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During the course of chemical kinetics while studying zero order or first order they consider rate of reaction to be dependent only on single Reactant For ex:- A--> product,Rate of Reaction=k[A]^0 or [A]^1 and then define half life of such reaction But it can be [A]^1[B]^-1 for zero order and for first order it can be something like [A]^3/2[B]^-1/2 So why we don't consider these cases?can we define half life of these cases?and pls tell what is the significance of half life of a Reaction?

Ronit
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Consider a reaction that follows the power law at constant volume, temperature, and pressure of the form:

$$\ce{aA +bB->cC +dD}$$

The rate of this reaction would be:

$$-r_\pu{A}=-\frac{\mathrm{d}C_\pu{A}}{\mathrm{d}t}=k\;C_\pu{A}^{\alpha}\;C_\pu{B}^{\beta}$$

We can express both concentrations in terms of conversion $X$:

$$C_\pu{A}=C_{\pu{Ao}}\;(1-X)$$

$$C_\pu{B}=C_{\pu{Ao}}\left(\frac{C_{\pu{Bo}}}{C_{\pu{Ao}}}-\frac{b}{a}X\right)$$

Substituting into the rate law:

$$-\frac{\mathrm{d}[C_{\pu{Ao}}\;(1-X)]}{\mathrm{d}t}=k\;C_{\pu{Ao}}^\alpha\;(1-X)^{\alpha}\;C_{\pu{Ao}}^\beta\left(\frac{C_{\pu{Bo}}}{C_{\pu{Ao}}}-\frac{b}{a}X\right)^{\beta}$$

Grouping up terms and re-arranging:

$$\pu{C_{Ao}}\frac{\mathrm{d}X}{\mathrm{d}t}=k\;\pu{C_{Ao}}^{\alpha+\beta}\;(1-X)^\alpha\;\left(\frac{\pu{C_{Bo}}}{\pu{C_{Ao}}}-\frac{b}{a}X\right)^\beta$$

$$\frac{\mathrm{d}X}{(1-X)^\alpha\;\left(\frac{\pu{C_{Bo}}}{\pu{C_{Ao}}}-\frac{b}{a}X\right)^\beta}=k\;\pu{C_{Ao}}^{\alpha+\beta-1}\;\mathrm{d}t$$

Half-life is defined as the time it takes for the concentration of limiting reagent $\pu{A}$ to reach half of its initial value:

$$t=t_{1/2}\implies\pu{C_A}=0.5\;\pu{C_{Ao}}$$

And conversion is also defined as:

$$X=\frac{\pu{C_{Ao}}-\pu{C_{A}}}{\pu{C_{Ao}}}=\frac{\pu{C_{Ao}}-0.5\;\pu{C_{Ao}}}{\pu{C_{Ao}}}=0.5$$

For reaction half-life to make sense when 2 or more reactants are involved, two additional conditions must be met:

$$\pu{C_{Ao}}=\pu{C_{Bo}}$$

$$a=b$$

This guarantees that both concentrations of $\pu{A}$ and $\pu{B}$ start at the same initial values and get consumed at the same rate, since it follows that:

$$\pu{C_A=C_B}$$

$$\pu{-r_A=-r_B}$$

So if you want to find the half-life for a reaction that meets all conditions mentioned, you would have to integrate:

$$\int_0^{0.5} \frac{\mathrm{d}X}{(1-X)^{\alpha+\beta}}=\int_0^{t_{1/2}}k\;\pu{C_{Ao}}^{\alpha+\beta-1}\;\mathrm{d}t$$

You will obtain an equation that you can solve for to calculate half-life $t_{1/2}$.

If the overall reaction order is not 1:

$$\alpha+\beta≠1\implies\pu{C_{Ao}}^{\alpha+\beta-1}≠1\implies t_{1/2}=f(\pu{C_{Ao}})$$

So half-life would depend on initial concentration of A.

Conversely, when the overall reaction order is 1:

$$\alpha+\beta=1\implies\pu{C_{Ao}}^{\alpha+\beta-1}=1\implies t_{1/2}≠f(\pu{C_{Ao}})$$

So half-life would be independent of initial concentration of A.

Sam202
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  • But does it make sense to define half life of a reaction ?half life of a species is understable for A-->Product. But for a reaction for a half life to be defined both reactants need to be halved at the same time. Which is usually not the case? Can you pls explain this – Ronit Nov 15 '22 at 06:33
  • In my book it was also written that for a reaction involving two or more reactants,half life of reaction is defined only when the initial concentration of reactants were taken in their stoichiometric ratio. So does that mean we can't define half of these reaction always??? – Ronit Nov 15 '22 at 06:36
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    One of the reactants is usually the limiting reagent, which means it will be totally consumed first before the others get a chance to. Conversion $X$ is defined for such limiting reagent $A$, as shown above. – Sam202 Nov 15 '22 at 06:45
  • http://goldbook.iupac.org/terms/view/H02716 this is link of ipuac book in which it is written that half life can only be defined when reactants are in their stoichiometric coefficient – Ronit Nov 15 '22 at 07:43
  • If such condition is necessary, then: $\frac{C_{Bo}}{C_{Ao}}=\frac{b}{a}$, which means $C_A=C_B$, and both concentrations will experience the same consumption rate. In other words, stoichiometric initial concentrations guarantees equal change in both reactant $A$ and $B$ after half-life $t_{1/2}$ elapsed. – Sam202 Nov 15 '22 at 07:56
  • @Ronit Even if reactants are in their stoichiometric ratio, the half-life of other then the 1st order reactions is concentration dependent. While it can be formally defined and practically used, e.g. for controlling of the synthesis, it is not a constant, but scenario specific parameter. – Poutnik Nov 15 '22 at 10:33