Real gas effects — dissociation, recombination

WHAT is happening?

WHY does it matter for a rocket?

Rocket performance is governed by exhaust velocity: $v_e = \sqrt{\frac{2\gamma}{\gamma-1}\,\frac{R_u T_c}{M}\left[1-\left(\frac{p_e}{p_c}\right)^{\frac{\gamma-1}{\gamma}}\right]}$

Dissociation attacks this in two ways:

1. It lowers $T_c$ — energy that should heat the gas instead breaks bonds, so the flame temperature is cooler than a naive "all the fuel burns to final products" estimate.
2. It lowers the molar mass $M$ — fewer big molecules, more light atoms/radicals. This helps $v_e$ (which $\propto 1/\sqrt{M}$).

So there is a genuine tug-of-war. The net effect on $v_e$ depends on whether recombination happens fast enough in the nozzle.

HOW: deriving the dissociation energy balance from first principles

Take a single dissociating reaction, e.g. $\text{A}_2 \rightleftharpoons 2\text{A}$, with bond dissociation enthalpy $\Delta H_d$ per mole of $\text{A}_2$ broken.

Step 1 — Energy conservation in the chamber. The heat released by combustion, $Q_{comb}$, must both raise temperature and pay for dissociation: $Q_{comb} = \underbrace{n\,\bar{c}_p\,(T_c - T_0)}_{\text{heats the gas}} + \underbrace{\alpha\, n\,\Delta H_d}_{\text{spent breaking bonds}}$ Why this step? Energy can't vanish; every joule going into breaking a bond is a joule not raising $T_c$.

Step 2 — Define degree of dissociation. $\alpha = \frac{\text{moles dissociated}}{\text{moles initially present}}, \qquad 0\le \alpha \le 1$ Why? We need one number that says "how much has fallen apart."

Step 3 — Equilibrium fixes $\alpha$. From chemical equilibrium for $\text{A}_2\rightleftharpoons 2\text{A}$ at total pressure $p$: $K_p(T) = \frac{p_A^2}{p_{A_2}} = \frac{(2\alpha)^2}{(1-\alpha)(1+\alpha)}\,\frac{p}{p^\circ}$ Why? $K_p$ from thermodynamics ($\ln K_p = -\Delta G^\circ/R_uT$) tells us the balance point; solving it gives $\alpha(T,p)$.

Key consequence — pressure suppresses dissociation. Because dissociation increases the number of moles, Le Chatelier says raising $p$ pushes the equilibrium back toward molecules ($\alpha \downarrow$). This is why high chamber pressure is good: less dissociation → hotter, more complete combustion.

Frozen vs. Equilibrium flow — the crux

As gas accelerates down the nozzle it cools. Two idealized limits:

Reality lies between. The controlling number is the Damköhler number: $Da = \frac{\tau_{flow}}{\tau_{chem}} = \frac{\text{residence time in nozzle}}{\text{chemical reaction time}}$

• $Da \gg 1$: plenty of time → equilibrium flow.
• $Da \ll 1$: no time → frozen flow.

Why this matters: Real nozzle performance is a few % below equilibrium and above frozen. Ignoring dissociation overpredicts $T_c$ and $v_e$; assuming frozen flow underpredicts recovered energy.

Worked examples

Common mistakes (steel-manned)

Active recall

Connections

• Chemical Equilibrium and Kp
• Adiabatic Flame Temperature
• Nozzle Flow and Exhaust Velocity
• Specific Impulse
• Le Chatelier's Principle
• Damköhler Number

Concept Map

Hinglish (regional understanding)