18-4 Absorption of Heat

18-4 Absorption of Heat#

Prompts

  • What is heat \(Q\)? How does it differ from temperature? From thermal (internal) energy?

  • Write \(Q = mc\,\Delta T\). What is specific heat \(c\)? Why does water have a high specific heat?

  • Convert 1 cal to joules. How much heat is needed to raise 1 kg of water by 1°C?

  • During a phase change (e.g., ice melting), does the temperature change? Write the equation relating \(Q\), latent heat \(L\), and mass \(m\).

  • You add heat to ice at \(-10°C\) until you have water at \(20°C\). What steps must you calculate? In what order?

Lecture Notes#

Overview#

  • Heat \(Q\) is energy transferred between a system and its surroundings because of a temperature difference.

  • Temperature change (no phase change): \(Q = mc\,\Delta T\), where \(c\) is the specific heat.

  • Phase change (melting, boiling): temperature stays constant; \(Q = Lm\), where \(L\) is the latent heat.

Heat, temperature, and thermal energy#

  • Thermal energy: internal energy from random motions of atoms/molecules.

  • Temperature: measure of “hotness”; related to average kinetic energy.

  • Heat \(Q\): energy transferred as heat. \(Q > 0\) = absorbed; \(Q < 0\) = released.

Units: J (joules), cal (calories), kcal. 1 cal ≈ 4.19 J; 1 kcal = 1000 cal.

Heat capacity and specific heat#

When heat \(Q\) is added to an object without a phase change:

(154)#\[ Q = C\,\Delta T \quad \text{or} \quad Q = mc\,\Delta T \]

  • \(C\): heat capacity (J/K or cal/°C).

  • \(c\): specific heat—heat capacity per unit mass (J/(kg·K) or cal/(g·°C)).

  • Water: \(c \approx 4186\) J/(kg·K) = 1 cal/(g·°C)—unusually high; water stores heat well.

Substance

\(c\) (J/(kg·K))

Water

4187

Aluminum

900

Copper

386

Lead

128

Poll: Which phase for largest \(\Delta T\)?

Suppose you have 1 J of energy to give to a substance via heat. A temperature vs. heat-added graph for the substance is given. Which phase should you add the energy to in order to get the largest \(\Delta T\)?

[FIGURE: T vs Q graph showing solid (steep slope), liquid (flat during melting), liquid (moderate slope), gas (shallow slope)—typical pattern where gas phase has smallest slope = smallest dT/dQ = largest dT per dQ for given mass]

  • (A) Solid

  • (B) Liquid

  • (C) Gas

  • (D) Cannot tell

Latent heat (heat of transformation)#

During a phase change, temperature stays constant while energy is added or removed:

(155)#\[ Q = Lm \]

  • \(L\): heat of transformation (J/kg or cal/g).

  • \(L_F\) (fusion): solid ↔ liquid (melting/freezing).

  • \(L_V\) (vaporization): liquid ↔ gas (boiling/condensing).

Water: \(L_F \approx 333\) kJ/kg; \(L_V \approx 2256\) kJ/kg. Vaporization requires much more energy per kg than fusion.

Heat transfer across a phase change#

When heating from one phase to another (e.g., ice at \(-10°C\) → water at \(20°C\)), add the steps:

  1. Warm to phase-change temperature: \(Q_1 = mc\,\Delta T\) (ice \(-10°C\) → \(0°C\)).

  2. Phase change: \(Q_2 = Lm\) (melt ice at \(0°C\)).

  3. Warm beyond: \(Q_3 = mc\,\Delta T\) (water \(0°C\) → \(20°C\)).

(156)#\[ Q_{\text{total}} = Q_1 + Q_2 + Q_3 \]

Use the appropriate \(c\) and \(L\) for each step.

Poll: Ice-water mixture equilibrium

The specific heat of water is 4.19 kJ/(kg·K) and the heat of fusion of water is 333 kJ/kg. You have 1 kg of ice at its melting point and 1 kg of water at temperature \(T_i\). What should \(T_i\) be so that when mixed, you have water in equilibrium at its freezing point?

  • (A) 7°C

  • (B) 13°C

  • (C) 20°C

  • (D) 93°C

  • (E) None of these is close

Poll: Specific heat

The same heat \(Q\) warms 1 kg of aluminum by 10°C and 1 kg of water by 2°C. Which has the greater specific heat?

  • (A) Aluminum

  • (B) Water

  • (C) Equal

Example: T vs Q graph—extract parameters#

Example: T vs Q graph—extract parameters

An experiment measures the temperature of a 500 g substance while steadily supplying heat. The figure shows the results.

[FIGURE: T vs Q graph with labeled phases—solid warming, melting plateau, liquid warming, boiling plateau, gas warming; axes labeled]

Find (a) specific heat of the solid phase, (b) specific heat of the liquid phase, (c) melting and boiling temperatures, (d) heats of fusion and vaporization.

Solution: (a) \(c = Q/(m\,\Delta T)\) from slope in solid region. (b) Same in liquid region. (c) Read from plateaus. (d) \(L = Q/m\) from width of each plateau.

Summary#

  • \(Q = mc\,\Delta T\)—temperature change; \(c\) = specific heat.

  • \(Q = Lm\)—phase change; \(L\) = latent heat (fusion or vaporization).

  • 1 cal ≈ 4.19 J; water has high \(c\) and high \(L\).

  • For processes crossing phase boundaries: sum heat for each step (warm → phase change → warm).