Specific Heat Capacity - Physics for Pilot Assessments

Different materials need different amounts of energy to warm up by the same number of degrees, and this lesson puts a number on that. By the end of it you should be able to calculate the heat needed to change a known mass of a material by a known number of degrees, run the same relationship backwards to find the temperature rise, the mass or the material's own constant, and explain why water is unusually slow both to heat and to cool.

Same energy in, different temperature rise

Take two identical electric heaters, put one into a kilogram of water and the other into a kilogram of aluminium, and switch both on for one minute. Each has delivered the same number of joules. The aluminium will have climbed several times further up the thermometer than the water. Nothing is wrong with either heater: the two materials simply differ in how much energy it takes to shift their temperature, and that is a property of the material in the same way that density is. Try it with a kilogram of lead and the rise is dramatic; try it with two kilograms of water instead of one and the rise is half what it was, because the same energy is being shared among twice as much material.

The same number of joules goes into each block, and the thermometer tells two different stories: the aluminium climbs almost five times further, because each of its degrees costs 900 joules against water's 4200. A high specific heat capacity therefore means a small rise for a given energy — slow to heat, not quick.

So the heat needed depends on three things, and it is worth naming each. It depends on how much material there is, which is the mass in kilograms. It depends on how far you want the temperature to move, which is the temperature change — and from the temperature lesson a change may be quoted in degrees Celsius or in kelvin indifferently, because the two units have the same size of step. And it depends on which material it is, which is captured in a single number for that substance: its specific heat capacity, the energy needed to raise one kilogram of it by one degree. The word specific is doing real work there — it means per kilogram, so the figure is a property of the substance and not of the lump in front of you.

Heat needed for a temperature change

Q = mc\,\Delta T

$Q$: the heat energy supplied or removed, in joules (J)
$m$: the mass of material being heated, in kilograms (kg)
$c$: the specific heat capacity of that material, in joules per kilogram per degree (J/kg°C, identical to J/kg K)
$\Delta T$: the temperature change, in degrees Celsius (°C) or kelvin (K) — the same number either way

The unit of c looks awkward written out and says exactly what the quantity is: joules, per kilogram, per degree. Water's value is 4200 joules per kilogram per degree, so raising one kilogram of water by one degree takes 4200 joules — and since a joule is roughly the energy to lift an apple a metre, warming a kettleful of water by a single degree is a surprisingly large amount of energy.

Rearranging it three ways

Any one of the four quantities can be the unknown, so derive the rearrangements once rather than memorising three separate facts. Start from heat = mass × specific heat capacity × temperature change. To find the temperature change, divide both sides by mass × specific heat capacity; on the right those two cancel and leave the temperature change alone, so $\Delta T = \frac{Q}{mc}$ . To find the specific heat capacity of an unknown material, divide both sides instead by mass × temperature change, giving $c = \frac{Q}{m\,\Delta T}$ . And to find the mass, divide both sides by specific heat capacity × temperature change, giving $m = \frac{Q}{c\,\Delta T}$ . In every case the trick is the same: divide by everything on the right that you do not want.

Values worth carrying

You will not be asked to recall these to three figures, but the ranking matters, because items are built on comparisons. Water is the outlier at 4200, several times higher than almost anything solid. Ice is about 2100, half water's value, which is worth noticing on its own: freezing water changes its specific heat capacity, so the solid and the liquid form of the same substance are two different materials as far as this calculation is concerned. Air is around 1000, aluminium 900, glass and concrete about 840 and 800, iron and steel about 450, copper 385, and lead only 130. Read the bottom of that list carefully: lead takes about a thirtieth of the energy water does, per kilogram per degree, which is why a small blowtorch will melt a lead pipe and will not boil a saucepan.

Why water is slow to heat and slow to cool

The same constant governs the journey in both directions. Taking energy out of a kilogram of water lowers its temperature by one degree per 4200 joules removed, exactly as putting energy in raises it by one degree per 4200 joules supplied, so a material that is slow to warm is equally slow to cool. Water is therefore doubly awkward and doubly useful. It is awkward when you want it hot in a hurry, which is why a kettle draws a couple of kilowatts and still takes a couple of minutes. It is useful whenever you want to move energy about or hold a temperature steady: the coolant in an engine carries a great deal of heat away from the cylinder block for a modest rise in its own temperature, a hot water bottle stays warm for hours, and a stone-and-water building keeps its temperature through a hot afternoon.

The same property shapes climate, and this is the version the tests like. A coastal town has milder winters and cooler summers than a town the same distance north but far inland, because the sea alongside it absorbs and releases enormous quantities of energy while changing its own temperature very little. The land, with a specific heat capacity around 800 and no ability to mix, heats and cools quickly by comparison. It is also why the sea is at its warmest weeks after the sun is at its highest, and coldest weeks after midwinter: it lags, because there is so much energy to shift before its temperature moves.

Example 2 — running it backwards to a temperature rise

90 000 joules are supplied to 2 kilograms of aluminium, whose specific heat capacity is 900 joules per kilogram per degree. By how much does its temperature rise?

Start from heat = mass × specific heat capacity × temperature change and divide both sides by mass × specific heat capacity. Those two cancel on the right, leaving $\Delta T = \frac{Q}{mc}$ .
Work out the bottom of the fraction first: $m c = 2 \times 900 = 1800$
So the temperature change is $\frac{90\,000}{1800}$ . Both numbers divide by 100, giving $\frac{900}{18}$ .
$\frac{900}{18} = 50$ , so the temperature rises by 50 °C.
Check it forwards: 2 × 900 × 50 = 90 000 joules, which is what was supplied.

Example 3 — identifying an unknown material

10 000 joules are supplied to a 2-kilogram block of metal, and its temperature rises by 25 °C, with no energy lost to the surroundings. What is its specific heat capacity, and which metal from the list above is it likely to be?

Rearrange for the constant: divide both sides of heat = mass × specific heat capacity × temperature change by mass × temperature change, giving $c = \frac{Q}{m\,\Delta T}$ .
The bottom of the fraction is 2 × 25 = 50.
So the specific heat capacity is $\frac{10\,000}{50} = 200$ joules per kilogram per degree.
Comparing against the values above: 200 is nowhere near copper's 385 or steel's 450, and is closest to lead at 130. It is a dense, low-capacity metal of that kind.
The phrase "with no energy lost to the surroundings" is doing real work in the question. In a real experiment some heat always escapes, so the measured rise is smaller than it should be and the calculated capacity comes out too high.

Example 4 — the same rise in two different materials

You want to raise the temperature of 1 kilogram of water by 10 °C, and separately 1 kilogram of steel by 10 °C. Steel's specific heat capacity is 450 joules per kilogram per degree. How much energy does each need?

For the water: $Q = 1 \times 4200 \times 10 = 42\,000\ \text{J}$
For the steel: $Q = 1 \times 450 \times 10 = 4500\ \text{J}$
The water needs more than nine times as much energy for the same rise, purely because of the constant — the masses and the temperature changes were identical.
Turn that round and it is the statement people get backwards. Supply both with the same 4500 joules and the steel rises 10 °C while the water rises only about 1 °C. The high figure means hard to heat, not quick to heat.

Why it matters in aviation

Anything on an aircraft whose job is to absorb heat is chosen and sized around this constant. Engine oil is asked to carry heat away from bearings and gears at least as fast as it appears there, and how much a given flow of oil can remove for an acceptable rise in its own temperature is exactly mass times specific heat capacity times temperature change. Brakes make the point more sharply. Bringing an aircraft to a stop turns a great deal of kinetic energy into heat in a matter of seconds, and almost all of it lands in the brake units as a temperature rise.

Because a given mass of material can only take up so much energy per degree, that rise is large after a heavy landing, and the aircraft has to wait on the ground until the brakes have shed the energy to the surrounding air before it is fit to depart again. The waiting time is heat capacity and cooling rate, nothing more. Water's unusually high figure has a consequence for airfields too: a large lake or a stretch of sea alongside one changes its own temperature so slowly that it holds the local air temperature steadier, day to night and season to season, than open ground inland does.

What the test asks

Two traps, and the second is the one that costs marks. The first is dropping the mass: an item quotes a specific heat capacity and a temperature rise and offers the product of those two alone as an answer, which is the heat for one kilogram and not for the mass you were given. Check that all three factors are in your sum before dividing anything. The second is reading a high specific heat capacity as meaning a material heats up easily — the exact opposite of what it says. Water's 4200 is the highest figure on the list precisely because water is the hardest of those materials to warm up. If an item asks which of several substances will show the largest temperature rise for a given amount of energy, it is asking for the one with the smallest specific heat capacity. Expect too an item where the material is ice rather than water, and the whole trap is using 4200 instead of 2100.