Foundations — Thermodynamic processes — isothermal (T const), isochoric (V const), isobaric (P const), adiabatic (Q = 0)

This page assumes you have seen nothing. Every symbol the parent note (parent topic) uses is built here from scratch, in the order that lets each one lean on the last.

1. The gas in a box — our whole universe

Everything in this chapter happens inside one picture: a cylinder of gas with a piston (a sliding lid) on top.

Why start here? Because pressure, volume, temperature and energy are all properties of this trapped gas. Look at the figure: the gas is the shaded region, the piston is the movable wall, and the arrows show the two things that can cross the boundary — heat and work. Hold that image; every symbol below lives in it.

2. Volume $V$ — how much room the gas fills

Why the topic needs it: two of the four processes are defined by what $V$ does. "Isochoric" = $V$ frozen; expansion = $V$ increasing. If you can point to $V$ in the picture (the shaded height), you can read every process.

$V=A\times h(\text{face area}\times \text{column height})$

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

Why "force over area" and not just "force"? Because the same pushing gas exerts more total force on a wide piston than a narrow one — but the pressure (the push per patch of wall) is the same. Pressure is the honest, size-independent measure of how hard the gas shoves. Look at the figure: same particle-drumming, but the wide piston catches more hits, so total force is bigger while $P$ stays fixed.

$P=\frac{F}{A}(\text{force}÷\text{area it acts on})$

The parent note's phrase "$P$ held fixed" (isobaric) just means: keep the drumming intensity constant, e.g. by putting a fixed weight on the piston.

4. Temperature $T$ — how fast the particles jiggle

Why the topic needs it: "isothermal" = $T$ frozen. And $T$ is the one number that controls the hidden internal energy (Section 8).

5. The amount of gas: $n$ and the constant $R$

Why needed: every formula in the parent — $PV=nRT$, $C_P=C_V+R$, the work integrals — carries $R$ as this bridge.

6. The Ideal Gas Law — tying $P$, $V$, $T$ together

This is the master link. It says the four numbers of the gas are not independent — fix any three and the fourth is forced. See Ideal Gas Law for the full build.

7. Heat $Q$ — energy crossing the wall because of a temperature gap

8. Internal energy $U$ and the symbol $\Delta$

Why the topic leans on this: it's why isothermal gives $\Delta U=0$ (hence $Q=W$), and why $\Delta U=n{C}_V\Delta T$ works in every process, not just constant volume.

9. Work $W$ — energy the gas spends by pushing the piston

The figure shows the key move: push the piston out by a tiny slice $dx$. The gas force is $F=PA$, and the swept volume is $dV=Adx$. So the tiny work is

$dW=Fdx=PAdx=P(Adx)=PdV.$

Adding up all the slices gives the parent's boxed result, $W=\int PdV$ — the area under the curve on a pressure–volume graph. (What "$\int$" and "$dV$" mean is unpacked next.) Full treatment: Work done by gas — PV diagrams.

10. The calculus symbols: $d$, $\int$, and $\ln$

The parent uses three pieces of calculus notation. Here is each as a picture — no prior calculus needed.

Why this tool and not simple "$P\times \Delta V$"? Because along most processes the pressure changes as the gas expands (look at the curved graph). Plain multiplication only works when $P$ is flat (the isobaric rectangle). The integral is the honest way to add up a changing height — that's exactly the question "what's the area under a curve?" and $\int$ is its answer.

Why $\ln$ and nothing else? The isothermal curve has height proportional to $1/V$. The area under a $1/V$ shape is, by definition, the natural log. No other function measures that area — so $\ln$ is forced on us, not chosen for style.

11. Heat capacities $C_V$, $C_P$ and the ratio $\gamma$

Why the topic needs it: $\gamma$ is the single knob controlling the adiabat's shape and the adiabatic work formula.

12. The First Law — the balance sheet that never fails

Every symbol here is now defined: $\Delta U$ (§8), $Q$ (§7), $W$ (§9). This single line, combined with $PV=nRT$ and one frozen variable, generates all four processes. See First Law of Thermodynamics.

13. How it all feeds the topic

Read top to bottom: the four state numbers build the Ideal Gas Law; temperature and $C_V$ build internal energy; heat and work build the First Law; $\gamma$ and $\ln$ shape the curves — and everything pours into the four named processes.

Equipment checklist

Test yourself — cover the right side and answer aloud.