chemical reaction
First, let assume an reversable reaction:
\[
\ce{ bB +cC <=> dD + eE}
\]
equilibrium constant
equilibrium of the reaction above can be expressed as:
\[
K = \frac{a_D^d \cdot a_E^e}{a_B^b \cdot a_C^c}
\]
- \(a_i\) means thermodynamic activities, powered by their own stoichiometric number.
reaction quotient
reaction quotient is expressed as symbol \(Q\),it has the same calculation as equilibrium constant(\(K\)):
\[
Q = \frac{a_D^d \cdot a_E^e}{a_B^b \cdot a_C^c}
\]
\(Q\) vs. \(K\)
- equilibrium constant is the reation quotient when a reaction approachs its equilibrium。
- activities \(a_i\) in the equation of \(Q\) are all instantaneous activities: as reaction goes on, these activities will change.
- \(Q > k\):reaction shift toward the left side (reactants)
- \(Q < k\):reaction shift toward the right side (products)
- \(Q = k\):equilibrium
concentration quotient and equilibrium constant
\[
K_c = \frac{[D]^d \cdot [E]^e}{[B]^b \cdot [C]^c}
\]
[...]
stands for concentration.
sometimes, when activities are unavailable and the solution is dilute, reaction quotient is used to represent equilibrium constant approximately (when equilibrium):
\[
K_c = \frac{[D]^d \cdot [E]^e}{[B]^b \cdot [C]^c}\approx
\frac{a_D^d \cdot a_E^e}{a_B^b \cdot a_C^c}
\]
However, according to the defination of thermodynamic activity:
\[
a_i = \gamma_i x_i
\]
- \(\gamma_i\) :activity coefficient of species \(i\)
- \(x_i\):concentration of species \(i\) (can be molarity,molality,etc.)
we obtain the relationship of equilibrium constant and concentration quotient:
\[
K = K_c \cdot \Gamma = \frac{[D]^d [E]^e}{[B]^b [C]^c} \cdot \frac{\gamma_D^d \gamma_E^e}{\gamma_B^b \gamma_C^c} =
\frac{a_D^d \cdot a_E^e}{a_B^b \cdot a_C^c}
\]
summary:in very dilute solution(\(\gamma_i \approx 1, \Gamma \approx 1\)), when it reachs equilibrium, concentration quotient equals to equilibrium constant: \(K \approx K_c\)
activity coefficient quotient
in the equation above,\(\frac{\gamma_D^d \gamma_E^e}{\gamma_B^b \gamma_C^c}\) is called activity coefficient quotient, represented by uppercase Gamma symbol \(\Gamma\) :
\[
\Gamma=\frac{\gamma_D^d \gamma_E^e}{\gamma_B^b \gamma_C^c}
\]
difference between Dissociation Constant and Equilibrium Constant
Based on above analysis, We would know the ifference between Dissociation Constant and Equilibrium Constant: it essentially the differenc of concentration quotient (when equilibrium) and equilibrium constant of a reaction.
take a weak acid as an example:
\[
\ce{HA <=> H+ + A-}
\]
dissociation constant \(K_a\) is:
\[
K_a = \ce{\frac{[H+][A-]}{[HA]}}
\]
whereas equilibrium constant \(K\) is:
\[
K = \frac{a_{H^+} \cdot a_{A^-}}{a_{HA}}
\]
their relationship is:
\[
\begin{align*}
K &= K_a \Gamma \\
&= \ce{\frac{[H+][A-]}{[HA]}} \cdot \frac{\gamma_{H^+}\gamma_{C^-}}{\gamma_{HA}} \\
&= \frac{a_{H^+} \cdot a_{A^-}}{a_{HA}}
\end{align*}
\]