The Kettle, the Altitude and the Efficiency Claim
A pupil in Almaty measures the efficiency of her family's electric kettle for a physics project. She records:
- mass of water: 1.00 kg (measured on a kitchen scale, ±0.01 kg);
- initial temperature: 20.0 °C (±0.5 °C);
- the water boils at 97.0 °C, because the city is about 800 m above sea level (±0.5 °C on the thermometer);
- rated power of the kettle: 2000 W;
- time from switching on to vigorous boiling: 200 s (±2 s).
She takes the specific heat capacity of water as 4200 J/(kg·°C) and ignores the kettle body.
She calculates an efficiency of about 81% and writes in her report: 'The remaining 19% is lost as heat to the room.' Her teacher points to the label on the kettle, which claims an efficiency of 85%, and asks her whether her measurement contradicts the manufacturer.
1. Reproduce her calculation and state the result with an uncertainty. Does the measurement contradict the 85% claim?
Read the worked answer
Energy delivered to the water: Q = m·c·ΔT = 1.00 × 4200 × (97.0 − 20.0) = 1.00 × 4200 × 77.0 = 323,400 J.
Electrical energy supplied: E = P·t = 2000 × 200 = 400,000 J.
Efficiency: η = 323,400 / 400,000 = 0.8085 → 80.9%, i.e. about 81%.
Uncertainty. The relative uncertainties are: ΔT, 1.0/77.0 = 1.3% (the two ±0.5 °C errors add); mass, 0.01/1.00 = 1.0%; time, 2/200 = 1.0%. Added in quadrature: √(1.3² + 1.0² + 1.0²) = √3.69 ≈ 1.9%. So η = 81% ± 2%, that is, roughly 79–83%.
Does it contradict 85%? Not decisively. 85% lies just outside the stated range, but the range covers only random error. Several systematic effects all push the true value upwards:
- the kettle body and the heating element also absorb heat, which is useful output but is not counted;
- the rated 2000 W may not be delivered exactly at the actual mains voltage;
- the water may reach 97 °C slightly before 'vigorous boiling' is judged by eye.
Because the systematic errors are one-sided and unquantified, the honest conclusion is 'my result is consistent with a true efficiency in the low-to-mid 80s', not 'the manufacturer is wrong'. Her statement that the missing 19% is 'lost to the room' is the clearest error: part of it heated the kettle itself and part was never supplied if the rated power is optimistic.
2. She wants to reduce the systematic error. Describe two concrete improvements and say what each one fixes.
Read the worked answer
1. Include the kettle in the energy balance. Repeat the experiment with a known mass of water and then with a different known mass, keeping everything else the same. The difference between the two runs cancels the kettle's own heat capacity, because the kettle body is heated to the same final temperature both times. This removes the largest one-sided error.
2. Measure the electrical power directly. Put a power meter or a multimeter with a current clamp on the kettle and record the actual power, or measure the energy with a joule meter instead of trusting the label. This converts 'rated power' into 'measured power' and removes the second one-sided error.
Two further refinements worth mentioning:
- Stir and measure temperature properly, with the thermometer in the bulk of the water, so the recorded final temperature is not the temperature of the layer next to the element.
- Define the end point physically, e.g. stop at a set temperature such as 90.0 °C well below boiling, which avoids the judgement call about when boiling is 'vigorous' and also avoids evaporation losses during rolling boil.
A good answer recognises the pattern: random errors are handled by repeating and averaging; systematic errors are handled only by changing the method, never by repeating the same flawed procedure.
3. The family asks whether they can trust the 85% label when choosing between two kettles. What should the pupil tell them, and what single measurement would be most useful for that decision?
Read the worked answer
What she should tell them. The label figure cannot be verified to a percentage point with a kitchen setup, and the difference between an 81% and an 85% kettle is small in money terms: for a household boiling, say, 3 litres a day, the wasted electricity differs by only a few percent of the kettle's share of the bill. The dominant factors in real use are how much water is boiled each time and whether the kettle is descaled - scale on the element is a large, growing, and easily avoidable loss.
The most useful single measurement is not the peak efficiency but the energy per litre boiled, measured with a joule meter from a fixed starting temperature, keeping initial temperature and water volume identical for both kettles. That number folds in the kettle body's heat capacity, the real power drawn, and the switch-off behaviour, and it is directly comparable between the two appliances. Repeating it twice per kettle and averaging gives a decision-grade figure.
The judgement the pupil should demonstrate is separating three questions that get confused: Is the manufacturer's number false? (not established); Is the difference worth paying for? (probably small); What can I actually measure that matters? (energy per litre, in her own kitchen).