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
edited at: 01-18-2022: