To solve this problem, we need to apply the principles of thermal conduction and the concept of latent heat. Let's break it down step by step.
Understanding Thermal Conduction
Thermal conduction is the process by which heat energy is transferred through materials. The rate of heat transfer (Q) through a material can be calculated using Fourier's law of heat conduction, which states:
Q = k × A × (ΔT / L)
Where:
- Q = rate of heat transfer (in watts, W)
- k = thermal conductivity of the material (for copper, k ≈ 400 W/m·K)
- A = cross-sectional area (in m²)
- ΔT = temperature difference across the material (in °C or K)
- L = length of the material (in meters)
Given Data
From the problem, we have:
- Length of the rod, L = 1.2 m
- Cross-sectional area, A = 4.8 cm² = 4.8 × 10-4 m²
- Temperature difference, ΔT = 100 °C
- Thermal conductivity of copper, k ≈ 400 W/m·K
Calculating the Rate of Energy Conducted
Now, we can substitute the values into the formula:
Q = 400 W/m·K × 4.8 × 10-4 m² × (100 °C / 1.2 m)
Calculating this step-by-step:
- First, calculate the temperature gradient: ΔT / L = 100 °C / 1.2 m = 83.33 °C/m
- Now, substitute this back into the equation: Q = 400 × 4.8 × 10-4 × 83.33
- This gives us: Q ≈ 16 W
Energy Transfer Rate and Ice Melting
Now that we know the rate of energy conducted along the rod is approximately 16 watts, we can move on to the second part of the question regarding the melting of ice.
Melting Ice at the Cold End
The energy required to melt ice can be calculated using the formula:
Q = m × L_f
Where:
- m = mass of ice melted (in kg)
- L_f = latent heat of fusion for ice (approximately 334,000 J/kg)
We already calculated that the rate of heat transfer is 16 W, which means that every second, 16 joules of energy is being used to melt the ice. To find the mass of ice melted per second, we can rearrange the formula:
m = Q / L_f
Substituting the values:
m = 16 J/s / 334,000 J/kg
This gives:
m ≈ 0.0000479 kg/s
To express this in grams, we multiply by 1000:
m ≈ 0.0479 g/s
Summary of Results
In summary, the rate at which energy is conducted along the copper rod is approximately 16 watts, and the rate at which ice melts at the cold end is about 0.0479 grams per second. This illustrates the efficiency of thermal conduction in metals like copper and the energy dynamics involved in phase changes like melting ice.
