How many grams of ice at -29.7 ∘C can be completely converted to liquid at 21.7 ∘C if the available heat for this process is 5.42×103 kJ ?
For ice, use a specific heat of 2.01 J/(g⋅∘C) and ΔHfus=6.01kJ/mol .
the sum of heats gained =5.42e3 kj
heat to warm ice+heat to melt ice+heat to warm water=5.42e3
m*cice*29.7+ m*Hfice + m*cw*(21.7)=5.42e3
solve for mass m, you can look up specific heat of ice cw. You are given Hfice, and cw specific heat water.
To solve this problem, we need to calculate the amount of heat required to completely convert the given mass of ice at -29.7 ∘C to liquid at 0 ∘C, and then calculate the mass of ice that can be converted with the available heat of 5.42×103 kJ.
Here's how to do it step by step:
Step 1: Calculate the heat required to raise the temperature of the ice from -29.7 ∘C to 0 ∘C.
To do this, we use the equation: q = m * c * ΔT
where:
- q is the heat energy
- m is the mass of the substance
- c is the specific heat capacity of the substance
- ΔT is the change in temperature
Given:
- ΔT = 0 ∘C - (-29.7 ∘C) = 29.7 ∘C
- c (specific heat of ice) = 2.01 J/(g⋅∘C)
- m is unknown (let's call it x)
Substituting the values into the equation:
q1 = x * 2.01 J/(g⋅∘C) * 29.7 ∘C
Step 2: Calculate the heat required to convert the ice at 0 ∘C to liquid at 0 ∘C.
To do this, we use the equation: q = m * ΔHfus
where:
- ΔHfus is the molar heat of fusion (6.01 kJ/mol)
- m is the number of moles of the substance
To convert the mass of ice from grams to moles, we use the molar mass of water: 18.02 g/mol
Given:
- x is the mass of ice in grams (from step 1)
Converting the mass of ice to moles:
moles of ice = x / 18.02 g/mol
Substituting the values into the equation:
q2 = (x / 18.02 g/mol) * 6.01 kJ/mol
Step 3: Add up the total heat required:
Total heat required = q1 + q2
Given the available heat of 5.42×103 kJ, we can set up the equation:
5.42×103 kJ = q1 + q2
Now, we can solve the equation to find the mass of ice (x):
5.42×103 kJ = x * 2.01 J/(g⋅∘C) * 29.7 ∘C + (x / 18.02 g/mol) * 6.01 kJ/mol
Solve the equation for x, which represents the mass of ice in grams.