Relationship Between \(K_p\) and \(K_c\) for Gas-Phase Reactions
For reactions involving gases, the equilibrium constants written in terms of
partial pressures (\(K_p\)) and in terms of
molar concentrations (\(K_c\)) are related through the ideal-gas law.
The bridge is the change in the total number of moles of gaseous species,
\(\Delta n_{\mathrm{gas}}\).
Key Result
For a balanced gas-phase reaction
\[
\ce{aA(g) + bB(g) + ... <=> cC(g) + dD(g) + ...},
\]
define
\[
\begin{aligned}
\Delta n_{\mathrm{gas}}
&= \big(\text{sum of stoichiometric coefficients of gaseous products}\big) \\
&\quad - \big(\text{sum of stoichiometric coefficients of gaseous reactants}\big).
\end{aligned}
\]
Then the relationship is
\[
\begin{aligned}
K_p &= K_c\,(RT)^{\Delta n_{\mathrm{gas}}}, \\[4pt]
K_c &= \dfrac{K_p}{(RT)^{\Delta n_{\mathrm{gas}}}}.
\end{aligned}
\]
Only species in the gaseous phase are counted in \(\Delta n_{\mathrm{gas}}\); solids, pure liquids,
and solutes in \(\mathrm{aq}\) do not contribute.
Where It Comes From (Sketch)
Under ideal-gas behavior, for each gas \(i\):
\[
\begin{aligned}
P_i &= c_i\,R\,T,
\end{aligned}
\]
with \(c_i\) the molar concentration of \(i\) (e.g., mol·L\(^{-1}\)).
The equilibrium expressions are
\[
\begin{aligned}
K_c &= \prod_i \big[c_i\big]^{\nu_i}, \\[4pt]
K_p &= \prod_i \big[P_i\big]^{\nu_i},
\end{aligned}
\]
where \(\nu_i\) is positive for products and negative for reactants
(or, equivalently, use separate numerator/denominator with positive exponents).
Substituting \(P_i=c_iRT\) gives
\[
\begin{aligned}
K_p
&= \prod_i \big(c_i R T\big)^{\nu_i} \\[4pt]
&= \left(\prod_i c_i^{\nu_i}\right)\,(RT)^{\sum_i \nu_i} \\[4pt]
&= K_c\,(RT)^{\Delta n_{\mathrm{gas}}}.
\end{aligned}
\]
Choosing \(R\) and Units
- Use a value of \(R\) consistent with the pressure unit implied for \(K_p\):
- \(R = 0.082057\ \mathrm{L\;atm\;mol^{-1}\;K^{-1}}\) for atm,
- \(R = 0.08314\ \mathrm{L\;bar\;mol^{-1}\;K^{-1}}\) for bar.
- Always use the absolute temperature \(T\) in kelvin.
- In rigorous thermodynamics, equilibrium constants are defined using activities and
are dimensionless. In general-chemistry problem solving, it is common to use the
numerical values of \(K_p\) and \(K_c\) with the above relation; this calculator adopts that
convention.
How to Use the Relation (Procedure)
- Write and balance the reaction with phases.
- Compute \(\Delta n_{\mathrm{gas}}\):
\[
\Delta n_{\mathrm{gas}} = \sum \nu_{\text{products (g)}} - \sum \nu_{\text{reactants (g)}}.
\]
- Select \(R\) in atm or bar to match your preferred convention and enter \(T\) (K).
- Convert using
\[
\begin{aligned}
K_p &= K_c\,(RT)^{\Delta n_{\mathrm{gas}}} \quad \text{or} \quad
K_c = \dfrac{K_p}{(RT)^{\Delta n_{\mathrm{gas}}}}.
\end{aligned}
\]
Worked Example
At \(T=1000\ \mathrm{K}\), for
\(\ce{2 SO2(g) + O2(g) <=> 2 SO3(g)}\) and \(K_c = 2.80\times 10^{2}\),
find \(K_p\) (use \(R=0.08314\ \mathrm{L\;bar\;mol^{-1}\;K^{-1}}\)).
Count gaseous coefficients:
\[
\begin{aligned}
\Delta n_{\mathrm{gas}}
&= (2) - (2 + 1) \\[4pt]
&= -1.
\end{aligned}
\]
Compute \((RT)^{\Delta n_{\mathrm{gas}}}\):
\[
\begin{aligned}
(RT)^{\Delta n_{\mathrm{gas}}}
&= (0.08314 \times 1000)^{-1} \\[4pt]
&= \left(8.314\right)^{-1}.
\end{aligned}
\]
Convert:
\[
\begin{aligned}
K_p
&= K_c\,(RT)^{-1} \\[4pt]
&= \left(2.80\times 10^{2}\right)\times \left(8.314\right)^{-1} \\[4pt]
&\approx 3.37 \;\;(\text{often rounded to } 3.4).
\end{aligned}
\]
Interpretation Tips
- \(\Delta n_{\mathrm{gas}} = 0 \Rightarrow K_p = K_c\).
- \(\Delta n_{\mathrm{gas}} > 0 \Rightarrow K_p > K_c\) at a given \(T\) (because \((RT)^{\Delta n_{\mathrm{gas}}} > 1\)).
- \(\Delta n_{\mathrm{gas}} < 0 \Rightarrow K_p < K_c\).
Common Pitfalls
- Counting solids, liquids, or aqueous species in \(\Delta n_{\mathrm{gas}}\) — do not.
- Using °C instead of K for temperature.
- Mixing \(R\) units that do not match your implicit pressure convention.
- Sign mistakes in \(\Delta n_{\mathrm{gas}}\). Always do
“products minus reactants” for gas-phase stoichiometric coefficients.
This page follows the same conventions used in many general-chemistry texts: \(K_p\) and \(K_c\)
are treated as numerical values, and activities are approximated by partial pressures or
molar concentrations for ideal systems.
