# equilibrium constant, reaction quotient, dissociation constant-relationships and differences

created at 01-18-2022 views: 5

## 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. $$K$$¶

1. equilibrium constant is the reation quotient when a reaction approachs its equilibrium。
2. activities $$a_i$$ in the equation of $$Q$$ are all instantaneous activities: as reaction goes on, these activities will change.
3. $$Q > k$$：reaction shift toward the left side (reactants)
4. $$Q < k$$：reaction shift toward the right side (products)
5. $$Q = k$$：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*}
created at:01-18-2022 TwoPunch