Chapter 11: Thermal Transformation (Numerical Problems)

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Chapter Overview & SLOs

What formulas are used for thermal expansion? Linear expansion is calculated as ΔL = α L₀ ΔT, where α is coefficient of linear expansion, L₀ is original length, and ΔT is temperature change. Volume expansion uses ΔV = β V₀ ΔT, where β = 3α is the coefficient of volume expansion.

How do we calculate heat energy during temperature changes? For heating or cooling without phase change, Q = mcΔT, where m is mass, c is specific heat capacity, and ΔT is temperature change.

How do we calculate heat energy during state changes? For melting/freezing: Q = mLf, where Lf is latent heat of fusion (ice: 3.3 × 10⁵ J/kg). For vaporization/condensation: Q = mLv, where Lv is latent heat of vaporization (water: 2.26 × 10⁶ J/kg).

How do we calculate total energy for a complete phase transformation? For ice at subzero temperature to steam above 100°C, sum five steps:

1. Heating ice from initial T to 0°C: Q₁ = m c_ice ΔT
2. Melting ice at 0°C: Q₂ = m Lf
3. Heating water from 0°C to 100°C: Q₃ = m c_water ΔT
4. Vaporizing water at 100°C: Q₄ = m Lv
5. Heating steam from 100°C to final T: Q₅ = m c_steam ΔT

Total energy = Q₁ + Q₂ + Q₃ + Q₄ + Q₅

Example - Total energy calculation: For 4 kg of ice at -20°C becoming steam at 120°C, total energy ≈ 1.236 × 10⁷ J.

Example - Linear expansion of steel bar: For a 1.5 m steel bar heated by 90 K, ΔL = (1.2 × 10⁻⁵ K⁻¹)(1.5 m)(90 K) = 1.62 × 10⁻³ m = 1.62 mm.

Example - Volume expansion and overflow: For a 1000 cm³ solid cube with α = 9 × 10⁻⁶ K⁻¹, β = 3α = 27 × 10⁻⁶ K⁻¹. Volume increase over 100 K: ΔV = (27 × 10⁻⁶)(1000)(100) = 2.7 cm³.

Example - Heat required for melting ice: To melt 15 kg of ice, Q = (15 kg)(3.3 × 10⁵ J/kg) = 4.95 × 10⁶ J.

Example - Heat required for vaporizing water: To vaporize 7 kg of water, Q = (7 kg)(2.26 × 10⁶ J/kg) = 1.58 × 10⁷ J.

These notes are strictly aligned with the Student Learning Outcomes (SLOs) for the FBISE 2026 annual examination.

• How do we calculate the expansion of steel bars or railroad tracks? For a 1.5 m steel bar heated by 90 K, the increase in length is $\Delta L = (1.2 \times 10^{-5}\text{ K}^{-1})(1.5\text{ m})(90\text{ K}) = 1.62\text{ mm}$; the final length is the sum of the original length and this increase.
• How do we determine volume expansion and overflow? For a 1000 cm³ solid cube with $\alpha = 9 \times 10^{-6}\text{ K}^{-1}$, we first find $\beta = 3\alpha = 27 \times 10^{-6}\text{ K}^{-1}$; the volume increase over 100 K is 2.7 cm³. Similarly, orange juice in a bottle will overflow by its calculated volume expansion if the container expansion is negligible.
• How is the heat required for melting or boiling determined? To melt 15 kg of ice at its melting point, $Q = (15\text{ kg})(3.3 \times 10^5\text{ J/kg}) = 4.95 \times 10^6\text{ J}$. To vaporize 7 kg of water, $Q = (7\text{ kg})(2.26 \times 10^6\text{ J/kg}) = 1.58 \times 10^7\text{ J}$.
• How do we calculate the total energy for a complete phase transformation? For 4 kg of ice at -20°C becoming steam at 120°C, we sum the heat for five steps: heating ice, melting ice, heating water, vaporizing water, and heating steam, totaling approximately $1.236 \times 10^7\text{ J}$.

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