Calculate $\mathrm{ pH }$ of $\pu{ 0.05 mol }$ of $\ce{Zn(CH3COO)2}$ and $\pu{ 0.025 mol }$ of $\ce{ NaOH }$ dissolved in $\pu{ 1 L }$ of distilled water?
I tried to use the systematic method by forming the following equations to extract cubic equation which can be solved by computer : \begin{array}{ } [\ce{ CH3COO- }]_I &= 2\times {0.05} = \pu{ 0.1 M }\\ [\ce{Zn^{2+}}] &=\pu{ 0.05 M }\\ [\ce{Na+}] &=\pu{ 0.025 M }\\ 0.1 = [\ce{CH3COO-}]_I &= [\ce{CH3COO-}]_e + [\ce{CH3COOH}]_e\\ 2\ce{[Zn^{2+}] + [Na+] +[ H+]_e &= [CH3COO-]_e +[ OH-]_e} \\ K_\mathrm{a} &=\frac{ [\ce{ H+}]_e\cdot{ [\ce{CH3COO-}]_e}}{[\ce{CH3COOH}]_e}\\ [\ce{OH-}]_e &=\frac{ K_\mathrm{w}}{[\ce{H+}]_e} \end{array}
First question: Are the equations correct?
Second question: Is there a simple approximate method to solve the above problem? How?
I appreciate any help?
$\large \text{ --------- Maxw Notes ---------}$
Editing the answers with all the markup is painfully slow, so I added different analysis in a number of different answers to this problem. No doubt that will be confusing to someone looking at this problem. So let me summarize the answers.
Note: I am going to carry "extra" significant figures in the calculations to gauge the mathematical exactness, but this is of course is ridiculous for the chemistry since the equilibrium constants are not known to that precision. If the true solution values are within 5% of the mathematically calculated values that's really good.
Setup an ICE table and solved the nasty linear equations with Wolfram Alpha to get an "exact solution."
Setup an ICE table and then simplified the problem to create a "Simplified ICE Table". Went through some head pounding to create an iterative solution. Realized the original ICE table could be used for iterated solution so calculated that solution too.
Reducing the Problem to a Single Variable
I had tried to solve the problem first with what I am calling the Iterative Solution. I could tell all the species other than $\ce{[Na+]}$ were independently pH controlled, but I couldn't figure out how to iterate the solution. Using the charge balance finally occurred to me so I tabulated the data to show that the approach would work.
Used charge balance of solution, and one assumption to solve the simplified problem with a cubic equation with $\ce{[OH-]}$ as the variable. Solved the cubic equation with Wolfram Alpha.
Assumption:
- $\ce{[CH3COO-]_{equil} \gg [OH-]_{equil}}$
Solution with Quadratic Equation
Used charge balance of solution, and two assumptions to solve the simplified problem with a quadratic equation with $\ce{[OH-]}$ as the variable. Solved the quadratic equation with Wolfram Alpha, but is reasonable to assume that solution could have been done by hand.
Assumptions:
- $\ce{[CH3COO-]_{equil} \gg [OH-]_{equil}}$
- $\ce{[CH3COO-]_{equil} \gg K_b }$
- That the normalization tweak works. (I know that it did from the exact solution, but the calculations showed that it worked too.)