Question

The time required for 1 kW electric heater to raise the temperature of 10 litres of water through 10°C is

• A. 210 s
• B. 420 s
• C. 42 s
• D. 840 s

Hint:

Heating Time Formula: $t = \frac{m \cdot c_p \cdot \Delta T}{P} = \frac{10 \times 4184 \times 10}{1000} = 418.4\text{ s} \approx \mathbf{420\text{ s}}$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Thermal energy and electric power conversion thermodynamics: electrical energy input ($E = P \cdot t$) equals heat energy absorbed ($Q = m \cdot c_p \cdot \Delta T$). Using specific heat of water $c_p = 4,184 ext{ J/(kg}\cdot ext{K)}$ and density $ ho = 1 ext{ kg/L}$, heating 10 kg water by 10°C with 1 kW heater requires 418.4 seconds ($pprox 420 ext{ s}$).
Key Formula or Approach:
\[ Q = m \cdot c_p \cdot \Delta T = 10\text{ kg} \times 4,184\text{ J/(kg}\cdot\text{K)} \times 10\text{ K} = 418,400\text{ J} \]
\[ \text{Power } (P) = 1\text{ kW} = 1,000\text{ W} = 1,000\text{ J/s} \]
\[ \mathbf{Time \text{ } (t)} = \frac{Q}{P} = \frac{418,400\text{ J}}{1,000\text{ J/s}} = \mathbf{418.4 \text{ s} \approx 420 \text{ s}} \]

Step 2: Detailed Explanation:
Step-by-step mathematical calculation of water heating time:
1. Given Parameters:
- Volume of water: $V = 10\text{ litres} \implies \text{Mass } m = 10\text{ kg}$ (taking density of water $\rho = 1.0\text{ kg/L}$).
- Specific heat capacity of water: $c_p = 4.184\text{ kJ/(kg}\cdot^\circ\text{C)} = 4,184\text{ J/(kg}\cdot\text{K)}$.
- Temperature rise: $\Delta T = 10^\circ\text{C} = 10\text{ K}$.
- Electric Heater Power: $P = 1\text{ kW} = 1,000\text{ Watts} = 1,000\text{ J/s}$.
2. Calculate Total Thermal Energy Required ($Q$):
\[ Q = m \times c_p \times \Delta T = 10\text{ kg} \times 4,184\text{ J/(kg}\cdot\text{K)} \times 10\text{ K} = \mathbf{418,400\text{ Joules (418.4 kJ)}} \]
3. Calculate Time Required ($t$):
\[ t = \frac{Q}{P} = \frac{418,400\text{ J}}{1,000\text{ J/s}} = \mathbf{418.4\text{ seconds} \approx 420\text{ seconds}} \]
- Hence, the time required is approximately 420 s (B).

Step 3: Final Answer:
Hence, the time required is 420 s, matching option (B).