Work in expansion - reversible isothermal w = −nRT ln(V₂ - V₁), irreversible w = −P_ext ΔV: Foundations

Everything in the parent note — $w$ , $P_{ext}$ , $d V$ , the integral, the logarithm — is just machinery for measuring that swept area. Below we build each piece from nothing, in an order where every new symbol leans only on ones already explained.

1. Volume $V$ — the space the gas fills

Picture it: a gas trapped in a cylinder by a movable piston (a sliding lid). If the piston slides out, the gas has more room — $V$ grows. If it slides in, $V$ shrinks.

Figure — Work in expansion - reversible isothermal w = −nRT ln(V₂ - V₁), irreversible w = −P_ext ΔV

Why the topic needs it: "Expansion" means $V$ increases. Every formula on the parent page tracks how something behaves as $V$ changes, so $V$ is our fundamental moving part.

$V_{1}$ ::: the starting volume (before expansion)
$V_{2}$ ::: the final volume (after expansion)
$Δ V$ ::: the change in volume, defined as $Δ V = V_{2} - V_{1}$

2. Pressure $P$ — how hard the gas pushes

Picture it: the gas molecules constantly bang against the piston. Add up all those tiny hits over the piston's area and you get one steady outward push — that push, per square metre, is the pressure.

The topic uses two different pressures, and confusing them is the #1 error:

Why two? Whether the piston moves — and how much work is done — is decided by the contest between these two. This tug-of-war picture is the heart of "reversible vs irreversible":

If $P_{gas}$ is only a hair above $P_{ext}$ ::: expansion is slow and gentle — reversible
If $P_{gas}$ is far above $P_{ext}$ ::: expansion is a sudden burst — irreversible

3. The infinitesimal $d V$ and the integral $\int$

Why this tool and not plain multiplication? Multiplication ( $P \times Δ V$ ) gives the area of a rectangle — correct only when pressure is a flat horizontal line (constant $P_{ext}$ , the irreversible case). When pressure follows a curve (the reversible case, where $P_{ext}$ tracks the falling $P_{gas}$ ), you need the integral to get the area under that curve. This is exactly why the two cases have different formulas.

4. Work $w$ — energy delivered by pushing

Now that $d V$ and $\int$ are defined, we can write the work formula honestly.

Picture it — where the area comes from: the tiny force on the piston is (pressure) × (piston area). When the piston creeps out by a tiny slice, the volume grows by a tiny $d V$ , and the little bit of work is $P_{ext} d V$ — exactly a thin sliver of area under a pressure-vs-volume graph. Summing all slivers from $V_{1}$ to $V_{2}$ gives the total:

Gas expands ( $Δ V > 0$ ) ::: $w < 0$ (system loses energy doing work)
Gas compressed ( $Δ V < 0$ ) ::: $w > 0$ (surroundings put energy into the gas)

We link this convention and the energy balance it feeds to the First Law of Thermodynamics.

5. The ideal gas law $P V = n R T$

Why the topic needs it: in a reversible expansion we set $P_{ext} = P_{gas}$ , and the ideal gas law lets us write that pressure as $P_{gas} = \frac{n R T}{V}$ . That substitution is the only reason a logarithm appears in $w_{rev}$ . Deep dive: Ideal Gas Law.

Rearranged for pressure ::: $P = \frac{n R T}{V}$ — pressure and volume are inversely linked at fixed $T$
On a $P$ -vs- $V$ graph at fixed $T$ ::: this traces a smooth downward-sloping curve (a hyperbola)

6. Isothermal — temperature held constant

Why the topic needs it: keeping $T$ constant is what lets us pull $n R T$ outside the integral as a constant, turning a hard integral into $- n R T \int \frac{1}{V} d V$ . See Isothermal Process.

7. The natural logarithm $ln$

This second rule collapses $ln V_{2} - ln V_{1}$ into $ln (\frac{V _{2}}{V _{1}})$ — the compact form of the reversible result.

If $V_{2} > V_{1}$ (expansion) ::: $ln (V_{2} / V_{1}) > 0$ , so $w_{rev} = - n R T ln (V_{2} / V_{1}) < 0$
If $V_{2} = V_{1}$ (nothing moves) ::: $ln (1) = 0$ , so $w = 0$ — no volume change, no work
If $V_{2} < V_{1}$ (compression) ::: $ln (V_{2} / V_{1}) < 0$ , so $w > 0$

8. The two work formulas these foundations build to

With every symbol now earned, here are the two destinations of the whole topic. Both come from the same work integral $w = - \int_{V_{1}}^{V_{2}} P_{ext} d V$ — they differ only in what $P_{ext}$ does along the path.

The full step-by-step derivation of each lives on the parent note; here you now hold every symbol they use.

9. Reading a PV diagram

This picture is what makes "reversible work > irreversible work" obvious: the curved reversible path encloses more area than the low flat irreversible rectangle. Explore further in PV Diagram.

Prerequisite map

The diagram below shows the dependency order: an arrow "X → Y" means "you must understand X before Y makes sense." Read it top to bottom — the raw ideas (volume, pressure, the integral) flow into the work integral, which then splits into the reversible and irreversible formulas that make up the whole topic. If the diagram fails to render, the same order is exactly sections 1 → 9 above.

Equipment checklist

What does the $Δ$ in $Δ V$ mean? ::: Final value minus initial value: $Δ V = V_{2} - V_{1}$ .
What is the difference between $P_{gas}$ and $P_{ext}$ ? ::: $P_{gas}$ pushes the piston out from inside; $P_{ext}$ pushes it in from outside.
Why is expansion work negative under the IUPAC convention? ::: Energy leaves the system when the gas does work on the surroundings, and leaving energy is counted as negative.
Why do we integrate instead of just multiplying $P \times Δ V$ ? ::: Because pressure can change during expansion; the integral adds up thin slivers where pressure is momentarily constant.
Geometrically, what does $\int_{V_{1}}^{V_{2}} P d V$ represent? ::: The area under the pressure-versus-volume path.
Write the full work integral with its limits. ::: $w = - \int_{V_{1}}^{V_{2}} P_{ext} d V$ .
State the reversible isothermal work formula. ::: $w_{rev} = - n R T ln (V_{2} / V_{1})$ .
State the constant-pressure irreversible work formula. ::: $w_{irrev} = - P_{ext} Δ V$ .
Where does the logarithm in $w_{rev}$ come from? ::: From summing the $1/ V$ slivers, i.e. $\int \frac{1}{V} d V = ln V$ .
What does "isothermal" let us do to $n R T$ inside the integral? ::: Treat it as a constant and pull it outside the integral.
What is the value of $R$ and its SI units? ::: $R = 8.314 J mol^{- 1} K^{- 1}$ .
Rewrite $ln V_{2} - ln V_{1}$ as a single log. ::: $ln (\frac{V _{2}}{V _{1}})$ .
On a PV diagram, why is reversible work larger than irreversible? ::: The reversible curved path encloses more area than the lower flat constant- $P_{ext}$ rectangle.

Related destinations once these foundations are solid: Entropy and Irreversibility, Free Expansion, Carnot Cycle, Heat Capacity.