Disappear alpha, concentrations [closed]








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I have a doubt that torments me constantly ... And I hope your answer so that you can solve it. When they give me $K\_{a}$ or $K\_{b}$, the concentrations and they ask me to find alpha, when can I discard alpha?


Example:


$K\_{b} = 1.79·10^{-5}$, $(NH\_{3}) = 0.74M$, $\alpha =$


Okay, with these data I arrive at the following expression:


$K\_{b} = (0.74·\alpha ^ 2) / (1-\alpha)$, here, in the "$1-\alpha$", can I discard?


I think the answer is no, but could you tell me why not? And in what cases can I do it?




I assume that by "despise" you actually mean discard the value of alpha. As in making the approximation: $$(1-\alpha ) \approx 1$$
To answer your question we need to consider the meaning of alpha and the value of the equilibrium constant.


$\alpha$ represents the degree of ionization, i.e., the amount that dissociates from 1. If you say that $\alpha$ is 0.1, for example, then what you are saying is that for every 1M of concentration 0.1M dissociates. If you have a different concentration than 1M, then the amount that dissociates is $\alpha M$ where M is the concentration of your starting substance.


The equilibrium constant, K, tells you something about the relation between products and reactants. If K is greater than 1 then products are favored and if K is less than one reactants are favored (or, the reaction does not "like" to go towards the right).


Look at the value of your constant $K\_b = 1.79 . 10^{-5}$. It is small. That should tell you that very little amount of your starting reactant will dissociate to yield products. Thus, we would expect the value of $\alpha$ to be small when compared to one. If this is the case then we can say that $$(1-\alpha ) \approx 1$$
If you solve the equation with the approximation you find $\alpha = 0.004906$ . If you solve the equation without approximation you get $\alpha = 0.004918$
Thus, it seems that the approximation is justified particularly if you don't need great precision in your answer.


So, when do you approximate? There is no a set answer to this. But, if your equilibrium constant is small ($ < 10^{-4}$) usually) and the concentration of your initial substance is big enough you can try and see what happens. The rule of thumb that I was told was to accept the approximation if the final result is less than 5% of the value of the approximated value. In this case, we obtained 0.004918 over a value of 1. This represents 0.5% and the approximation is justified.


Make sure you ask your professor what is the criterion he or she uses to justify an approximation.


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