Response quotient

The reaction quotient for a chemical reaction is given as the quotient of the instantaneous (currently ) existing activities of the substances: ${\displaystyle Q_{r}(t)}$ ${\displaystyle {\ce {\alpha \,{\rm {A{}+\beta \,{\rm {B{}<=>\rho \,{\rm {R{}+\sigma \,{\rm {S{}}}}}}}}}}}}$ ${\displaystyle t}$ ${\displaystyle a}$

${\displaystyle Q_{r}(t)=\prod _{i}a_{i}(t)^{\nu _{i}}}$

where are the stoichiometric coefficients of the substances involved . ${\displaystyle \nu _{i}}$ ${\displaystyle i}$

In the case of chemical equilibrium , the reaction quotient changes into the equilibrium constant , which is known from the law of mass action . While the equilibrium constant is only defined for the case of chemical equilibrium, the reaction quotient can always be specified and describes the state of the system (in equilibrium or not). ${\displaystyle \lim _{t\to \infty }Q_{r}(t)}$ ${\displaystyle K}$

The reaction quotient can be used to predict which reaction direction must increasingly take place in order to reach equilibrium:

${\displaystyle Q<K}$ the activity of the products is reduced compared to the equilibrium, in order to achieve the equilibrium the forward reaction has to take place more frequently

${\displaystyle Q>K}$ the activity of the products is increased compared to the equilibrium, in order to achieve equilibrium the reverse reaction has to take place more frequently

${\displaystyle Q=K}$ (chemical equilibrium reached)

So far it has not been specified in more detail whether relative activities or absolute activities are used. All of the above is true for both relative and absolute activities.

Case of using relative activities

The case of the use of relative activities, which is often encountered, is discussed here.

For the absolute free enthalpy change (at constant pressure, constant temperature, without external fields) the following applies:

${\displaystyle \Delta G=\sum _{i}\nu _{i}\mu _{i}=\sum _{i}\nu _{i}(\mu _{i}-\mu _{i}^{\circ }+\mu _{i}^{\circ })=\sum _{i}\nu _{i}\mu _{i}^{\circ }+\sum _{i}\nu _{i}(\mu _{i}-\mu _{i}^{\circ }),}$

where is the absolute chemical potential . or in other words: ${\displaystyle \mu _{i}}$

${\displaystyle \exp(\beta \Delta G)=\prod _{i}\exp(\beta (\mu _{i}-\mu _{i}^{\circ }))^{\nu _{i}}\exp(\beta \mu _{i}^{\circ })^{\nu _{i}}}$,

where is. Using the definition of relative activity one gets: ${\displaystyle \beta =1/(k_{B}T)}$

${\displaystyle \exp(\beta \Delta G)=\prod _{i}a_{i}^{\nu _{i}}\exp(\beta \mu _{i}^{\circ })^{\nu _{i}}}$

Therefore then:

${\displaystyle \exp(\beta \Delta G)=Q_{r}\exp \left(\beta \sum _{i}\nu _{i}\mu _{i}^{\circ }\right)}$, or:

${\displaystyle Q_{r}=\exp(\beta (\Delta G-\Delta G^{\circ }))}$

If one wants to consider equilibrium ( ), one has achieved this if , in this case corresponds to that one is in equilibrium. Thus: ${\displaystyle \Delta G=0}$ ${\displaystyle Q_{r}=\exp(-\beta \Delta G^{\circ })}$ ${\displaystyle Q_{r}=K}$

${\displaystyle K=\exp \left(-{\frac {\Delta G^{\circ }}{k_{B}T}}\right)}$.

The relationship between the reaction quotient and the equilibrium constant is:

${\displaystyle \Delta G(t)=k_{B}T\ln \left({\frac {Q_{r}(t)}{K}}\right)}$

Case of using absolute activities

In the event that absolute activities are used to establish the reaction quotient, the following applies in equilibrium (as in the case of relative activities ): Therefore: ${\displaystyle \lambda _{i}}$ ${\displaystyle \Delta G=0}$

${\displaystyle K=Q_{r}=\prod _{i}\lambda _{i}^{\nu _{i}}=\exp(\beta \underbrace {\Delta G} _{=0})=1}$

Is in balance and always . ${\displaystyle \Delta G=0}$ ${\displaystyle Q=K=1}$

Note, in the case of using relative activities, (relative activities) does not mean, as here, that one is in equilibrium. ${\displaystyle Q_{r}=1}$

Individual evidence

1. ↑ Peter W. Atkins, Julio de Paula: Physical chemistry . 5th edition. Wiley-VCH, Weinheim 2013, ISBN 978-3-527-33247-2 , pp. 224 ff .
2. ↑ Richard E. Dickerson: Principles of Chemistry. Walter de Gruyter, 1988, ISBN 978-3-110-09969-0 , p. 815.
3. ↑ Richard E. Dickerson: Principles of Chemistry. Walter de Gruyter, 1988, ISBN 978-3-110-09969-0 , p. 196 ( limited preview in the Google book search).