An ice cube at 0°C measures 14.1 cm on a side. It sits on top of a copper block with a square cross section 14.1 cm on a side and a length of 18.9 cm. The block is partially immersed in a large pool of water at 89.3°C. How long does it take the ice cube to melt? Assume that only the part in contact with the copper liquefies; that is, the cube gets shorter as it melts. The density of ice is 0.917 g/cm3
for heat flow rate (conduction) through the copper, wouldn't it matter how "partially immersed" it is?
To find the time it takes for the ice cube to melt, we need to calculate the amount of energy needed to melt the ice and then determine the rate at which this energy is transferred to the ice cube.
1. Calculate the heat energy required to melt the ice cube:
The heat energy required to melt a substance can be calculated using the formula:
Q = m * L
where Q is the heat energy, m is the mass, and L is the latent heat of fusion.
The mass of the ice cube can be calculated using its volume and density:
V = side^3
m = density * V
Given:
side = 14.1 cm (0.141 m)
density of ice = 0.917 g/cm3 (0.917 kg/m3)
Calculate the mass of the ice cube:
V = side^3 = 0.141^3 = 0.0027 m3
m = density * V = 0.917 * 0.0027 = 0.0025 kg
The latent heat of fusion for ice is 334,000 J/kg.
Calculate the heat energy required to melt the ice:
Q = m * L = 0.0025 * 334,000 = 835 J
2. Calculate the rate of heat transfer to the ice cube:
The rate of heat transfer is given by Newton's law of cooling:
Q/t = k * A * ΔT / d
where Q/t is the rate of heat transfer, k is the thermal conductivity of copper, A is the contact area between the ice cube and the copper block, ΔT is the temperature difference, and d is the thickness of the copper block.
Given:
k (thermal conductivity of copper) = 386 W/(m·K)
A (contact area) = side^2 = 0.141^2 = 0.0199 m2
ΔT (temperature difference) = 89.3°C - 0°C = 89.3 K
d (thickness of the copper block) = 18.9 cm (0.189 m)
Calculate the rate of heat transfer to the ice cube:
Q/t = k * A * ΔT / d = 386 * 0.0199 * 89.3 / 0.189 = 918 W
3. Calculate the time required for the ice cube to melt:
Now that we have the heat energy required to melt the ice (835 J) and the rate of heat transfer to the ice cube (918 W), we can calculate the time it takes to transfer the required heat energy.
t = Q / Q' = 835 / 918 ≈ 0.91 s
Therefore, it takes approximately 0.91 seconds for the ice cube to melt when in contact with the copper block.