3
$\begingroup$

This is the exact question I faced on an exam:

Calculate the $\mathrm{pH}$ of $\pu{0.05 M}~\ce{Na2CO3}. \\(\ce{H2CO3}: K_\mathrm{a_1}= 4\times 10^{-7},~ K_\mathrm{a_2}= 4.7\times 10^{-11})$

Solution

$$\ce{Na2CO3 ->2Na+ + CO3^2-}$$

I suppose nothing that can contribute to the $\mathrm{pH}$ of a solution happens to $\ce{Na+}$ ions and we proceed with $\ce{CO3^2-}$ which has a concentration of $\pu{0.05M}$

$$\ce{CO3^2- + H2O <=> HCO3- + OH-}$$

To calculate $\mathrm{pH}$, I need to first figure out concentration of $\ce{OH-}$ ions, and to do so I have to know $K_\mathrm{b}$ dissociation constant for $\ce{CO3^2-}$

On equilibrium concentration of species are as follows: $$[\ce{CO3^2-}] = \pu{0.05 M}-x,[\ce{HCO3-}] = [\ce{OH-}] = x$$

So, we have:

$$\dfrac{x^2}{0.05-x}=K_\mathrm b$$

All I need to know is $K_\mathrm{b}$ value for $\ce{CO3^2-}$ ion. I tried to derive $K_\mathrm{b}$ from $K_\mathrm{a}$ values using $K_\mathrm{a} \times K_\mathrm{b} = 1\times 10^{-14}$ but apparently obtained the incorrect answer. What method should I use to do it the right way?

$\endgroup$
2
  • 3
    $\begingroup$ You should use that method, but you messed up somewhere. Without more work, we can't tell you what you did wrong. $\endgroup$
    – Zhe
    Jul 2, 2018 at 20:59
  • $\begingroup$ @Zhe Thank you,i rechecked my work,found mistake and got the correct result. $\endgroup$ Jul 2, 2018 at 21:18

1 Answer 1

1
$\begingroup$

The carbonate ion is the Conjugate base of the weak acid $\ce{HCO_3^-}\ (K={4.7\times10^{-11}})$, so this solution will alkaline. Given the concentration of this solution ,the pH should be sufficiently high to preclude the formation of any significant amount of $\ce{H_2CO_3}$ , so the solution of this problem as a solution of a monoprotic weak base: $\ce{CO_3^{-2} + H_2O <=> HCO_3^- + OH^-}$ $$\ce{K_b}=\frac{[OH^-][HCO_3^-]}{[CO_3^{-2}]} =\frac{K_w}{Ka}=\frac{10^{-14}}{4.7\times 10^{-11}}=\ce{10^{-3.7}}$$ Neglecting the $\ce{OH^-}$ produced by the autoprotolysis of water, it is valid to make the usual assumption that $\ce{[OH^-]}={[HCO_3^-]}$,and thus $$\dfrac{[OH^-]^2}{0.05-{[OH^-]}}=K_b= \ce{10^{-3.7}}$$ The equilibrium expression must be solved as a quadratic and yields the root $\ce{[OH^-]}$=0.00306 Which corresponds to pOH = 2.5 or pH = 11.5

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.