Rate Constants and Concentration-Time Equations: Exploring aA→B

[sodium.dioxid]

Every book I look at, they state

rate = -Δ[A]/Δt = k[A] , for A → B

From there, they go on to derive the concentration-time equation.

Well, my concern is what if we have: aA → B

Shouldn't "a" be accounted for in the derivation.

In other words, why don't we derive a more general equation using rate = -Δ[A]/aΔt = k[A]?

It seems like the book wants me to use ln[A]_t = -kt + ln[A]_0 even when I have aA → B

For some reason, it's always the chemistry books horrible at explaining things (unlike Biology and Physics).

 

[Borek]

In other words, why don't we derive a more general equation using rate = -Δ[A]/aΔt = k[A]?

I don't see how it is more general. Speed is a change of concentration per time unit, period.

 

[epenguin]

Maybe he means bimolecular reactions sUch as maybe 2NO₂ → N₂O₄ ?

The answer is that treatments are found in the texbooks for a=2. For a>2 they assentially do not happen becuase of the high improbablilty of a productive trimolecular collision. Of course soichiometries a>2 exist, but the fundamental kinetic laws for elementary i.e. essentially single-step reactions which you are styuding now are only ever straight first order or second order.
There are then more complicated mechanisms which you will come across with all sorts of kinetic laws, but they are all made of several elementary reactions which each follow first or second order kinetic law.

 

[MrSid]

Na:O2

Have you checked the wiki page for derivations of rate equations? the general problem is set up as aA + bB --> cC + dD and the equations are then resultant of several assumptions that simplify solving the differential equations and integrating the results.

It is these simplifying assumptions that always have to be met by the experiments if one can attempt to use the simple 0, 1st, and 2nd order integrated equations under steady state approximations. Experimental data fits to the equations are then used to model the system. Bad fits usually mean new experiments with different simplifying criteria, and then a reanalysis.

see wikipedia Rate_equation_(chemistry) and in the english article use the link for Steady_state_approximation

 

[LudusRex]

I can understand your concern about the derivation of the concentration-time equation for the reaction aA→B. It is important to note that the rate constant (k) is specific to a particular reaction and does not depend on the stoichiometric coefficient (a) of the reactant. This is because the rate of a reaction is determined by the number of collisions between reactant molecules, not the number of molecules involved in the reaction.

In other words, the rate of a reaction is proportional to the concentration of the reactant, regardless of its stoichiometric coefficient. This is why the general equation for aA→B can still be written as rate = -Δ[A]/Δt = k[A], as the stoichiometric coefficient does not affect the proportionality constant (k).

However, it is important to keep in mind that the stoichiometric coefficient does play a role in determining the initial concentration of the reactant and therefore, should be considered when deriving the concentration-time equation. This is why in some cases, the equation may be written as ln[A]_t = -kt + ln[aA]_0, to account for the initial concentration of the reactant.

In summary, the general equation rate = -Δ[A]/aΔt = k[A] can be used for the reaction aA→B, but it is important to consider the initial concentration of the reactant when deriving the concentration-time equation. I would suggest consulting multiple sources and seeking clarification from your instructor if you are still unsure about the derivation.