A gas gets hotter when you squeeze it and colder when you let it expand, with no flame and no refrigerator anywhere in the story, and solids change size when their temperature changes. By the end of this lesson you should be able to explain both of those gas effects in terms of work rather than heat, calculate the work done when a gas is compressed or allowed to expand, work out how much a solid object grows for a given temperature rise and run that backwards, and say which way a bimetallic strip bends and why.
Squeezing a gas warms it
Take a syringe full of air, block the nozzle, and push the plunger in hard and fast. The air inside comes out measurably warmer, and nothing was heated. To see why, follow one particle as it bounces off the plunger while the plunger is moving inwards. A ball bouncing off a stationary wall comes away at the speed it arrived; a ball bouncing off a bat that is swinging towards it comes away faster, because the bat has pushed on it while they were in contact. The advancing plunger is that bat. Every particle that strikes it while it is moving inwards leaves faster than it arrived, and with the plunger travelling in and countless collisions happening, the average kinetic energy of the whole population rises — which is a rise in temperature.
The accounting is the more general way to see it, and it uses conservation of energy from the energy module. You pushed the plunger against the gas pressure through some distance, so you did work; work is a transfer of energy, so that energy went somewhere; it did not leave the sealed cylinder, so it is now in the gas, as internal energy. The temperature rose because energy was added — the only unfamiliar part is that it was added by doing work rather than by heating. How much work depends on the pressure being pushed against and on how much the volume changes, and there is a neat way to see the formula for it. The force on a piston of area A at pressure p is p times A, because pressure is force divided by area. Work is force times distance, so pushing that piston through a distance d does work equal to p times A times d — and A times d is precisely the change in the gas's volume.
Work done on or by a gas at constant pressure
- the work done, in joules (J)
- the pressure the piston moves against, in pascals (Pa)
- the change in the gas's volume, in cubic metres (m³)
That expression assumes the pressure stays roughly constant during the change, which is a fair approximation for a small change and is all these items ever ask for.
One honest qualification about the warming, because it decides whether the effect is noticeable. If you compress the gas slowly, in a thin metal cylinder that conducts well, the heat produced leaks away into the cylinder and the room as fast as it appears, and the gas ends up at very nearly the temperature it started at. If you compress it quickly, there is no time for any of that, and the full temperature rise shows up. That is why a bicycle pump warms up when you are pumping hard and not when you are dawdling, and it is why a diesel engine works at all: it compresses its air so fast and so far that the air reaches several hundred degrees, hot enough to ignite fuel sprayed into it with no spark plug anywhere in the design.
Letting a gas expand cools it
Run the argument backwards and it comes out the other way. A particle bouncing off a piston that is retreating comes away slower than it arrived, because the piston gave way while they were in contact instead of pushing back fully. The gas is doing work on the piston rather than receiving work from it, and if no heat is coming in, that work has to be paid for out of the gas's own internal energy. Its particles slow down, and its temperature falls. Hold a can of compressed gas and discharge it for a while and the can becomes cold enough to frost over; open a large gas cylinder quickly and the valve does the same. Nothing was refrigerated, and no heat was removed — the gas cooled because it did work on its way out.
The atmosphere runs the same process on a very large scale, and it is the mechanism referred to in the previous lesson. A parcel of air that rises finds itself surrounded by lower pressure, so it expands, and expanding means pushing the surrounding air outwards, which is work done by the parcel on its surroundings. That work comes out of the parcel's internal energy, so it cools as it climbs even though no heat has been taken from it. Air that descends is compressed by the increasing pressure around it, has work done on it, and warms for the same reason in reverse. This is why the lapse rate exists at all, and why the cooling of rising air is not a matter of the air losing heat to anything.
Thermal expansion of solids and liquids
Solids and liquids change size with temperature too, and for a reason that follows directly from the particle picture. In a solid the particles vibrate about fixed positions; heat it and they vibrate through a larger swing, and because the forces holding them apart stiffen faster than the forces pulling them together, the average distance between neighbouring particles increases slightly. Multiply that tiny increase by the enormous number of particles across the length of an object and the object measurably grows — in every direction at once, so a hole in a metal plate grows as well as the plate itself. The particles themselves do not get bigger; nothing about them changes except how far apart they sit on average.
How much an object grows depends on how long it was to start with, on the temperature change, and on which material it is — the same three-ingredient pattern as the heat calculations earlier in this module. The material's contribution is a number called the coefficient of linear expansion, written with the Greek letter alpha, and it is the fraction of its own length that the material gains per degree.
Linear thermal expansion
- the increase in length, in metres (m)
- the coefficient of linear expansion of the material, per degree Celsius (1/°C)
- the original length, in metres (m)
- the temperature change, in degrees Celsius (°C) or kelvin (K)
The coefficients are all very small numbers, which is why expansion goes unnoticed until something is either very long or very tightly fitted. Steel is 0.000012 per degree, which is twelve millionths — so a steel bar gains twelve millionths of its length for every degree it warms. Concrete is almost identical at about twelve millionths, which is a genuine piece of luck and the reason steel-reinforced concrete does not tear itself apart in the sun. Glass is about nine millionths, brass nineteen, and aluminium twenty-three, so aluminium expands roughly twice as much as steel for the same rise.
Liquids expand considerably more than solids for the same temperature rise, because their particles are not held in a lattice and have more freedom to move apart, and gases expand more again — enormously more, since a gas at constant pressure gains about a three-hundredth of its volume per degree, as Charles's law says. That ordering explains the ordinary liquid-in-glass thermometer: the liquid and the glass both expand, but the liquid expands so much more that the column visibly climbs the tube, and if the two expanded equally the instrument would read nothing at all.
Water is the famous exception and deserves a sentence, because it explains something covered in the matter and fluids module. Between 0 °C and 4 °C water contracts as it warms rather than expanding, so it is at its densest at 4 °C, and above that temperature it behaves normally. It also expands when it freezes rather than contracting, which is why ice is less dense than water and floats, why a pond freezes from the top down and stays liquid underneath, and why water left in a pipe over a hard winter splits the pipe.
The everyday consequences of expansion are all about leaving room for it. Bridges carry expansion joints — deliberate gaps with interlocking teeth — because a deck fixed rigidly at both ends would either buckle in summer or tear in winter. Railway track is either laid with small gaps or welded into continuous lengths held under carefully managed tension. Pipework carrying hot fluid is run with loops or bellows in it rather than dead straight. Overhead cables are strung with deliberate slack so that they tighten rather than snap in cold weather, and they sag noticeably in summer. And glass cracks when heated or cooled unevenly, because the part that changed temperature tries to change size while the part that has not yet holds it, so the two pull against each other.
The bimetallic strip
The neatest use of unequal expansion is to convert a temperature change into a movement. Bond a strip of brass and a strip of steel together along their whole length, so that neither can slide relative to the other, and warm the pair. The brass tries to grow by nineteen millionths of its length per degree while the steel manages only twelve, and since they cannot slide, the only way both can be accommodated is for the whole assembly to curve — with the brass taking the longer path round the outside of the curve and the steel the shorter path on the inside. Cool the strip back down and it straightens; cool it below its starting temperature and it curves the other way. So the rule to remember is that the strip bends away from the metal that expands more, or equivalently that the more expansive metal ends up on the outside.
A movement that is produced by a temperature change and reverses itself when the temperature returns is exactly what a thermostat needs. Mount one end of the strip and let the free end carry an electrical contact, and the contact closes or opens at a set temperature; adjust how far the contact sits from the strip and you have adjusted the temperature at which it switches. The same component appears as a thermal cut-out that breaks a circuit if something overheats, in the older style of flashing vehicle indicator, and in simple oven and iron controls.
Why it matters in aviation
A gas turbine is this lesson turned into a machine. Air is taken in and compressed through a series of stages, and the work done on it raises its temperature substantially before any fuel has been burned — the syringe effect, scaled up and running continuously. Fuel is then burned in that already hot air, and the hot gas expands through the turbine and out of the nozzle, doing work on the turbine blades and cooling as it goes. Part of that work drives the compressor and the remainder is what the engine exists to produce.
Two consequences are worth drawing out. The temperature limits a crew monitors exist because compression has already used part of the material's margin before combustion adds anything, so how much heat may be added is set by what the metal downstream will tolerate rather than by how much fuel will fit. And the cooling that accompanies expansion is put to work in the other direction: air bled off hot and at high pressure is expanded through a turbine in an air-conditioning pack, which cools it enough to be mixed and delivered to the cabin, using nothing but the effect described in this lesson.
What the test asks
The central misconception here is that a temperature can only be changed by adding or removing heat. An item will describe a gas being compressed rapidly in an insulated cylinder and offer the answer that its temperature is unchanged because no heat was supplied — which mistakes one of the two routes for both of them. Work does the job just as well, and the diesel engine is the proof. The mirror-image version has a gas expanding and asks where the cooling came from, with an option saying heat was lost to the surroundings; it was not, the gas did work. On expansion, the standard distractor claims that the particles themselves expand, and a second one claims that a hole in a heated metal plate gets smaller as the metal around it grows — it gets larger, because every dimension grows. And on the bimetallic strip, expect both bending directions offered as options, so have the rule ready: the more expansive metal takes the outside of the curve.