Pressure Changes in a Rigid Reactor

Introduction

Gas pressure in a rigid vessel can change because a reaction alters the number of gas molecules, because the temperature changes, or both. A pressure rise therefore does not by itself prove that the reaction made more gas. Compare initial and final ideal-gas states, then use the balanced equation to calculate the new gas amount. This gives a quantitative way to interpret a reactor pressure trace.

Core explanation

For an ideal-gas mixture in a sealed rigid vessel, P = n gas RT/V. Since V is the same before and after, P final/P initial = (n gas,final/n gas,initial)(T final/T initial), using absolute temperatures and pressures. If the reactor is returned to the same temperature, the temperature ratio is one, so pressure ratio equals gas-mole ratio. If temperature rises during reaction, both factors matter.

For a reaction with extent ξ, total gas moles change by Δn gas = Δν gas ξ, where Δν gas is the sum of gas-product coefficients minus the sum of gas-reactant coefficients. Solids and liquids are omitted from this gas-mole change , although they remain in atom and mass balances. For N₂O₄(g) ⇌ 2NO₂(g), Δν gas = 2 − 1 = +1. Starting from 1.00 mol N₂O₄ and reaching an extent of 0.300 mol gives 0.700 mol N₂O₄ and 0.600 mol NO₂, or 1.300 mol gas total. At constant volume and the same temperature, pressure is 1.300 times its initial value. This calculation does not claim that a specific equilibrium extent is 0.300 mol; that amount is stipulated for the example.

If the temperature also increases from 300 K to 330 K, the pressure ratio becomes 1.300 × (330/300) = 1.430. It would be wrong to infer ξ = 0.430 mol from the 43% pressure increase alone. First correct for temperature, then use the gas-mole relation. Conversely, if temperature is held constant and an initial pressure is measured, a final pressure can sometimes reveal ξ through P final/P initial = 1 + Δν gas ξ/n gas,initial. This works only when the vessel is closed, volume is fixed, and gases behave adequately ideally.

When Δν gas = 0, reaction alone does not change total ideal-gas pressure at fixed T and V, even though composition may change completely. For H₂(g) + Cl₂(g) → 2HCl(g), two gas moles become two gas moles per extent. A pressure gauge alone cannot track conversion in that idealised isothermal case. It may still respond to reaction heat, a side reaction, or real-gas effects.

In practice, condensation, dissolution, leaks, or external gas feed change gas amount independently of the displayed stoichiometric Δν gas. A reacting gas mixture can also be nonideal at high pressure. The equation is a transparent model whose assumptions should be checked against the question's setting.

Step-by-step reasoning

1. Record initial and final absolute temperatures, pressure reference and the vessel's fixed volume. 2. Balance the reaction with phases and find Δν gas from gaseous coefficients only. 3. Determine ξ from supplied conversion, equilibrium information or limiting-reagent conditions. 4. Calculate final gas moles as initial gas moles plus Δν gas ξ, including any inert gas. 5. Apply P final/P initial = (n final/n initial)(T final/T initial) and check the sign of each effect.

Visual explanation

Draw a sealed box of unchanged size. Before reaction it contains one N₂O₄ pair icon. After a partial reaction, show 0.7 pair icons and 0.6 separate NO₂ icons per initial mole, representing 1.3 gas moles total. Keep the box size fixed and write “same T → 1.3 times pressure.” A second arrow marked 300 K to 330 K adds the factor 330/300.

Real-world analogy

Consider a fixed-size room with people entering or leaving and with their activity level changing. More occupants and more vigorous motion can both increase wall impacts. Counting occupants alone would miss the activity effect; monitoring activity alone would miss the population change. Reactor pressure similarly combines gas amount and temperature.

Real-world example

A laboratory studies dissociation in a sealed heated vessel. The total pressure measured after the vessel stabilises at a known temperature can help estimate how much of a gaseous dimer separated into monomers. The calculation requires a closed vessel, its temperature, and the reaction's gas-mole change. Chemical analysis may still be needed to rule out side products.

Why?

Why do inert gas moles matter in the pressure ratio? An inert component contributes to the initial and final total gas moles even though its own amount does not change. A given ξ therefore represents a smaller fractional change in total amount when more inert gas was present at the start.

Common misconception

“A pressure rise means the balanced reaction has positive Δν gas.” Heating can raise pressure even if Δν gas is negative. Separate n and T factors. Also, use absolute pressure; a gauge-pressure ratio is not interchangeable with an absolute-pressure ratio.

Worked example

A sealed rigid vessel initially contains 1.00 mol N₂O₄ at 300 K and 1.00 atm absolute pressure. At an observed later state, 0.300 mol N₂O₄ has dissociated according to N₂O₄ → 2NO₂, and temperature is 330 K. Final amounts are 0.700 mol N₂O₄ and 0.600 mol NO₂, total 1.300 mol. Then P final = (1.00 atm)(1.300/1.000)(330/300) = 1.43 atm. If the vessel were instead cooled back to 300 K without changing composition, pressure would be 1.30 atm. The 0.13 atm difference between those results reflects temperature rather than additional dissociation.

Quick check

1. At fixed V and T, a reaction changes total ideal-gas moles from 2.0 to 1.5. What is P final/P initial? Answer: The ratio is n final/n initial = 1.5/2.0 = 0.75, so final pressure is three-quarters of initial pressure.

Exam focus

Use gaseous coefficients only for Δν gas, but include inert gases in both total-mole inventories. State whether temperatures are equal before cancelling them. A measured pressure ratio determines extent only when other causes of gas-mole or temperature change are controlled.

Advanced insight

For N₂O₄ ⇌ 2NO₂ starting with pure dimer and dissociation fraction α, gas moles scale as 1 + α, so P/P₀ = 1 + α at equal T and V under the ideal model. At a changed temperature, α can still be inferred only after dividing the pressure ratio by T/T₀. This relation connects pressure data to composition but not automatically to equilibrium constants without partial pressures.

Summary

In a rigid closed ideal-gas vessel, pressure ratio is gas-mole ratio times Kelvin-temperature ratio. Reaction extent changes gas moles by Δν gas ξ. Account for inert gas, phase changes and temperature before interpreting a pressure measurement as reaction progress.

Practice questions

1. For 2A(g) → B(g), what is Δν gas per mole of reaction as written? Answer: One product gas mole minus two reactant gas moles gives Δν gas = −1. 2. At fixed V and T, total gas moles drop from 4.0 to 3.0. If initial pressure is 2.0 atm, find final pressure. Answer: P final = 2.0(3.0/4.0) = 1.5 atm. 3. In that example, if final temperature were 360 K instead of the initial 300 K, what would final pressure be? Answer: P final = 2.0(3.0/4.0)(360/300) = 1.8 atm.