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Heating water

Heat and matter · level 4, lesson 4 of 6

A question to start

A 2,000 W kettle holds 1.5 kg of water. How long does it take to boil? And could a tank of hot water keep a Mars home warm through the night?

The idea

A heater of power \(P\) (in watts, joules per second) running for a time \(t\) gives energy

\[Q = Pt.\]

Put that together with \(Q = mc\Delta T\), and you can find how long heating takes:

\[t = \frac{mc\,\Delta T}{P}.\]

Real heaters lose some heat to their surroundings, so the real time is a little longer.

Water is also a good store of heat, because its \(c\) is so large. Warm a big tank by day, and it gives the heat back as it cools at night.

day: heatednight: gives heat out60 °C30 °C

Worked examples

1. The kettle. From 20 °C to 100 °C: \(Q = 1.5 \times 4180 \times 80 = 501{,}600\) J. \(t = 501{,}600/2000 = 251\) s, a little over 4 minutes.

2. A shower. 50 kg of water warmed from 10 °C to 40 °C: \(50 \times 4180 \times 30 = 6.3\times10^6\) J, about 1.7 kWh.

3. A tank for the night. A Mars home needs 3 kW of heat through a 12 h 20 min night: \(3000 \times 44{,}400 = 1.3\times10^8\) J. A tank cooling from 60 °C to 30 °C gives \(4180 \times 30 = 125{,}000\) J per kg, so it needs \(1.3\times10^8 / 125{,}000 \approx 1{,}070\) kg: about a cubic metre of water.

A night's heat store Mars Earth
Night at the equator 12 h 20 min 12 h
Water tank for 3 kW, 60 °C to 30 °C about 1,070 kg about 1,040 kg

Try it

  1. How much energy does a 1,500 W heater give in 10 minutes?
  2. How long does a 2,000 W kettle take to warm 1 kg of water from 20 °C to 100 °C?
  3. Why does the real kettle take a little longer?
  4. Why is water better than the same mass of steel for storing heat?
Answers
  1. \(1500 \times 600 = 900{,}000\) J.
  2. \(1 \times 4180 \times 80/2000 = 167\) s, nearly 3 minutes.
  3. Some heat warms the kettle itself and escapes to the air.
  4. Its specific heat capacity is about nine times bigger, so it holds far more heat for each degree.

Common mistake

Forgetting to change minutes to seconds. Watts are joules per second. 10 minutes is 600 s.

Where it leads