If 5.40 kcal of heat is added to 1.00 kg of ice at 0⁰C, how much liquid water at 0⁰C is produced? Show all calculations leading to an answer.
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To solve this problem, we can use the concept of specific heat and the heat of fusion of ice.
Step 1: Calculate the heat required to raise the temperature of ice from 0⁰C to its melting point.
The specific heat capacity of ice is 2.09 J/g·°C (or 2.09 kJ/kg·°C since 1 g = 0.001 kg), which means it takes 2.09 kJ of heat to raise the temperature of 1 kg of ice by 1°C.
Given that we have 1.00 kg of ice and we need to raise its temperature from 0⁰C to the melting point (0⁰C), the heat required is:
Heat = mass × specific heat capacity × temperature difference
Heat = 1.00 kg × 2.09 kJ/kg·°C × (0⁰C - 0⁰C)
Heat = 0 kJ
Since there is no change in temperature, no heat is required to raise the ice to its melting point.
Step 2: Calculate the heat required to convert the ice at 0⁰C to liquid water at 0⁰C.
The heat of fusion for ice is 333.5 kJ/kg. This means it takes 333.5 kJ of heat to convert 1 kg of ice at 0⁰C to 1 kg of liquid water at 0⁰C.
Given that we have 1.00 kg of ice, the heat required to convert it to liquid water is:
Heat = mass × heat of fusion
Heat = 1.00 kg × 333.5 kJ/kg
Heat = 333.5 kJ
Step 3: Add the heat required to raise the temperature of ice to its melting point and the heat required to convert the ice to liquid water.
Total heat = Heat to raise temperature + Heat of fusion
Total heat = 0 kJ + 333.5 kJ
Total heat = 333.5 kJ
So, a total of 333.5 kJ of heat is needed to convert 1.00 kg of ice at 0⁰C to liquid water at 0⁰C.
Step 4: Convert the given heat of 5.40 kcal to kJ.
1 kcal is equivalent to 4.18 kJ.
Given that 5.40 kcal of heat is added, the heat in kJ is:
Heat in kJ = 5.40 kcal × 4.18 kJ/kcal
Heat in kJ = 22.572 kJ
Step 5: Calculate the mass of water produced using the heat.
Now that we know the total heat required is 333.5 kJ and the given heat is 22.572 kJ, we can calculate the mass of water produced using the ratio of heats:
mass of water produced / Given heat = Total heat / Heat of fusion
mass of water produced = (Total heat / Heat of fusion) × mass of ice
mass of water produced = (22.572 kJ / 333.5 kJ/kg) × 1.00 kg
mass of water produced ≈ 0.07 kg
Therefore, approximately 0.07 kg (or 70 g) of liquid water at 0⁰C is produced when 5.40 kcal of heat is added to 1.00 kg of ice at 0⁰C.