Ice of mass 12.0 kg at 0.00° C is placed in an ice chest. The ice chest has 3.10 cm thick walls of thermal conductivity 1.00 10-5 kcal/s · m · C° and a surface area of 1.20 m2.
(a) How much heat must be absorbed by the ice before it melts?
(b) If the outer surface of the ice chest is at 29.0° C, how long will it take for the ice to melt?
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To answer these questions, we need to understand the basic principles of heat transfer, specifically conduction. Heat transfer by conduction is governed by Fourier's Law, which states that the rate of heat transfer through a material is directly proportional to the area, temperature difference, and thermal conductivity, and inversely proportional to the thickness of the material.
(a) To determine how much heat must be absorbed by the ice before it melts, we need to calculate the total amount of heat transferred through the ice chest walls.
The rate of heat transfer (Q) through a material can be calculated using the formula:
Q = (k * A * ΔT) / L
Where:
Q = Rate of heat transfer
k = Thermal conductivity of the material
A = Surface area through which heat is transferred
ΔT = Temperature difference across the material
L = Thickness of the material
Given:
Mass of ice (m) = 12.0 kg
Temperature of ice (T1) = 0.00°C = 0.00°C + 273.15K = 273.15K
Thickness of ice chest walls (L) = 3.10 cm = 3.10 cm * 0.01m/cm = 0.031m
Thermal conductivity of ice chest walls (k) = 1.00 × 10^-5 kcal/s · m · °C
Surface area of ice chest (A) = 1.20 m^2
First, we need to calculate the temperature difference across the ice chest walls. The outer surface temperature (T2) is given as 29.0°C = 29.0°C + 273.15K = 302.15K.
ΔT = T2 - T1
ΔT = 302.15K - 273.15K
ΔT = 29.0K
Now we can calculate the rate of heat transfer:
Q = (k * A * ΔT) / L
Q = (1.00 × 10^-5 kcal/s · m · °C) * (1.20m^2) * (29.0K) / (0.031m)
Calculating this will give us the rate of heat transfer through the ice chest walls.
(b) To determine how long it will take for the ice to melt, we need to know the total amount of heat required to melt the ice. This can be calculated using the equation:
Qtotal = m * ΔH
where Qtotal is the total heat required, m is the mass of the substance, and ΔH is the heat of fusion.
For ice, the heat of fusion is 334,000 J/kg.
Since we know the mass of the ice is 12.0 kg, we can now calculate the total heat required to melt the ice:
Qtotal = (12.0 kg) * (334,000 J/kg)
This will give us the total heat required to melt the ice.
To find out how long it will take for the ice to melt, we need to divide the total heat required by the rate of heat transfer calculated in part (a):
Time = Qtotal / Q
Calculating this will give us the time required for the ice to melt.
By following these steps, you should be able to calculate the answers to both parts (a) and (b) of the given problem.