Section 7.8: Review Examples

These examples integrate heat capacity and latent heat concepts to reinforce understanding and problem-solving skills.

Example 1: Heating and Melting Ice

Calculate the total energy needed to convert 0.1 kg of ice at -5 °C to water at 25 °C. (c_ice = 2.1×10³ J/kg·K, c_water = 4.18×10³ J/kg·K, L_fusion = 3.34×10⁵ J/kg)

Heat ice to 0 °C: \( Q_1 = 0.1 \cdot 2100 \cdot 5 = 1050 \, \text{J} \)
Melt ice: \( Q_2 = 0.1 \cdot 334{,}000 = 33{,}400 \, \text{J} \)
Heat water to 25 °C: \( Q_3 = 0.1 \cdot 4180 \cdot 25 = 10{,}450 \, \text{J} \)
Total energy: \( Q_\text{total} = 44{,}900 \, \text{J} \)

Example 2: Vaporization

Find the energy required to vaporize 0.05 kg of water at 100 °C. (L_vaporization = 2.26×10⁶ J/kg)

\( Q = m L = 0.05 \cdot 2{,}260{,}000 = 113{,}000 \, \text{J} \)
The energy required is 113,000 J.

Example 3: Calorimetry Problem

A 50 g piece of metal at 150 °C is placed in 0.2 kg of water at 25 °C. If final temperature is 30 °C, calculate specific heat of metal.

Energy lost by metal = Energy gained by water:
\( m_\text{metal} c_\text{metal} (T_i - T_f) = m_\text{water} c_\text{water} (T_f - T_i) \)
\( 0.05 \cdot c_\text{metal} \cdot (150-30) = 0.2 \cdot 4180 \cdot (30-25) \)
\( 6 c_\text{metal} = 4180 \times 1 \)
\( c_\text{metal} \approx 696.7 \, \text{J/kg·K} \)

Practice Problems

  1. Heat 0.25 kg of ice at -10 °C to water at 20 °C. Find total energy.
  2. Vaporize 100 g of water at 100 °C. Calculate required energy.
  3. A calorimeter problem: Metal at 200 °C in 0.5 kg water at 25 °C reaches 35 °C. Find c_metal.
  4. Calculate energy to heat 0.3 kg ice from -5 °C to 0 °C and then melt it.
  5. Energy to convert 0.2 kg ice at 0 °C to steam at 100 °C.
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