Rate of Reaction and half life [duplicate]








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[Why can't the half-life be determined in a reaction with more than one reactant?](/questions/167315/why-cant-the-half-life-be-determined-in-a-reaction-with-more-than-one-reactant)

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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?




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.


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