Ranking: specific heats

Problem

Four materials are heated, each $1\ \text{kg}$ from $20\ ^\circ\text{C}$ to $30\ ^\circ\text{C}$:

• (A) water, $c = 4186\ \text{J/(kg·K)}$;
• (B) aluminium, $c = 900\ \text{J/(kg·K)}$;
• (C) iron, $c = 450\ \text{J/(kg·K)}$;
• (D) lead, $c = 130\ \text{J/(kg·K)}$.

Rank the amount of heat needed from smallest to largest.

Solution

Formula. The heat needed to raise the temperature without a change of state is

$Q = m\,c\,\Delta T$

Here the mass $m = 1\ \text{kg}$ and the change $\Delta T = 10\ \text{K}$ are the same for all materials, so $Q$ is proportional only to the specific heat $c$: $Q \propto c$.

Comparing the specific heats.

$130 < 450 < 900 < 4186$

so lead requires the least heat and water much more (water has an exceptionally high specific heat).

Numerical values (with $m\,\Delta T = 10\ \text{kg·K}$):

• $Q_D = 10\cdot 130 = 1300\ \text{J}$
• $Q_C = 10\cdot 450 = 4500\ \text{J}$
• $Q_B = 10\cdot 900 = 9000\ \text{J}$
• $Q_A = 10\cdot 4186 = 41860\ \text{J}$

Ranking.

$\ev{D < C < B < A}$

Links

Topics: Thermodynamics Concepts: Specific heat and heat capacity Skills: Ranking reasoning