Chemistry
Relation and similarity of equilibrium constants
Step-by-step chemistry solution: Relation and similarity of equilibrium constants
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1. What the question is asking – plain‑language restatement
The student wonders why different “equilibrium constants” ( (K_p, K_c, K_a, K_x,\dots) ) that are defined in terms of pressure, concentration, activity, mole fraction, etc. are all said to be “the same” thermodynamic quantity
[ K = e^{-\Delta G^{\circ}/RT} ]
yet the numerical values of (K_p) and (K_c) (or the others) are not always identical.
In other words:
- How does a single thermodynamic equilibrium constant (K) give rise to many different expressions?
- Why do those expressions sometimes give different numbers (e.g. (K_p \neq K_c) when (\Delta n \neq 0))?
The solution must start from the definition of the thermodynamic constant and derive the familiar forms, showing where the differences come from (standard states, unit conventions, the ideal‑gas relation, etc.).
2. Step‑by‑step derivation
2.1 Thermodynamic definition of the equilibrium constant
For a generic reaction written in the stoichiometric form
[ \sum_{i} \nu_i \, \mathrm{A}_i = 0 ]
the Gibbs free‑energy change at any state is
[ \Delta G = \Delta G^{\circ} + RT\ln Q \tag{1} ]
where
- (\Delta G^{\circ}) – standard Gibbs free energy (all reactants/products at their standard states),
- (Q) – the reaction quotient, defined as the product of activities (a_i) raised to their stoichiometric coefficients:
[ Q = \prod_i a_i^{\;\nu_i} \tag{2} ]
At equilibrium, (\Delta G = 0) and therefore
[ 0 = \Delta G^{\circ} + RT\ln K \qquad\Longrightarrow\qquad K \equiv Q_{\text{eq}} = e^{-\Delta G^{\circ}/RT} \tag{3} ]
The symbol (K) here is the thermodynamic equilibrium constant. It is dimensionless because activities are defined relative to a reference state.
2.2 What is an activity?
For a component (i) the activity is
[ a_i = \frac{f_i}{f_i^{\circ}} \quad \text{(for gases)} \qquad\text{or}\qquad a_i = \frac{\gamma_i\,c_i}{c_i^{\circ}} \quad \text{(for solutes)} ]
- (f_i) = fugacity (≈ partial pressure (p_i) for an ideal gas)
- (f_i^{\circ}=1\;\text{bar}) (standard pressure)
- (c_i) = molar concentration (mol L(^{-1}))
- (c_i^{\circ}=1\;\text{mol L}^{-1}) (standard concentration)
- (\gamma_i) = activity coefficient (≈ 1 for ideal solutions)
Thus, activities are already “normalized” by their standard states, making them dimensionless.
2.3 From activities to measurable quantities
2.3.1 Gas‑phase reactions → (K_p)
For an ideal gas, fugacity ≈ partial pressure (p_i).
Using the standard pressure (p^{\circ}=1\;\text{bar}),
[ a_i = \frac{p_i}{p^{\circ}} ]
Insert into (2):
[ K = \prod_i \left(\frac{p_i}{p^{\circ}}\right)^{\nu_i} \equiv K_p \quad\text{(by definition)} \tag{4} ]
(K_p) is dimensionless; in practice chemists often drop the (p^{\circ}) and write
[ K_p^{\text{(numeric)}} = \prod_i p_i^{\;\nu_i} ]
which carries units of (\text{(bar)}^{\Delta n}) where (\Delta n = \sum \nu_i) for gases. The apparent discrepancy between (K) and the numeric (K_p) is only a matter of bookkeeping the standard‑state factor.
2.3.2 Solution‑phase reactions → (K_c)
For solutes in an (ideal) solution,
[ a_i = \frac{c_i}{c^{\circ}}, \qquad c^{\circ}=1\;\text{mol L}^{-1} ]
Hence
[ K = \prod_i \left(\frac{c_i}{c^{\circ}}\right)^{\nu_i} \equiv K_c \quad\text{(dimensionless)} \tag{5} ]
Again, the numeric (K_c) that appears in textbooks (with units of (\text{(mol L}^{-1})^{\Delta n})) is just the product of concentrations without dividing by the standard concentration.
2.3.3 Acid–base equilibria → (K_a)
Acid dissociation is a special case of (5) where the reaction is
[ \mathrm{HA} \rightleftharpoons \mathrm{H^{+}} + \mathrm{A^{-}} ]
The thermodynamic constant is
[ K = \frac{a_{\mathrm{H^{+}}}\,a_{\mathrm{A^{-}}}}{a_{\mathrm{HA}}} ]
If we replace activities by concentrations (ideal solution) we obtain the acid‑dissociation constant
[ K_a = \frac{[ \mathrm{H^{+}} ][ \mathrm{A^{-}} ]}{[ \mathrm{HA} ]} \tag{6} ]
The same reasoning about the standard concentration applies, so the true (K) is dimensionless while the numeric (K_a) may be quoted with units of concentration.
2.3.4 Mole‑fraction based constant → (K_x)
When the standard state is the ideal‑gas mixture at 1 bar expressed in mole fractions, the activity of a gas (i) is simply its mole fraction (x_i) (because (p_i = x_i\,P_{\text{tot}}) and the factor (P_{\text{tot}}/p^{\circ}) cancels).
Thus
[ K = \prod_i x_i^{\;\nu_i} \equiv K_x \tag{7} ]
Again dimensionless.
2.4 Why (K_p) and (K_c) are not numerically identical
For a gas‑phase reaction we can relate pressure and concentration through the ideal‑gas law
[ p_i = c_i RT \qquad\text{or}\qquad c_i = \frac{p_i}{RT} \tag{8} ]
Insert (8) into the expression for (K_c) (5):
[ K_c = \prod_i \left(\frac{p_i}{RT\,c^{\circ}}\right)^{\nu_i} = \left(\frac{1}{RT\,c^{\circ}}\right)^{\Delta n} \prod_i p_i^{\;\nu_i} ]
Recognizing the product of pressures as the numeric (K_p),
[ K_c = \left(\frac{1}{RT\,c^{\circ}}\right)^{\Delta n} K_p^{\text{(numeric)}} \tag{9} ]
or, in dimensionless form (including the standard pressure (p^{\circ})),
[ K_c = K_p\;(RT)^{-\Delta n}\;(c^{\circ})^{-\Delta n} \;(p^{\circ})^{\Delta n} ]
Because (\Delta n = \sum_{\text{gases}} \nu_i) may be non‑zero, a factor ((RT)^{\Delta n}) appears. Consequently the numerical values of (K_p) and (K_c) differ unless (\Delta n = 0).
2.5 General conversion formula
For a reaction that involves only gases, the dimensionless constants obey
[ K_p = K_c\;(RT)^{\Delta n} \tag{10} ]
where
- (\Delta n = \displaystyle\sum_{\text{products}} \nu_i - \sum_{\text{reactants}} \nu_i) (gas moles)
- (R = 0.08314\;\text{L·bar·mol}^{-1}\text{K}^{-1}) if pressures are in bar, or the appropriate value for the chosen unit system.
If the reaction also contains solutes, the same idea holds but the conversion involves the standard concentration (c^{\circ}=1\;\text{mol L}^{-1}) as shown in (9).
2.6 Summary of the hierarchy
| Symbol | Measured quantity | Definition (dimensionless) | Typical “numeric” form |
|---|---|---|---|
| (K) (thermo) | – | (\displaystyle K = \prod_i a_i^{\nu_i}=e^{-\Delta G^{\circ}/RT}) | – |
| (K_p) | Partial pressures of gases | (\displaystyle K_p = \prod_i \left(\frac{p_i}{p^{\circ}}\right)^{\nu_i}) | (\displaystyle \prod_i p_i^{\nu_i}) (units (\text{bar}^{\Delta n})) |
| (K_c) | Molar concentrations (solutions) | (\displaystyle K_c = \prod_i \left(\frac{c_i}{c^{\circ}}\right)^{\nu_i}) | (\displaystyle \prod_i c_i^{\nu_i}) (units ((\text{mol L}^{-1})^{\Delta n})) |
| (K_a) | Acid‑base species | Same as (K_c) for the specific reaction | (\displaystyle \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]}) |
| (K_x) | Mole fractions in gas mixture | (\displaystyle K_x = \prod_i x_i^{\nu_i}) | Same numeric expression (dimensionless) |
All of them reduce to the same thermodynamic constant; the apparent differences arise solely from the choice of standard state and the relationship between the experimentally convenient variable (p, c, x, …) and the activity.
3. Final answer
The equilibrium constant that appears in the relation
[ K = e^{-\Delta G^{\circ}/RT} ]
is a single, dimensionless number defined in terms of activities. When we replace activities by measurable quantities—partial pressures, concentrations, or mole fractions—we must divide each quantity by its standard‑state reference (1 bar, 1 mol L(^{-1}), etc.). The resulting expressions are called (K_p, K_c, K_a, K_x,\dots). Because the conversion between, say, pressure and concentration introduces the factor ((RT)^{\Delta n}) (or (c^{\circ}) and (p^{\circ}) factors), the numeric values of (K_p) and (K_c) are generally different unless the net change in gas moles (\Delta n) is zero. Nevertheless, all of these constants are mathematically equivalent to the single thermodynamic constant (K).
4. Common Mistakes
| Mistake | Why it’s wrong | How to avoid it |
|---|---|---|
| Treating (K_p) or (K_c) as dimensionless without the standard‑state factor. | The quoted “(K_p)” in many textbooks omits the (p^{\circ}) (or (c^{\circ})) division, so its numerical value carries units. | Always write the activity form ((p_i/p^{\circ})^{\nu_i}) or ((c_i/c^{\circ})^{\nu_i}); remember the constant itself is dimensionless. |
| Using (K_p = K_c) for any gas‑phase reaction. | The conversion factor ((RT)^{\Delta n}) is only unity when (\Delta n = 0). | Check (\Delta n) first; apply (K_p = K_c (RT)^{\Delta n}). |
| Mixing units (e.g., bar vs atm) without adjusting (p^{\circ}). | The standard pressure is 1 bar; if you use atm, the numerical value of (K_p) changes. | Keep a consistent standard state; if you work in atm, set (p^{\circ}=1\;\text{atm}) and remember the thermodynamic definition still uses the same reference. |
| Assuming activity coefficients are always 1. | In real (non‑ideal) gases or solutions, (\gamma_i \neq 1); neglecting them gives an incorrect (K). | State |
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