Calculation pH from concentration

[yang09]

Homework Statement

Shells of mussels are mainly of calcium carbonate. IUsually the shell is stable and its content is in equilibrium with the surrounding. Mussels need to build up or reduce their shell, however, for additional protection or during growth. They achieve this by changing the pH-value of the seawater in their surrounding.
Assume that the concentration of Ca²⁺ ions in seawater and in the vicinity of a mussel bank is 0.01 M and that the total concentration of "inorganic carbon" in seawater is 0.0023 M.
Calculate the pH(correct to 2 decimals) of the water in the vicinity of mussels so that their shell remains stable

Homework Equations

pH = -log[H]
Not sure if this is relevant: pH = -log[Activity of Hydrogen]

The Attempt at a Solution

Not sure on what to do. I know that that to find pH, we need to find the hydrogen concentration. I am thinking that activity of hydrogen might be involved but not sure. Any help is appreciated. Thanks

 

[Borek]

You have to find out minimum concentration of CO₃^2- at which calcium carbonate will not dissolve (use K_sp of CaCO₃ for that), then assume "inorganic carbon" means just all forms of carbonic acid present in the sea water.

 

[yang09]

To find the minimum concentration, what I did was I set up the Ksp of CaCO3: CaCO3 <--> [Ca2+][CO3 2-] and solved for CO3 2-.
The Ksp of CaCO3 was 4.8 X 10^-9 so here was my setup: 4.8 X 10^-9 = [0.01][x]. I solved x, which is the concentration of Carbon Dioxide, to be 4.8 X 10^-7 M.
From there, I used the first acid dissociation of carbonic acid: H2CO3 <--> HCO₃^- + H⁺. Would I use the 0.0023 M as the concentration of HCO₃^- and solve for H+ and then calculate pH? Or can I skip the first dissociation and just use this set up: H2CO3 <--> CO3 2- + 2H+?

 

[Borek]

x is not a concentration of carbon dioxide, but of CO₃^2-, and you should use second dissociation constant, not the first one. You are interested mostly in the HCO₃^-/CO₃^2- equilibrium.

 

[yang09]

Sorry. My "x" value is of Carbonate, not dioxide. I mixed them up. Is my calculation of the minimum concentration of CO3 2- correct or atleast in the right ballpark? In order to solve for the [H] value from the second dissociation( HCO3 -<--> CO3 2- + H+), I would have use the first dissociation step and solve for HCO3- (H2CO3 - <--> HCO3 - + H+). When doing the first dissociation step, the value I used for the carbonic acid was the concentration of inorganic carbon" they gave us in the question(0.0023 M). From there, I calculated a value of 3.31 X 10^-4 for HCO3-. With this value, I went on to the second dissociation step( HCO3 - <--> CO3 2- + H+). The value I used for CO3 2- was the minimum concentration I had calculated using the Ksp of CaCO3( 4.8 X 10^-7). I solved for H+ and finally calculated a pH of 7.5. Does that seem reasonable? How would I check myself to see if the pH value I calculated was correct aside from redoing my calculation? Is there a shortcut for double checking my answer without going through the calculation again?

 

[Borek]

Your concentration of CO₃^2- looks reasonable.

As for the rest of your calculations - 'total inorganic carbon' means all forms, not just dissolved carbon dioxide. That means you can't use this number directly, you have to introduce it into calculation as an equation telling you what is sum of concentrations of all carbonate forms. Depending on the final result it may happen that concentrations of other forms are negligibly small, and your assumption is correct - but you have to check it.

 

[yang09]

That means you can't use this number directly, you have to introduce it into calculation as an equation telling you what is sum of concentrations of all carbonate forms.

Can you elaborate on what you mean here?
What I did was that I did an I.C.E. Table ( I = initial concentration, C = Change in concentration, E = equation) and set up a quadratic equation with the 0.0023 M as the initial concentration of carbonic acid. Is that what you mean when you say introduce it into calculation as an equation?

 

[Borek]

Ah, OK. I thought you treated given number as the equilibrium concentration. What you did seems to be a correct approach - with one important remark. You are making an indirect assumption about the HCO₃^- concentration - you should check, if it holds.

 

[yang09]

What do you mean by an indirect assumption about the HCO3 ^- concentration? The way I solved for it was through the first dissociation step[ H2CO3--> HCO3^- + H⁺]. I had an initial concentration of 0 for both HCO3- and H+ and solved the quadratic system for x, which was my HCO3- concentration.

 

[Borek]

You used calculated [HCO₃^-] and known [CO₃^2-] in your calculation of pH - but [HCO₃^-] is a source of [CO₃^2-], so its concentration is not constant. You should check if it didn't change too much through the second dissociation step.

 

[yang09]

Thanks for the help Borek and explaining step by step on how to go about solving the problem. I got an answer now. I appreciate it