Avogadro's Law Simulator

What Is the Avogadro's Law Simulator?

How to use the Avogadro's law simulator

1. Load a case, or start from Reset. The five buttons under Load a case — One mole at 1 atm, Blowing up a balloon, A breath at body heat, The same gas in a furnace and Letting gas out of a gas holder — write all four sliders together and print a name such as “Letting gas out of a gas holder” on the case line. Move any slider and that line reverts to “custom setting”. Reset returns to 1.00 mol, 2.00 mol, 100 kPa, 0 °C and air.
2. Set the two amounts. Amount before and Amount after each run from 0.10 to 5.00 mol in steps of 0.01, reading back as “1.00 mol” and “2.00 mol”. They are the only controls that move Volume ratio, Gas added and Molecules after, and they redraw both gas columns and both marked points on the plot.
3. Choose the pressure the pistons work against. Pressure covers 20 to 500 kPa in steps of 0.025, which is fine enough to land exactly on 101.325 kPa as well as 100 kPa. Raising it squeezes Molar volume and both volume readings without touching the ratio. Holding the amount still and squeezing instead is a different law, the one the Boyle's law simulator draws as a pressure-against-volume curve.
4. Choose the temperature. Temperature runs from -100 to 500 °C in whole degrees, and the Temperature stat converts it for you, reading “273.15 K” at 0 °C. Warming the gas raises the molar volume without moving the ratio; that effect on its own, at a fixed amount of gas, is what the Charles's law simulator plots against temperature in kelvin.
5. Press a different gas. Helium, Air and Carbon dioxide change Molar mass, Gas density, Mass of the gas, the sentence naming the other two gases and the colour of the dots. They change no volume, no ratio, no column height and no dot count — which is the whole experiment on this page.
6. Read the three sentence lines under the canvas. Why restates the current setting in words, The other two gases weighs the same amount of the two gases you did not pick, and What to notice changes as you cross the landmarks — equal amounts, a cold carbon-dioxide setting, a few hundred kPa, or -50 °C and below. Pause stops the jiggling without moving a single number.

Worked example: change one thing at a time

Every row below starts from the state the lab boots in — 1.00 mol becoming 2.00 mol of air, at 100 kPa and 0 °C — and changes exactly one thing about it. Each cell is a string the running lab printed at those control positions; where a cell and the lab disagree, the lab is right.

Rows 2 and 3 are the reason this lab exists. Swapping air for helium and then for carbon dioxide leaves six of the seven columns byte for byte where they were, and moves only the mass: “8.006 g”, “57.92 g”, “88.02 g”. Alongside the table, Molar mass steps 4.003, 28.96 and 44.01 g/mol and Gas density steps 0.1763, 1.275 and 1.938 g/L, while Molecules after sits at “1.204e24 molecules” through all three. The dots change colour and not number.

Row 4 separates the molar volume from the law. Five times the pressure cuts Molar volume from “22.711 L/mol” to “4.542 L/mol” and both volumes fall with it, yet Volume ratio does not stir from “2.000”. The straight line on the plot tilts down towards the axis; it does not bend, and it still starts at the origin. That is the difference between the equation of state, which the pressure and temperature control, and Avogadro's law, which they cannot reach.

Row 5 adds heat and gets the same answer. At 500 kPa and 500 °C the molar volume climbs back to “12.857 L/mol” and the volume after reads “25.71 L”, with the ratio still “2.000” to the last digit. If you would rather run that ratio on measurements of your own, the Avogadro's law calculator rearranges the same relation for whichever of the four quantities you are missing and asks for no pressure and no temperature at all. It is the identical arithmetic with the molar volume divided out, not a second opinion.

Formula and symbol reference

The lab works out the molar volume from the pressure and the temperature, multiplies it by each amount for the two volumes, and takes the ratio of the amounts for the number the two cylinders are drawn to. The mass, the density and the molecule count hang off the side of that and never feed back into it. Ranges marked “in this lab” are the controls' own ends and the strings the lab prints there.

Two of the figures above are worth a second look when you meet them on screen. The 22.711 L/mol the lab opens on belongs to 100 kPa, and the 22.414 L/mol the One mole at 1 atm button produces belongs to 101.325 kPa; neither is the molar volume, and the standards they come from are taken apart in the full guide to Avogadro's law. Air's 28.96 g/mol is a mean over a mixture, so the mass and density readings for air describe a sample of ordinary air rather than a substance.

The physics: why the volume follows the molecule count

Molecules keep a gas at pressure by hitting the walls, and every molecule hits just as hard as any other of the same speed whatever it is built from. Add more of them behind a piston that can rise and the piston rises until the blows are as thinly spread as they were before. That is why both cylinders here are drawn with a free piston and a pressure arrow on top: take the freedom away and there is no law left to watch.

The two columns share one vertical scale, and the larger of them always fills the same fraction of the box. So the height you see on the right divided by the height on the left is Volume ratio, which is also the ratio of the two amounts — the lab does not have a second scale to hide a discrepancy in. Drag either amount slider and the dots rebuild with it, twelve per mole rounded to a whole number, with a floor of two so the smallest settings still show something.

The plot sharing the canvas with the cylinders puts volume against amount of gas from zero to 5.00 mol, and what it draws is a straight line climbing from the origin at a slope equal to the molar volume, labelled in the plot as the same line for every gas. The pale ring sits on the before point and the gold ring on the after point, with dotted guides down to both axes. Move the pressure or the temperature and the line pivots about the origin, because Vm = R T / P is the only thing changing.

That expression is the ideal-gas equation with the amount divided out, which is why this lab needs the two quantities the ratio does not. The guide to the ideal gas law sets out where it comes from, and the ideal gas law calculator solves it for any one of pressure, volume, amount and temperature when you need a number rather than a picture. Here the lab prints the molar volume to three decimals and the two volumes to four significant figures.

Molecules after is the second amount multiplied by Avogadro's number, so “1.204e24 molecules” at 2.00 mol is exact arithmetic on a defined constant rather than anything measured. Gas density comes from the other side of the same molar volume, as the molar mass divided by it: helium reads “0.1763 g/L” against air's “1.275 g/L” on the opening setting, which is the whole of why a helium balloon rises. Working a density out from conditions of your own is what the gas density calculator is for.

Where Avogadro's law breaks down

The lab solves its own model exactly, so nothing on screen ever fails. Every limit below is a limit of that model, of the apparatus it stands for, or of the drawing, and each item says what the lab does about it.

A vessel whose walls will not move

Every container in this lab can grow: both pistons ride free, the arrow above each carries the same pressure, and all five case buttons describe a piston, a balloon skin, a chest or the roof of a holder. That is deliberate, because the mistake made most often with this law is to apply it to a sealed steel bottle, where the walls settle the volume and the gauge climbs instead. Nothing on these sliders can describe that; what covers it is the full equation of state with the volume fixed.

A few hundred kilopascals and upwards

Take Pressure to 300 kPa or beyond and What to notice changes to say that the straight line is close but no longer exact, because real molecules take up room and pull on one another. The lab keeps drawing a perfectly straight line through the origin, since that is what an ideal gas does. No percentage is given for the departure, here or anywhere on this page, because nothing in this build computed one.

Cold, and the approach to condensation

At -50 °C and below the same line reports that cold and dense is where real gases stray furthest, and that near their boiling points they leave the straight line altogether by condensing. Only one of these warnings shows at a time, and they rank: equal amounts first, then carbon dioxide freezing, then pressure, then cold — so -60 °C at 400 kPa prints the pressure line, not this one. The readings carry on regardless: at -100 °C the molar volume prints “14.396 L/mol” and both volumes follow it down. Treat everything at that end as the volume an ideal gas would occupy, not a prediction about a real one.

Carbon dioxide at the cold end of the slider

Choose Carbon dioxide and set -78 °C or colder and the lab names the problem outright: carbon dioxide turns solid near -78 °C at ordinary pressures, so a real sample would have frozen out before the piston moved. The boundary is exact rather than approximate — -77 °C does not trigger it — and helium at the same -100 °C gets the general cold warning instead, because it has no such transition anywhere near there.

Gas that leaks away, dissolves or is used up

The amount you set with Amount after is the amount still present as gas. Helium creeping out through a balloon skin, a gas going into solution in the liquid beneath it, or one eaten by a reaction all shift that figure with nobody touching a valve, and the lab cannot represent any of them. Gas added reports the difference between the two sliders and says nothing about how the gas got in or out.

The dots are a drawing rule, not a molecule count

Twelve dots per mole, rounded, is a rule chosen so the two columns look honest beside each other: 12 dots before and 24 after on the opening setting. The real count is the one in Molecules after, and the gap between the two is a factor of about 5e22. Because the rule is the same for every gas, the dots also make the headline invariant visible — they change colour when you change gas and they never change number.

What the drawing leaves out when there is no room

Every string on the canvas is measured before it is drawn, then shortened or dropped rather than overprinted. What survives depends on the width of its band, not the width of your screen. As a band narrows the caption under the plot sheds its clause first, “a straight line through the origin: double the moles, double the volume” becoming “double the moles, double the volume”, and the one under the cylinders follows, “same pressure, same temperature: 1.00 mol and 2.00 mol” becoming “1.00 mol and 2.00 mol”. Narrower still, in an article column, the chart caption goes altogether, and the in-plot label shortens to “the same for every gas”.

The lab's own arithmetic and display

Three decimals on the molar volume, four significant figures on the volumes, the masses and the density, three decimals on the ratio, two on the amounts. A number that looks exact on the panel has usually been rounded to get there, so a factor worked out by dividing two printed readings is not the factor the physics gives. Take the ratio from Volume ratio, which the lab computes from the amounts themselves.

Where Avogadro's law is actually used

Collecting gas in a syringe or a burette

Reading a graduated barrel means reading a volume, and in a laboratory held at one pressure and temperature that reading stands in for an amount of substance. The lab makes the substitution visible: the two cylinders are exactly the same apparatus twice over, and the only thing separating their readings is how much gas is inside. Collect twice as much and the plunger travels twice as far, which is the straight line on the plot with its two rings.

Breathing, and how much air a chest moves

Press A breath at body heat: 0.10 mol at 1 atm and 37 °C occupies 2.545 L, and doubling it to 0.20 mol takes 5.090 L. Those are the conditions a lungful is measured at, and the reason a respiratory volume has to be quoted with a pressure and a temperature beside it — the same 0.20 mol at 0 °C and 100 kPa would fill rather less.

Metering gas, and correcting it to a standard

What a meter counts is the volume that went past it under whatever conditions the pipe was in, while the bill has to be written in a standard volume. The two figures the lab prints for one mole, “22.711 L/mol” at 100 kPa and “22.414 L/mol” at 101.325 kPa, are exactly the sort of gap that step has to close. Press One mole at 1 atm and then One mole at IUPAC STP to see it happen with nothing else moving.

Why a helium balloon rises

A balloon and the air it shoved aside share a pressure, a temperature and a volume, so by this law they hold equal numbers of molecules; what differs is what those molecules weigh. On the opening setting the lab reads “0.1763 g/L” for helium against “1.275 g/L” for air, and the lift is that difference multiplied by the volume. It is gross lift, with the skin, the string and anything being carried still to come off it.

Flue gas, ducts and chimneys

Press The same gas in a furnace and one mole needs “64.283 L/mol” at 500 °C instead of the “22.711 L/mol” it takes at 0 °C, which is why hot gas needs such generous ducting. That factor of about 2.83 between the two printed molar volumes is Charles's law, though, not this one: Avogadro's law is the part that stayed at “2.000” while the temperature moved.

Reading a balanced equation off two volumes

Where gases are measured at one pressure and temperature, a ratio of volumes is already a ratio of moles, so the coefficients of a reaction can be settled without weighing anything. Air is a mixture, and the law treats it no differently from a pure gas, which is what makes the Air button as legitimate a choice here as either of the other two. The ideal gas law simulator is the place to go when the pressure and the temperature will not sit still between the two measurements.

Where to go next

The full account — the definition, why the textbook says 22.4 L while this tool says 22.711, the difference between the law and the number, and seven worked problems — is in Avogadro's Law (V1/n1 = V2/n2). For your own numbers rather than the sliders', the Avogadro's law calculator solves the ratio in four directions and shows its working, and the ideal gas law is the equation all of these fold back into.

Next door, Boyle's law holds the amount of gas still and squeezes it, while Charles's law holds it still and heats it; their labs, the Boyle's law simulator and the Charles's law simulator, each move one quantity at a time as this one does. The gas density calculator takes the density side further, and the rest are in the library of physics simulations.