Before we begin, there are two implicit assumptions:
- That there is not a significant change in the volume of the solution by either adding $\ce{Na2CrO4}$ nor the precipitation of the $\ce{BaCrO4}$
- That concentrations can be used instead of activities.
First of all, in equation $(1)$ why did they put $\ce{[Sr^2+]}$ same as the initial concentration (before the addition of $\ce{Na2CrO4}$)?
The problem statement asks for the $\ce{[Ba^{2+}]}$ when $\ce{SrCrO4}$ "begins to precipitate." Given that "some" $\ce{SrCrO4}$ must precipitate to reach the point where $\ce{SrCrO4}$ has already begun to precipitate, let's define some variables:
$\ce{[Sr^{2+}]_{init}}$ -- The initial concentration of $\ce{Sr^{2+}}$, 0.10 molar.
$\ce{[Sr^{2+}]_{ppt}}$ -- the concentration of $\ce{Sr^{2+}}$ removed by ppt.
$\ce{[Sr^{2+}]_{final} = [Sr^{2+}]_{init} - [Sr^{2+}]_{ppt}}$ -- the concentration of $\ce{Sr^{2+}}$ after some $\ce{SrCrO4}$ has precipitated.
Now here comes the rub. Chemistry isn't like math. The value of $\pi$ has been calculated to millions of digits. $\ce{[Sr^{2+}]_{init}}$ was given as 0.10 molar. So there are only two significant figures in this value. If 0.01% of the $\ce{Sr^{2+}}$ is used to form the initial precipitate then $\ce{[Sr^{2+}]_{final} \approx ([Sr^{2+}]_{init}} = \pu{0.10 M})$
Note that the problem could have been written differently. The problem could have asked for $\ce{[Ba^{2+}]}$ when $\ce{SrCrO4}$ reaches its $\mathrm{K_{sp}}$ (or some variation thereof). This would be right at the equilibrium point, before any $\ce{SrCrO4}$ has formed.
When $\ce{Na2CrO4}$ is added, $\ce{[Sr^2+]}$ will react with $\ce{CrO4^2-}$ (obtained from $\ce{Na2CrO4}$) to form precipitate. The concentration of $\ce{[Sr^2+]}$ will undoubtedly change until the equilibrium is established.
Obviously when $\ce{SrCrO4}$ nucleates there is some finite size of the $\ce{SrCrO4}$ particle which will be stable in an aqueous solution. So if we use a nanoliter of the cation solution then the given answer will be wrong. Thus there is a implied assumption that a macro volume of the solution is being used. "Macro" in this case meaning that creating a detectable amount of the $\ce{SrCrO4}$ does not change the $\ce{[Sr^2+]}$ from its initial nominal value of 0.10 molar.
I would appreciate if you could tell me the proper way of solving this problem.
The book gives the proper answer albeit without a full explanation of the additional assumptions.
The notion is that $\ce{BaCrO4}$ is more insoluble than $\ce{SrCrO4}$, so most of the $\ce{Ba^{2+}}$ will be removed before any $\ce{Sr^{2+}}$ starts to precipitate.
I'll add that this assumption isn't really true. As the $\ce{BaCrO4}$ precipitates, it will incorporate some of the $\ce{Sr^{2+}}$ into the $\ce{BaCrO4}$ precipitate. Again, not an appreciable amount compare to what is left in solution since initially $\ce{[Sr^{2+}] = [Ba^{2+}]}$. However if initially $\ce{[Sr^{2+}] = 0.01\cdot [Ba^{2+}]}$ then there would be a significant loss of $\ce{Sr^{2+}}$ in the solution.