A cast-iron saucepan of mass 1.4kg is filled with 5L of water at 15°C, placed on an electric hotplate, and bought up to boiling point. What is the cost of the energy used if the electricity costs 10 cents per kWh and the hotplate has an efficiency of 70%.
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To calculate the cost of the energy used, we need to determine the total energy consumed by the electric hotplate, taking into account its efficiency.
First, let's calculate the energy required to heat the water in the saucepan from 15°C to its boiling point. We can use the specific heat capacity of water to calculate this.
The specific heat capacity of water is approximately 4.184 J/g°C. Since we have 5L of water, which is equivalent to 5000 grams (1L of water is approximately 1000 grams), the amount of energy required to heat the water can be calculated as follows:
Energy = mass * specific heat capacity * temperature change
Temperature change = boiling point - initial temperature = 100°C - 15°C = 85°C
Energy = 5000g * 4.184 J/g°C * 85°C = 178,900 J
Next, we need to account for the efficiency of the hotplate, which is 70%. This means that only 70% of the energy consumed by the hotplate is actually transferred to heat the water. To calculate the total energy consumed by the hotplate, we can use the formula:
Total energy consumed = Energy required / Efficiency
Total energy consumed = 178,900 J / 0.70 = 255,571 J
Now, let's convert this energy to kilowatt-hours (kWh) since the electricity cost is given per kWh. To convert joules to kWh, we can use the conversion factor:
1 kWh = 3.6 × 10^6 J
Energy in kWh = Total energy consumed / (3.6 × 10^6 J)
Energy in kWh = 255,571 J / (3.6 × 10^6 J)
Energy in kWh ≈ 0.071 kWh
Finally, we can calculate the cost of the energy used by multiplying the energy consumption in kWh by the cost per kWh:
Cost of energy used = Energy in kWh * Cost per kWh
Given that the electricity costs 10 cents per kWh:
Cost of energy used = 0.071 kWh * $0.10/kWh
Cost of energy used ≈ $0.0071
Hence, the cost of the energy used is approximately 0.71 cents.