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Water from ice

Heat and matter · level 6, lesson 2 of 6

A question to start

On Earth, water comes out of a tap. On Mars, a base will dig up ice at about −60 °C. How much energy does it take to turn a kilogram of that ice into a glass of water at room temperature?

The idea

Three steps, each with its own formula:

  1. Warm the ice from −60 °C to 0 °C: \(Q_1 = m\,c_\text{ice}\,\Delta T\).
  2. Melt it at 0 °C: \(Q_2 = mL_f\).
  3. Warm the water from 0 °C to 20 °C: \(Q_3 = m\,c_\text{water}\,\Delta T\).
warm the ice126 kJmelt it: 334 kJwarm water84 kJ total: 544 kJ, about 0.15 kWh
Getting 1 kg of water at 20 °C Mars Earth
Where it starts ice dug from the ground, about −60 °C a tap
Heat needed, about 544 kJ (0.15 kWh) almost none

That does not count the energy to dig the ice, carry it and clean the water.

Worked examples

1. One kilogram. \(Q_1 = 1 \times 2100 \times 60 = 126{,}000\) J. \(Q_2 = 1 \times 334{,}000 = 334{,}000\) J. \(Q_3 = 1 \times 4180 \times 20 = 83{,}600\) J. Total 543,600 J, about 0.15 kWh.

2. A day's top-up. Most of a base's water would be cleaned and used again. Suppose it must still melt 1,000 kg of new ice a sol: \(1000 \times 0.15 = 150\) kWh, about 6 kW day and night.

3. Using waste heat. A reactor or power unit makes far more spare heat than this. Piping it to the ice-melter turns waste into water.

Try it

  1. Which of the three steps takes the most energy?
  2. Find the energy to turn 3 kg of ice at −20 °C into water at 0 °C.
  3. How much energy turns 1 kg of ice at −60 °C into steam at 100 °C? (Use \(L_v\) = 2,260 kJ/kg.)
  4. Why would a Mars base clean and reuse its water rather than melt new ice?
Answers
  1. Melting: 334 kJ of the 544 kJ.
  2. \(3 \times 2100 \times 20 + 3 \times 334{,}000 = 126{,}000 + 1{,}002{,}000 = 1.13\times10^6\) J.
  3. \(126 + 334 + 418 + 2260 = 3{,}138\) kJ.
  4. Melting, digging and cleaning take a lot of energy and work; cleaning used water is cheaper.

Common mistake

Missing a step. Ice at −60 °C must first be warmed to 0 °C before it can melt. Leaving out a step, or using the wrong \(c\) (ice's is half of water's), gives the wrong total.

Where it leads