Relating Kp and Kc

Result

Relating Kp and Kc

T-047Home CU-203Threads equilibrium

Statement

Gas-phase equilibrium constants and the gas law.

Why it matters

law-of-mass-action established the equilibrium constant expression itself, but for gas-phase reactions, that expression can be written equally validly in terms of partial pressures (\(K_p\)) or in terms of molar concentrations (\(K_c\)), and experimental data is sometimes reported in one form and sometimes the other. This result supplies the direct conversion between the two, built entirely from the ideal gas law, so that a single measured or tabulated equilibrium constant can always be converted to whichever form a particular calculation, or a particular piece of literature data, happens to require.

Hypotheses

All reacting species behave as ideal gases under the conditions of the equilibrium.The derivation below substitutes the ideal gas law, \(PV=nRT\), directly for each gaseous species; at high pressure or low temperature, where real-gas deviations become significant, the simple relation derived here becomes correspondingly less accurate, and a fugacity-based (non-ideal) treatment would be required instead. \(\Delta n\), the change in moles of gas across the reaction, is computed correctly from the balanced equation (moles of gaseous products minus moles of gaseous reactants).Any non-gaseous species (pure solids or liquids) participating in the equilibrium do not appear in either \(K_p\) or \(K_c\) at all (their activities are conventionally taken as \(1\)), and must therefore also be excluded when computing \(\Delta n\), which counts only the gas-phase species' stoichiometric coefficients.

Proof

PV=nRT \ \Longrightarrow\ P=\frac{n}{V}RT=[X]RT

Rearranging the ideal gas law for a single gaseous species \(X\) shows that its partial pressure is directly proportional to its molar concentration \([X]=n/V\), with the proportionality constant \(RT\) — the essential substitution used throughout the rest of the derivation. A

K_p = \prod_i P_i^{\nu_i}

The equilibrium constant \(K_p\), by definition, is the product of each gaseous species' equilibrium partial pressure raised to its stoichiometric coefficient \(\nu_i\) (positive for products, negative for reactants) — simply the law of mass action expressed in the pressure-based form appropriate for gas-phase equilibria. A

K_p = \prod_i \left([X_i]RT\right)^{\nu_i} = \left(\prod_i[X_i]^{\nu_i}\right)(RT)^{\sum_i\nu_i} = K_c\,(RT)^{\Delta n}

Substituting Step 1's relation for every gaseous species into Step 2's product, the concentration-dependent factors collect exactly into \(K_c\) (the identical product expressed in concentrations rather than pressures), while every remaining \(RT\) factor collects into a single power, with exponent \(\Delta n=\sum_i\nu_i\), the net change in moles of gas across the reaction. A

\Delta n = 0 \ \Longrightarrow\ K_p=K_c\ \text{(numerically, though with different units/dimensions in general)}

In the special case where a reaction has equal moles of gaseous product and reactant, the \((RT)^{\Delta n}\) factor in Step 3 reduces to \((RT)^0=1\), so \(K_p\) and \(K_c\) coincide numerically for that specific reaction, even though they remain, in general, differently defined and differently dimensioned quantities. A

Result

K_p = K_c\,(RT)^{\Delta n}

Reading. The two forms of a gas-phase equilibrium constant convert into one another by a single factor of \((RT)^{\Delta n}\), where \(\Delta n\) is simply the net change in moles of gas as the reaction is written, computed directly from the balanced equation's stoichiometric coefficients.

Scope. Strictly valid for ideal gases (Hypotheses); \(R\) and \(T\) must be used consistently with the pressure and concentration units chosen for \(K_p\) and \(K_c\) respectively, and only gas-phase species are included in \(\Delta n\), never any solid or liquid participants in a heterogeneous equilibrium.

Corollaries & converses

le-chatelier's qualitative prediction that increasing total pressure (by compressing the system) shifts a gas-phase equilibrium toward the side with fewer moles of gas is the same \(\Delta n\) that appears explicitly, quantitatively, in this Result's conversion exponent — the two results describe the identical underlying stoichiometric fact from complementary angles.
reaction-quotient's \(Q_p\) and \(Q_c\) obey the identical conversion relation, \(Q_p=Q_c(RT)^{\Delta n}\), at any point during a reaction's approach to equilibrium, not only exactly at equilibrium; the Result's derivation (Steps 1–3) never actually assumed equilibrium at all, only ideal-gas behaviour.
Converse: given \(K_p\) and the reaction's \(\Delta n\), the Result is immediately solved for \(K_c\) (or vice versa), so tabulated data reported in either form is always usable for a calculation requiring the other.

Fails without

Drop the ideal-gas hypothesis at high pressure or low temperature, where real-gas non-ideality is significant: the simple substitution of Step 1 no longer holds exactly, since real partial pressures and concentrations are related through a fugacity coefficient rather than the ideal gas law alone; the Result then only approximately, not exactly, converts between \(K_p\) and \(K_c\).
Include a solid or liquid participant's stoichiometric coefficient in \(\Delta n\): since non-gaseous species do not appear in either \(K_p\) or \(K_c\) at all (their activities are conventionally \(1\), Hypotheses), incorrectly including their coefficients in \(\Delta n\) gives the wrong exponent in the Result and a systematically incorrect conversion.

Common errors

Computing \(\Delta n\) from the overall balanced equation's total stoichiometric coefficients, including any solid or liquid species, rather than only the gas-phase species (Fails without, second bullet).
Using inconsistent units for \(R\) relative to the pressure and concentration units chosen for \(K_p\) and \(K_c\) (e.g. \(R=8.314\,\text{J/mol/K}\) paired with pressure in atm rather than pascals, without an appropriate unit-consistent value of \(R\)).
Assuming \(K_p\) and \(K_c\) are always numerically equal; this holds only in the special case \(\Delta n=0\) (Step 4), not generally.
Forgetting that temperature \(T\) in the Result must be in absolute (kelvin) units, since it enters directly through the ideal gas law of Step 1.

Discussion

The relation between \(K_p\) and \(K_c\) is a direct, purely mathematical consequence of the ideal gas law rather than an independent empirical discovery in its own right; its significance lies less in any historical novelty and more in its everyday practical necessity, since gas-phase equilibrium data in the chemical literature and in industrial process design is reported in both forms depending on context (partial pressures are natural for many industrial gas-phase processes run at controlled total pressure, such as ammonia synthesis, while concentrations are often more natural in a laboratory solution-phase or kinetics context).

The same \((RT)^{\Delta n}\) conversion factor reappears, with an entirely analogous derivation, whenever any two equivalent descriptions of a gas-phase quantity related through the ideal gas law need to be interconverted — for instance, between pressure-based and concentration-based rate constants for gas-phase kinetics, an exact structural parallel to the equilibrium case developed here.

Common misconception: that \(K_p\) and \(K_c\) are simply two different symbols for the identical numerical quantity. Except in the special \(\Delta n=0\) case (Step 4), they are numerically different (and, in general, dimensionally different) quantities, related by the specific, calculable conversion factor of the Result rather than being interchangeable by definition.

Worked examples

\text{N}_2(g)+3\text{H}_2(g)\rightleftharpoons2\text{NH}_3(g), \quad K_c = 0.500\ \text{(at } T=700\,\text{K)}

Computing \(\Delta n\) for this reaction from the gas-phase stoichiometric coefficients alone: \(\Delta n = 2-(1+3) = 2-4 = -2\). A

K_p = K_c(RT)^{\Delta n} = 0.500\times(0.08206\times700)^{-2} = 0.500\times(57.44)^{-2} = 0.500\times3.03\times10^{-4} = 1.52\times10^{-4}

Substituting \(\Delta n=-2\), \(R=0.08206\,\text{L atm mol}^{-1}\text{K}^{-1}\) (chosen to match \(K_c\)'s molar concentration units against \(K_p\) in atm), and \(T=700\,\text{K}\) into the Result gives the corresponding \(K_p\), substantially different in magnitude from \(K_c\) precisely because \(\Delta n\neq0\) here. A

K_p \approx 1.52\times10^{-4}\ (\text{vs. } K_c=0.500)

Reading. The large negative \(\Delta n\) for ammonia synthesis (four moles of gas becoming two) makes \(K_p\) and \(K_c\) differ by several orders of magnitude at this temperature, illustrating why the conversion factor cannot be neglected for reactions with substantial \(\Delta n\).

Scope. The identical substitution, with the reaction's own \(\Delta n\) and the temperature of interest, converts any gas-phase \(K_c\) to \(K_p\) or vice versa.

Problems

For the reaction \(\text{H}_2(g)+\text{I}_2(g)\rightleftharpoons2\text{HI}(g)\), find \(\Delta n\) and state the relationship between \(K_p\) and \(K_c\) for this specific reaction.
Solution
\(\Delta n = 2-(1+1)=0\). By Step 4, \(K_p=K_c(RT)^0=K_c\): the two constants are numerically equal for this particular reaction, since equal moles of gas appear on both sides.
For the decomposition \(\text{CaCO}_3(s)\rightleftharpoons\text{CaO}(s)+\text{CO}_2(g)\), find \(\Delta n\) to use in the Result, explaining which species are included.
Solution
Only gas-phase species are counted (Hypotheses); \(\text{CaCO}_3(s)\) and \(\text{CaO}(s)\) are both solids and do not appear in \(K_p\), \(K_c\), or \(\Delta n\) at all. The only gaseous species is \(\text{CO}_2(g)\), appearing as a product with coefficient \(1\) and no gaseous reactants, so \(\Delta n=1-0=1\).
Given \(K_p=2.50\times10^{-3}\) for a reaction with \(\Delta n=+1\) at \(T=500\,\text{K}\), find \(K_c\) (use \(R=0.08206\,\text{L atm mol}^{-1}\text{K}^{-1}\)).
Solution
Rearranging the Result: \(K_c = K_p(RT)^{-\Delta n} = K_p/(RT)^{\Delta n}\). \(RT = 0.08206\times500=41.03\). \(K_c = 2.50\times10^{-3}/41.03 = 6.09\times10^{-5}\).