Specific heat capacity of water = 4186 J kg^-1 K^-1 Specific heat capacity of ice = 2090 J kg^-1 K^-1 Latent heat of fusion of ice = 335 kJ kg^-1 Latent heat of vaporization of water = 2260 kJ kg^-1 Question: You have 3 kg of ice at -20°C. (i) How much heat is needed to raise the temperature of the ice from -20°C to 0°C? (ii) How much heat is needed to melt all the ice at 0°C to water? (iii) If 1×10^6 J of heat is added to the ice at -20°C, will the ice melt? Explain:
The Correct Answer and Explanation is:
Let’s break down each part of the problem step by step.
(i) Heat needed to raise the temperature of ice from -20°C to 0°C
The heat required to change the temperature of a substance is given by the equation: Q=mcΔTQ = mc\Delta TQ=mcΔT
Where:
- QQQ is the heat required (in joules, J),
- mmm is the mass of the substance (in kg),
- ccc is the specific heat capacity of the substance (in J kg−1^{-1}−1 K−1^{-1}−1),
- ΔT\Delta TΔT is the change in temperature (in °C or K).
Here:
- Mass of ice, m=3m = 3m=3 kg,
- Specific heat capacity of ice, c=2090c = 2090c=2090 J kg−1^{-1}−1 K−1^{-1}−1,
- Change in temperature, ΔT=0°C−(−20°C)=20°C\Delta T = 0°C – (-20°C) = 20°CΔT=0°C−(−20°C)=20°C.
Substituting these values into the formula: Q=3×2090×20=125400 JQ = 3 \times 2090 \times 20 = 125400 \text{ J}Q=3×2090×20=125400 J
So, the heat required to raise the temperature of the ice from -20°C to 0°C is 125400 J.
(ii) Heat needed to melt all the ice at 0°C
The heat required to melt ice is given by the equation: Q=mLfQ = mL_fQ=mLf
Where:
- QQQ is the heat required (in joules, J),
- mmm is the mass of the substance (in kg),
- LfL_fLf is the latent heat of fusion (in J kg−1^{-1}−1).
Here:
- Mass of ice, m=3m = 3m=3 kg,
- Latent heat of fusion of ice, Lf=335×103L_f = 335 \times 10^3Lf=335×103 J kg−1^{-1}−1.
Substituting these values into the formula: Q=3×335×103=1005000 JQ = 3 \times 335 \times 10^3 = 1005000 \text{ J}Q=3×335×103=1005000 J
So, the heat required to melt all the ice at 0°C is 1005000 J.
(iii) Will the ice melt if 1×10^6 J of heat is added?
To check if the ice will melt, we need to compare the total heat required to raise the temperature of the ice to 0°C and then melt it. From part (i) and part (ii), we have:
- Heat required to raise the temperature from -20°C to 0°C: 125400 J,
- Heat required to melt the ice at 0°C: 1005000 J.
Total heat required: 125400 J+1005000 J=1130400 J125400 \text{ J} + 1005000 \text{ J} = 1130400 \text{ J}125400 J+1005000 J=1130400 J
Now, if only 1 \times 10^6 J of heat is provided, it is less than the 1130400 J required. Therefore, the ice will not melt completely with 1 million joules of heat.
Conclusion: 1 million joules of heat will be enough to raise the temperature of the ice from -20°C to 0°C, but it will not be enough to melt all the ice. The remaining heat will still be required to overcome the latent heat of fusion and melt the ice.
