Gas-Phase Dissociation

01 / UNDERSTAND & APPLY

1. Alpha is a fraction of the original reactant, not of the final mixture

The degree of dissociation α is the fraction of an initially specified amount of a parent species that dissociates through the stated reaction. If n₀ moles of A are initially present and n₀α dissociate, the unreacted parent amount is n₀(1−α). For a single dissociation starting from parent alone, α lies between zero and one. Its numerical value depends on temperature and composition constraints.

For A⇌B+C, the equilibrium amounts from pure A are n₀(1−α), n₀α and n₀α. The total is n₀(1+α). Consequently the mole fraction of B is α/(1+α), not α. Dissociation creates additional particles, so the final-mixture denominator differs from the original-parent denominator.

For A⇌2B, the remaining parent amount is the same, but the product amount is 2n₀α. The total is again n₀(1+α), and the B mole fraction is 2α/(1+α). Equal total particle multiplication does not imply identical equilibrium expressions: the product coefficient and concentration power still differ.

When products are initially present, define the extent explicitly rather than forcing a pure-parent formula. If the reaction shifts backward, a forward extent can be negative within the physical bounds. A label “degree of dissociation” must identify the initial parent basis and direction; otherwise two people may attach different denominators to the same mixture.

02 / UNDERSTAND & APPLY

2. Derive Kp for one parent making two different products

For ideal gases, partial pressure p_i=y_iP, where P is total pressure of the mixture. Starting with pure A for A⇌B+C gives p_A=P(1−α)/(1+α) and p_B=p_C=Pα/(1+α). Substitute these expressions into the reaction quotient rather than replacing every partial pressure with the total pressure.

Using the familiar unnormalized pressure convention, K_p=p_Bp_C/p_A=Pα²/(1−α²). With a dimensionless standard-state constant, the corresponding expression is K=(P/p°)α²/(1−α²). The algebra is the same, but the pressure convention must match the supplied numerical constant. The temperature is fixed for a given K.

Rearranging the textbook expression gives α²=K_p/(P+K_p), with P and K_p expressed consistently in that convention. The result shows lower total pressure favors dissociation in this particular one-to-two gas reaction. It does not imply that changing pressure changes K at constant temperature.

This compact expression assumes pure A initially, only the one gas reaction, no inert gas included in P, and ideal behavior. Each assumption matters. A mixture initially containing B, or total pressure partly contributed by an inert gas, requires a revised mole table and partial pressures.

03 / UNDERSTAND & APPLY

3. A⇌2B has a different coefficient factor

For A⇌2B starting with pure A, p_A=P(1−α)/(1+α) but p_B=2Pα/(1+α). Squaring the product pressure gives the factor four: K_p=p_B²/p_A=4Pα²/(1−α²). Forgetting that factor means the stoichiometric coefficient was counted in the total moles but lost in the quotient.

The resulting degree is α²=K_p/(4P+K_p). N₂O₄⇌2NO₂ is a familiar structural type of this equilibrium, but use whatever constant and temperature are supplied rather than assuming a universal dissociation fraction. Brown color indicates NO₂ presence qualitatively; a quantitative color-based inference would need a calibrated relation and path length.

For a fixed-volume concentration description, let C=n₀/V be the original parent concentration. Then [A]=C(1−α), [B]=2Cα and K_c=4Cα²/(1−α). This expression has no 1+α denominator because concentration uses the fixed vessel volume, not the changing total mole count.

The difference between the K_p and K_c formulas is not a contradiction. Total pressure P is itself C(1+α)RT at equilibrium for this pure-parent fixed-volume model. Substituting that relation into K_p recovers K_p=K_cRT in the compatible conventional units because Δn_gas=1.

04 / UNDERSTAND & APPLY

4. Total-pressure change can measure alpha under controlled conditions

At fixed temperature and volume, ideal-gas pressure is proportional to total moles. For a pure parent that creates ν gas particles per dissociated parent molecule, n_total=n₀[1+(ν−1)α]. If P₀ is the hypothetical or measured undissociated initial pressure under the same T,V conditions, P/P₀=1+(ν−1)α.

For a one-to-two dissociation, α=P/P₀−1. This relation does not require the two product particles to be different, because it uses total particle count only. The equilibrium constant calculation still requires the individual product stoichiometry. Pressure alone can identify the extent when the reaction is known, but it cannot identify unknown product chemistry.

The temperature and volume restrictions are essential. Heating the vessel raises pressure even without dissociation, and changing volume changes pressure independently of particle count. If P₀ was measured at a different temperature, first correct the ideal-gas relation or formulate the full mole balance. An uncorrected pressure increase is not automatically a dissociation signal.

An inert-gas baseline also contributes pressure. Subtract its known contribution or include it in the mole table before inferring α. If multiple reactions change particle number, one total-pressure observation generally cannot determine every individual extent without additional information.

05 / UNDERSTAND & APPLY

5. Dissociation lowers the gas mixture’s mean molar mass

For an ideal gas mixture, density ρ and mean molar mass M̄ satisfy M̄=ρRT/P. If a fixed mass of pure parent with formula molar mass M₀ dissociates, the total mass remains constant while the number of particles increases. For a one-parent-to-ν-particles process, M̄=M₀/[1+(ν−1)α].

Thus α=(M₀/M̄−1)/(ν−1). This is a particle-count inference under the specified stoichiometry, not evidence that the mass of individual atoms has changed. The gas now contains a mixture of parent and products whose number-average molar mass is lower than the intact parent’s.

Older vapor-density conventions compare a gas density with a reference gas at the same temperature and pressure. If the reference is hydrogen and the approximation M(H₂)=2 g mol⁻¹ is intended, a numerical relation M̄≈2D is often used. State the reference rather than assume every quantity called vapor density uses hydrogen. Relative to air, the conversion factor is different.

Nonideal compressibility, impurities, condensation or unknown parallel reactions can invalidate a simple molar-mass inference. A calculated α outside [0,1] is a warning to check the model and measurements, not a number to round into range. For complete one-to-two dissociation, M̄ cannot fall below M₀/2 in this model.

06 / UNDERSTAND & APPLY

6. Association reduces particle count

If monomers associate as nA⇌A_n and α is the fraction of initial monomer units incorporated into aggregates, unassociated monomer amount is n₀(1−α) and aggregate amount is n₀α/n. Total particles are n₀[1−α+α/n]. Be precise: α counts monomer units consumed, not the mole fraction of aggregates.

The corresponding mean molar mass is M̄=M₀/[1−α(1−1/n)]. For dimerization, M̄=M₀/(1−α/2). Complete dimerization gives 2M₀, while no association gives M₀. The same counting principle appears in dilute-solution colligative measurements under their ideal-particle assumptions.

Association and dissociation change particle count in opposite directions. A measured apparent molar mass above the formula mass suggests association within this restricted model; a value below suggests dissociation. It is not a unique diagnosis without excluding nonideality, contamination and other chemistry.

If both monomers, dimers and higher aggregates coexist, a single apparent molar mass generally yields only a combined particle-count average. It cannot uniquely determine all aggregation fractions. Additional equilibria or independent measurements are needed rather than inventing a single α that hides multiple species.

07 / UNDERSTAND & APPLY

7. Rebuild the mole table when the starting mixture changes

For A⇌B+C with initial amounts a, b and c, use a reaction extent ξ: equilibrium amounts are a−ξ, b+ξ and c+ξ. The total becomes a+b+c+ξ, plus any inert-gas amount. Partial pressures follow by dividing each reacting amount by that total and multiplying by total pressure. This general method is more reliable than trying to patch a pure-A shortcut.

Adding product generally suppresses forward dissociation under otherwise comparable conditions because it increases the reaction quotient. The effect should be evaluated at the actual imposed volume or pressure. If adding a gas also changes the volume, the combined composition and dilution effects must be included.

At constant temperature and volume, adding an ideal inert gas does not change reacting partial pressures p_i=n_iRT/V, so it does not shift the ideal gas equilibrium. At constant total pressure, inert-gas addition expands the mixture and lowers reacting partial pressures, favoring the side with more gas particles for a one-to-two dissociation. Total pressure alone does not identify the perturbation.

When solving for ξ, enforce nonnegative species amounts: ξ≤a and ξ≥−min(b,c) in this reaction. A polynomial may have additional mathematical roots outside these bounds. Substitute the accepted root back into both the equilibrium expression and the mole balance.

08 / UNDERSTAND & APPLY

8. The same alpha idea applies to weak electrolytes, with different equations

For a weak monoprotic acid HA in dilute water with initial analytical concentration C and no added conjugate base, define α as its ionized fraction. Neglecting water’s hydronium contribution gives [HA]=C(1−α), [H⁺]=[A⁻]=Cα, so K_a=Cα²/(1−α). The square-root estimate α≈√(K_a/C) requires α≪1 and a negligible water contribution.

Dilution increases the degree of ionization in this simple weak-electrolyte model, while the equilibrium constant at fixed temperature remains unchanged. A larger α does not necessarily mean a lower pH: dilution can increase the fraction ionized while decreasing the actual hydronium concentration Cα. Fraction and concentration answer different questions.

For ideal particle counting, if one formula unit produces ν independent solute particles when it dissociates, the van ’t Hoff factor is i=1+(ν−1)α. Thus α=(i−1)/(ν−1). For association into n-mers, i=1−α(1−1/n). These relations assume the colligative measurement reflects ideal independent particle counts.

Real electrolytes have ion interactions and activity effects, so a measured i need not give a literal equilibrium dissociation fraction by this simple equation, especially for strong electrolytes at appreciable concentration. Use the relation when the problem explicitly adopts the ideal dissociation/association model. Do not combine constants or apparent masses from unrelated solvents and temperatures.

09 / UNDERSTAND & APPLY

9. Original dissociation and association practice

These original problems assume ideal gases or the explicitly stated ideal solute-particle model. Pressure constants use the unnormalized convention where a pressure unit is included in the supplied K_p. A thermodynamic dimensionless formulation would use pressure ratios to a standard pressure.

Begin with one mole of parent when only a fraction is needed. Write all equilibrium amounts and total particles before calculating a mole fraction, mean mass or pressure. Validate the result against the limits of no reaction and complete reaction.