A 0.299-m-thick sheet of ice covers a lake. The air temperature at the ice surface is -15.0 °C. In 4.90 minutes, the ice thickens by a small amount. Assume that no heat flows from the ground below into the water and that the added ice is very thin compared to 0.299 m. Calculate the number of millimeters by which the ice thickens. Do not enter unit.
To calculate the number of millimeters by which the ice thickens, we can use the formula for heat conduction through a material, which is given by:
Q = (kAΔT)t / d
Where:
Q is the heat transferred through the material,
k is the thermal conductivity of the material (which is 2.19 W/m·K for ice),
A is the surface area of the material through which heat is conducted,
ΔT is the temperature difference across the material,
t is the time for heat transfer, and
d is the thickness of the material.
In this case, we want to find the change in thickness of the ice, so we can rearrange the formula as follows:
d = (kAΔT)t / Q
We are given that the initial thickness of the ice is 0.299 m, the temperature difference is -15.0 °C (which we convert to Kelvin by adding 273.15), the time is 4.90 minutes (which we convert to seconds by multiplying by 60), and we need to find the value of Q.
Since we are assuming no heat flows from the ground below into the water, we can assume that the only heat transfer is due to the temperature difference at the top surface of the ice. Therefore, we can assume that the heat transferred is equal to the heat lost from the top surface of the ice.
To find the heat lost, we can use the formula for heat transfer through a material:
Q = ρcVΔT
Where:
ρ is the density of the material (which is 917 kg/m³ for ice),
c is the specific heat capacity of the material (which is 2.09 kJ/kg·K for ice),
V is the volume of the material, and
ΔT is the temperature difference across the material.
The volume V can be calculated by multiplying the surface area A by the thickness d.
Now, let's calculate the value of Q:
Q = ρcVΔT
= (917 kg/m³) * (2.09 kJ/kg·K) * (A * d) * (-15.0 + 273.15) K
Since the added ice is very thin compared to 0.299 m, we can assume that the change in thickness is negligible compared to the initial thickness. Therefore, the final thickness of the ice can be approximated as:
d_final ≈ d_initial + Δd
Where d_initial is the initial thickness of the ice and Δd is the change in thickness.
Substituting the values into the formula:
d_final ≈ 0.299 m + (kAΔT * t) / Q
Let's calculate Δd and then convert it to millimeters:
Δd = d_final - d_initial
= (kAΔT * t) / Q
Now, let's calculate Δd in millimeters:
Δd (in millimeters) = Δd * 1000
Use the given values to calculate the change in thickness of the ice.