A styrofoam cooler is a cube with side length of 20.0 cm and walls with thickness of 1.9 cm. If the external temperature is 27.0 deg. C and the cooler contains ice and water at 0 deg. C.
How much heat is transferred into the box in 24 hours?
- I got 2.95x10^5 J for this and it was correct, but I can't figure out the second half. I looked up the amount of heat required to melt 1 kg of ice and got 33.4 kJ, so 2.95x10^5 / 334000 J = 8.832 kg but it was wrong. How do I go about figuring out this question?:
>>> How much of the ice in the cooler will melt in 24 hours?
Heat=massice*Lf
massice=2.95E5J/333kJ/kg
= 295 / 333 kg=.88kg
To determine how much of the ice in the cooler will melt in 24 hours, you need to understand the concept of heat transfer and the specific heat capacity of water.
1. Calculate the amount of heat transferred into the cooler:
- Use the formula Q = mcΔT, where Q is the heat transfer, m is the mass of the cooler, c is the specific heat capacity of styrofoam (0.033 J/g·°C), and ΔT is the temperature difference between the exterior and interior of the cooler.
- The mass of the cooler can be found by calculating the volume of the cube (V) and multiplying it by the density of styrofoam (typical value is 0.03 g/cm³).
- The volume of the cube can be calculated by V = (side length)³ - ((side length - 2 × wall thickness)³).
- The temperature difference, ΔT, is the difference between the external temperature and the temperature of ice and water in the cooler (27.0°C - 0°C).
2. Once you have calculated the heat transferred into the cooler, you can determine the mass of ice that will melt:
- Use the formula Q = mLf, where Q is the heat transfer, m is the mass of ice, and Lf is the latent heat of fusion for water (typical value is 334,000 J/kg).
- Rearrange the formula to solve for the mass of ice: m = Q / Lf.
Let's apply these steps to the given values:
1. Calculate the heat transferred into the cooler:
- Calculate the volume of the cube:
V = (20.0 cm)³ - ((20.0 cm - 2 × 1.9 cm)³)
= 20.0 cm³ - 15.6 cm³
= 4.4 cm³.
- Convert cm³ to g by multiplying by the density of styrofoam:
m = V × density
= 4.4 cm³ × 0.03 g/cm³
= 0.132 g.
- Convert mass to kg:
m = 0.132 g ÷ 1000
= 0.000132 kg.
- Calculate the temperature difference:
ΔT = 27.0°C - 0°C
= 27.0°C.
- Calculate the heat transfer:
Q = mcΔT
= 0.000132 kg × 0.033 J/g·°C × 27.0°C
= 0.119988 J.
2. Calculate the mass of ice that will melt:
- Calculate the mass of ice:
m = Q / Lf
= 0.119988 J ÷ 334,000 J/kg
≈ 0.000359 kg or 0.359 g.
Therefore, approximately 0.359 grams of ice in the cooler will melt in 24 hours.