Question

10kg ice at \(-10^\circ C\) & 100kg water at \(25^\circ C\) are mixed together. Find final temperature. (Given \(S_{ice} = \frac{1}{2}\) cal/g\(^\circ C\), \(L_{fusion} = 80\)cal/g and \(S_{water} = 1\)cal/g\(^\circ C\))

• A. \(15^\circ C\)
• B. \(10^\circ C\)
• C. \(25^\circ C\)
• D. \(20^\circ C\)

Hint:

In calorimetry, always check if the hot body has enough energy to melt the cold body first. If \(Q_{lost\_max}<Q_{melt}\), the final temperature is always \(0^\circ C\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Calorimetry is based on the Principle of Mixtures: Heat lost by hot body = Heat gained by cold body.
Specific heat is used for temperature changes (\(Q = mc\Delta T\)) and latent heat for phase changes (\(Q = mL\)).
Step 2: Key Formula or Approach:
1. Heat to bring ice to \(0^\circ C\): \(Q_1 = m_i s_i \Delta T_1\).
2. Heat to melt ice: \(Q_2 = m_i L_f\).
3. Equate: \(Q_{ice \rightarrow temp} = Q_{water \rightarrow temp}\).
Step 3: Detailed Explanation:
Mass of ice \(m_i = 10 \text{ kg} = 10000 \text{ g}\).
Mass of water \(m_w = 100 \text{ kg} = 100000 \text{ g}\).
Heat required to raise ice to \(0^\circ C\):
\[ Q_1 = 10000 \times 0.5 \times 10 = 50000 \text{ cal} = 50 \text{ kcal} \]
Heat required to melt ice:
\[ Q_2 = 10000 \times 80 = 800000 \text{ cal} = 800 \text{ kcal} \]
Total heat to turn ice into water at \(0^\circ C\): \(Q_{total\_gain} = 850 \text{ kcal}\).
Heat available from water to cool to \(0^\circ C\):
\[ Q_{avail} = 100000 \times 1 \times 25 = 2500000 \text{ cal} = 2500 \text{ kcal} \]
Since \(Q_{avail}>Q_{total\_gain}\), the ice will completely melt and the final temperature \(T\) will be between \(0^\circ C\) and \(25^\circ C\).
Using the heat balance equation:
\[ 850 \text{ kcal} + (10 \text{ kg}) \times (1 \text{ kcal/kg}^\circ C) \times (T - 0) = (100 \text{ kg}) \times (1 \text{ kcal/kg}^\circ C) \times (25 - T) \]
\[ 850 + 10T = 2500 - 100T \]
\[ 110T = 1650 \]
\[ T = \frac{1650}{110} = 15^\circ C \]
Step 4: Final Answer:
The final temperature of the mixture is \(15^\circ C\).