2 · Q20Easy

Example 1 · Thermal Properties of Matter · Calorimetry, Latent Heat, and Heat Capacity

A lead bullet moving with velocity 'v' strikes a wall and stops. If 50% of its energy is converted into heat, then the increase in temperature is (s=s = specific heat of lead)

Averaging temperatures when ice is present

Melting absorbs heat without raising the temperature. 540 g of water at 80 °C gives up exactly the 43 200 cal that 540 g of ice needs to melt, so the mixture ends at 0 °C, not at an average.

Confusing heat capacity with specific heat

Specific heat c is per gram; heat capacity mc is for the whole body. 'Heat that raises the BODY by 1 °C' is heat capacity.
Drill 3 more on mixing, melting and heat from motion

Summary — formulas & gotchas at a glance

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Formulas (1)

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