Isothermal Process Simulator

What Is the Isothermal Process Simulator?

How to use the isothermal process simulator

1. Start from the state it opens in. The lab boots on Amount of gas n 1.0 mol, Temperature T (kelvin) 300 K, Volume before 10.0 L and Volume after 20.0 L, with the four cards reading 1728.8 J, 1728.8 J, 0.0 J and 5.763 J/K. Reset returns all four sliders to exactly that and clears free mode.
2. Move one volume and watch the gold region, not the number. Volume before and Volume after both run from 1.0 to 20.0 L in steps of 0.5, and the region shaded between them is what the work card is reporting. Widen it and the card climbs; narrow it and the card falls.
3. Put the volume after below the volume before. The two are never sorted for you. The arrow on the curve turns round, the vessel strip beneath the plot squeezes instead of stretching, and the work card goes negative — -1728.8 J for 20.0 back down to 10.0 L, which means that much work done on the gas.
4. Move the temperature and watch what does not change. Temperature T (kelvin) runs from 100 to 800 K in steps of 10. Take it up with the volumes untouched and the work card, both pressure cells and the area cell all move, while the Entropy change card stays where it is.
5. Move the amount of gas. Amount of gas n runs from 0.1 to 5.0 mol in steps of 0.1 and scales the work and the entropy change together: 5.0 mol on the opening volumes gives 8644.2 J and 28.81 J/K, 0.1 mol gives 172.9 J and 0.5763 J/K.
6. Read the four cards down the right. Work done by the gas carries W = nRT ln(V2/V1); Heat absorbed carries Q = W with the reason printed beside it; Change in internal energy is labelled exactly zero for an ideal gas; and Entropy change carries nR ln(V2/V1).
7. Read the six cells under the canvas. They give the area under the curve by numerical integration, the pressure before and after, the volume ratio, the pressure ratio and the temperature in degrees Celsius. The strip headed What is happening puts the current state into one sentence and is rewritten on every move.
8. Press Free expansion instead. The curve, the shading, the arrow and the faint second isotherm all go, the button relabels itself Back to the reversible path, and the work and heat cards fall to 0.0 J. The Entropy change card does not move at all.

The step worth repeating is the fourth one. Hold the volumes at 10.0 and 20.0 L and step the temperature through 100, 300 and 800 K: the work card reads 576.3, 1728.8 and 4610.3 J while the entropy card prints 5.763 J/K all three times. The work depends on the ratio and the temperature; the entropy change depends on the ratio alone.

If it is the heat you know rather than the volumes, this is the wrong lab and the first law of thermodynamics simulator is the right one: it holds the gas fixed and drives the change from a heat slider, so it answers “this much heat went in — where did the volume end up?”. This lab answers the opposite question, and neither can be asked on the other. The first law of thermodynamics guide sets out the ledger both of them are working inside.

Worked example: change one thing at a time

Every row below is one setting of the four sliders, and every cell is a string the running lab printed there. The first six are on the preset buttons above; the last six are a drag away. Where a cell and the lab ever part company, believe the lab.

Rows 4 and 5 are the pair to read together. They share the first row’s volumes and differ only in temperature, and all three print 5.763 J/K in the entropy column while the work column runs from 576.3 to 4610.3 J. Nothing has gone wrong: the temperature stands in front of the logarithm in the work and is absent from the entropy change altogether.

Rows 1 and 11 are the diminishing return. Doubling the volume buys 1728.8 J and going tenfold buys 5743.1 J at the same mole and temperature — more, but nowhere near five times more. Drag the volume before down to 1.0 L for a twentyfold expansion and the card reads 7472.0 J. The work goes as the logarithm of the ratio, so twenty times the room is worth ln 20 over ln 2, about 4.32 times a doubling, and doubling the work means squaring the ratio rather than doubling it.

Rows 2 and 9 carry the signs. Running a setting backwards flips the work, the heat and the entropy change together, which is the convention this lab uses: a positive work is work done by the gas, so a compression has to come out negative. The internal-energy card is not in the table because it reads 0.0 J in every one of these settings, and in all twelve free-mode settings too.

The last column is a measurement, not a copy. The area cell runs Simpson’s rule over 400 panels on the curve the canvas draws, and the work card calls a logarithm once; they are different computations that happen to agree to the digit they print in all twelve rows. They are not guaranteed to, as the limits further down quantify, so read them as two routes to one quantity rather than as a check on each other. If entropy is the part you came for and you already have a heat, the entropy change calculator takes it directly.

That table is the reason this lab exists. The gas starts in the same place and finishes in the same place down both columns, the internal-energy row is zero in both, and the entropy row is identical — yet one route does 1728.8 J of work and draws the same in as heat, and the other does nothing at all. Work and heat are properties of the journey; internal energy and entropy are properties of the two ends.

Formula and symbol reference

The lab works from one relation and its two consequences. The path is P = nRT/V, so the product of pressure and volume holds at nRT; the work is the area under that path, W = nRT ln(V2/V1); and the entropy change is the same logarithm without the temperature in front, nR ln(V2/V1). The heat follows from the first law once the internal-energy term is gone.

Only one constant is typed into the lab: R = 8.314 J/(mol·K), the rounded value behind every figure on this page rather than the full SI one. Everything else on the screen comes from your four slider positions. The pressures are absolute pressures, never gauge, and the relation PV = nRT itself belongs with the rest of the gas laws rather than to this page.

One row there is a formatting rule rather than a physical one. The two ratio cells are one quantity read two ways, and each is rounded from its own cell’s value to four significant figures, so on the tenfold setting one of them carries an extra decimal. Read either as the ratio; do not read one as a check on the other.

The physics: an area, a logarithm, and the route you did not take

Naming the curve is the easy half. PV = nRT tells you where the gas ends up, and this lab computes both pressures from it, which is precisely why watching them agree proves nothing: it is the definition of the curve restated. Move either volume slider and the pressure cells report the new state at once, but the gold region the canvas shades between the two volumes is the part that has to be swept out, and that area is what the expansion cost.

Step Volume after up one notch at a time and watch the work card: every half-litre adds less than the one before it, because each is swept against a lower pressure, and a running total whose increments fall as 1/V like that is a logarithm. That is where the shape of the work comes from, and it is why the returns fall away so fast: at 1.0 mol and 300 K a doubling gives 1728.8 J and a twentyfold expansion gives 7472.0 J, not twenty times as much.

The area cell is the only honest check on the screen. It measures the gold region numerically, panel by panel, while the work card evaluates a logarithm once; the two arrive at the same printed figure by genuinely different routes. That is worth more than any amount of multiplying the two pressures together, which is arithmetic the lab has already done for you.

The heat has no freedom here. For an ideal gas the internal energy depends on temperature alone, so holding the temperature still holds it still, and the card says 0.0 J for that reason rather than as a finding. The first law then has two terms where it had three, and the heat card can only repeat the work card — every joule the gas does is paid for in real time by the reservoir.

Now press the free-expansion button, and watch the drawing tell the truth. The gas goes from the same volume to the same volume, so both dots stay exactly where they were, but the curve between them is gone. A gas rushing into a vacuum is not in equilibrium on the way, so it has no pressure and no volume to plot in between, and a lab that shaded the region anyway and then printed 0.0 J would be lying with a picture.

The entropy card does not flinch, because entropy depends on the end points and those have not moved. What changes is where the entropy went: on the reversible route the reservoir loses exactly what the gas gains and the universe comes out level, while the free route takes nothing from anywhere, so the universe gains 5.763 J/K and the change cannot be undone. The full guide to the isothermal process works that argument through with the derivation and eight problems beside it.

Where the isothermal process simulator breaks down

The lab solves its own model exactly, so nothing on the screen ever fails. Everything below is a limit of that model, of the numbers you feed it, or of what the display can carry, and each item says what this lab does about it.

The gas has to be ideal for the internal-energy card to read zero

That card is an identity for an ideal gas, whose internal energy depends on temperature alone. A real gas carries an internal energy that depends on volume too, because its molecules pull on one another, so a real isothermal change does not have exactly zero there and the heat is not exactly the work. A real gas expanding freely also changes temperature rather than holding it. No figure here describes any particular real gas, because none was measured.

The area is the work only on a slow, reversible path

Every shaded region on this canvas assumes the gas stays in equilibrium the whole way, so that it has one pressure at each volume and the curve means something. Free expansion is exactly the case that breaks the assumption, which is why the lab stops drawing a curve rather than shading a region it cannot justify. Any real expansion fast enough to leave equilibrium sits somewhere between the two columns of that table.

Something has to be holding the temperature still

An expanding gas does work, and without a supply of heat it would cool. The lab assumes a reservoir large enough that the temperature never budges, and the heat card is the bill that reservoir pays. Take that away and the process is no longer isothermal at all, whatever the slider says, and the change becomes an adiabatic one with a different curve and a different area.

The amount of gas is held for the whole change

One slider sets the moles and it describes the gas before and after, so nothing leaks in or out between the two states. A vessel that is being filled or emptied is a different problem and this lab will quietly give you the wrong answer for it. Where the end state is the question rather than the cost of getting there, the Boyle’s law calculator is the tool.

Kelvin is not a preference here, it is a prefactor

The temperature multiplies the whole answer instead of appearing on both sides of a ratio, so the 273.15 between the two scales never cancels. The slider is in kelvin only and the Celsius figure is a reading beside it, which is the one design decision on this page that removes a mistake rather than warning about it. Across the whole 100 to 800 K band the error is exactly 273.15 divided by the kelvin temperature, so a Celsius number used in its place is at least 34.14 % wrong, and the sliders only reach that floor at their 800 K stop.

The two pressures on screen will not always multiply back out

Both cells are rounded to two decimals before you see them, and multiplying each by its own volume gives products that disagree in 5.99 % of the settings these sliders reach. It is right often enough to look dependable, which is what makes it a trap. It would not be a check even if it worked every time, because both pressures come from nRT divided by a volume in the first place.

The area cell and the work card are two computations, not one

They agree to the figure they print in all twelve settings tabulated above and in almost every other one, but the area is a sum over 400 panels and the work is a single logarithm, so their last digits can part company. That happens in 78 of 526,110 settings, which is rare rather than never. It is the ordinary signature of numerical integration and is worth knowing before you treat a match as a verification.

The work never lands on a round figure

The cards print one decimal and the underlying value is irrational for every slider position, so nothing here comes out tidy: in 42,600 sampled settings the work landed exactly on a printed tenth not once. If a textbook answer is round, it has been rounded. Compare to the precision on the card rather than expecting the digits to agree all the way down.

The sliders stop where they stop, and the published figures stop with them

The amount of gas runs 0.1 to 5.0 mol, the temperature 100 to 800 K and both volumes 1.0 to 20.0 L. Every extreme quoted on this page — the 99626.1 J at the top, the 2.1 J at the bottom — is the most those four ranges can do and nothing more. A wider range would give a different largest work, and this lab is not the place to find it.

Nothing here has been measured

The four sliders are figures you choose, the gas is an ideal one rather than any real substance, and no reading on this page describes a particular cylinder, compressor or experiment. The pressures are absolute pressures, so a gauge reading has to have the atmosphere added to it before it belongs on this axis. Verify anything you intend to rely on against your own data first.

Where isothermal work is actually used

Putting a floor under the cost of compressing a gas

Squeezing a gas slowly enough to stay at one temperature is the cheapest way to do it, and this lab gives that floor directly: set the volume after below the volume before and read the negative number on the work card. A real compressor runs too fast to stay cool and always needs more than that figure. The gap between the two is what intercooling is for.

Setting a question that almost everyone gets wrong first

“A gas doubles its volume at constant temperature. How much work does it do?” has no single answer until the route is named, and the two preset buttons at the end of the list settle it in about ten seconds. Ask for the entropy change as a follow-up and the surprise lands properly, because that one really is the same down both routes.

Telling a state function from a path function without algebra

Two cards move when you press the free-expansion button and two do not, on one screen, with the end points untouched. That is the whole distinction, drawn rather than defined. It is the reason the entropy of an irreversible change is still computed along an imaginary reversible path, which is a step that looks like cheating until you have watched these four cards behave.

Reading somebody else’s pressure-volume diagram

On a PV plot the work is an area, and areas are easy to misjudge by eye when one axis spans a factor of twenty. Load The widest expansion and the Pressure before P1 cell reads 33256.00 kPa, so the vertical axis has to stretch past that while the work card reports 99626.1 J. Read the axis labels before reading the shape, on this canvas and on any other.

Getting an entropy change without measuring any heat

The entropy card needs only the amount of gas and the ratio of the volumes, so it answers for the free route where there is no heat to measure at all. That is the standard way round a process nobody can follow: take the end points, invent a reversible path between them and compute along that. The dash in the work cell and a real figure in the entropy cell are that idea in two readouts.

Costing one leg of a cycle before drawing the whole thing

Engine cycles are built from legs, and the isothermal ones are where heat crosses the boundary in step with the work. Price a single leg here first, with its own sign, and the cycle diagram stops being a wall of symbols. The Carnot efficiency simulator takes over at the point where two of these legs are joined by two adiabatic ones.

Where to go next

For the derivation, the eight worked problems and the diagrams that go with them, read how to find the work in an isothermal process. If you would rather type figures than drag them, the isothermal process calculator takes the same four and also runs the relation backwards, returning either volume, the amount of gas or the temperature from a work you already have.

The neighbouring questions have tools of their own. Boyle’s law and its simulator own where the gas ends up; the guide to entropy and the entropy lab own the card this page leaves unchanged; and the ideal gas law with its calculator covers a single state rather than a process.

Further out, the complete guide to the gas laws places PV = nRT among the rest, the laws of thermodynamics and their lab set out the rules all of this obeys, and the guide to work done in physics covers the force-times-distance version of the quantity shaded here. The rest is in the library of physics simulations and on the blog, and the site search will find a topic by name.