Foundations — γ = Cp - Cv — for monatomic, diatomic, polyatomic

This page assumes nothing. Before you can read the parent note γ = Cp/Cv, you must be fluent in every letter it uses. We build them one at a time, each on top of the last.

1. Heat, $Q$ — energy in transit

The picture. Imagine a gas trapped in a cylinder with a flame underneath. The flame pushes energy through the metal wall into the gas. That flowing energy is $Q$.

Why the topic needs it. Every heat-capacity idea starts with the question "if I pour in this much heat $dQ$, what happens?" So $Q$ is the input we track.

2. Temperature $T$ — the average jiggle

The picture. In the figure above, faster arrows = hotter gas = higher $T$. Temperature is not the total energy — it's the average speediness per molecule.

Why the topic needs it. Heat capacity asks "how much heat raises the temperature by one degree?" — so we need both $Q$ and $T$ to even pose the question.

3. Volume $V$, Pressure $P$, and the piston

The picture. The cylinder has a lid that can slide — a piston. If the piston is bolted still, $V$ can't change ($dV=0$). If the piston is free and a fixed weight sits on it, the gas keeps a fixed push $P$ but is free to expand, so $V$ grows.

Why the topic needs it. The whole reason $C_p\ne C_v$ is these two setups — locked piston (constant $V$) versus loaded piston (constant $P$). You cannot understand Mayer's relation without picturing both.

4. Work $W$ — energy the gas spends pushing

Why the topic needs it. At constant pressure, some heat leaks out as this pushing-work instead of warming the gas — that "leak" is the whole reason $C_p>C_v$.

5. Internal energy $U$ — energy stored inside

The picture. If $T$ is the average speed per molecule, then $U$ is the grand total over every molecule and every kind of motion.

Why the topic needs it. The First Law (next) splits incoming heat into "goes into $U$" and "goes into work $W$." Everything hinges on $U$.

6. The First Law — energy bookkeeping

This is just energy accounting: every joule you pour in must be accounted for — it either stays inside (raises $U$) or gets spent pushing the piston ($PdV$). Built fully in First Law of Thermodynamics.

Why the topic needs it. This single equation, applied twice (once with $dV=0$, once with $P$ fixed), produces both $C_v$ and $C_p$.

7. Moles $n$ and the gas constant $R$

Why the topic needs it. At constant pressure, differentiating $PV=RT$ gives $PdV=RdT$ — the exact size of the work-leak. And Mayer's relation $C_p-C_v=R$ is stated per mole, so $n$ and $R$ must be crystal clear. See Mayer's Relation.

8. Heat capacities $C_v$ and $C_p$

Reading the notation. The big fraction $\frac{dQ}{dT}$ means "small heat divided by the small temperature-rise it caused" — a rate. The little subscript $V$ or $P$ says "keep this thing constant while you measure."

Why the topic needs it. These two numbers ARE the topic. $\gamma$ is their ratio.

9. Degrees of freedom $f$ and equipartition

Full build in Degrees of Freedom & Equipartition Theorem and Internal Energy of Ideal Gas.

Why the topic needs it. This is what turns $\gamma$ from an abstract ratio into a molecule-shape detector: $\gamma =1+2/f$.

10. The ratio $\gamma$ itself

Once you have $C_v=\frac{f}{2}R$ and $C_p=C_v+R$, the ratio collapses to $\gamma =1+2/f$. Downstream, $\gamma$ powers adiabatic $P{V}^{\gamma }=\text{const}$ and the speed of sound $v=\sqrt{\gamma RT/M}$.

How these feed the topic

Equipment checklist

Connections

• First Law of Thermodynamics — where $dQ=dU+PdV$ comes from
• Degrees of Freedom & Equipartition Theorem — gives $C_v=\frac{f}{2}R$
• Internal Energy of Ideal Gas — $U=\frac{f}{2}RT$
• Mayer's Relation — $C_p-C_v=R$
• Adiabatic Process — uses $\gamma$ downstream
• Speed of Sound in Gas — $v=\sqrt{\gamma RT/M}$