Chapter 10, Problem 10.11P

(a)

Interpretation Introduction

Interpretation:

The expressions for ${\text{K}}_{\text{H}}{\text{, K}}_{\text{a1 }}{\text{and K}}_{\text{a2}}$ combined and then find an expression for $\left[{\text{CO}}_{\text{3}}^{\text{2-}}\right]$ in terms of   ${\text{P}}_{{\text{CO}}_{\text{2}}}\text{and}\left[{\text{H}}^{\text{+}}\right]$ has to be derived.

Explanation of Solution

The expressions for ${\text{K}}_{\text{H}}{\text{, K}}_{\text{a1 }}{\text{and K}}_{\text{a2}}$ as derive as

$\begin{array}{l}\text{equation}\text{C}\text{,}\left[{\text{CO}}_{\text{3}}^{\text{2-}}\right]\text{=}\frac{{\text{K}}_{\text{a2}}\left[{\text{HCO}}_{\text{3}}^{\text{-}}\right]}{\left[{\text{H}}^{\text{+}}\right]}\text{(F)}\\ \text{equation}\text{B}\text{,}\left[{\text{HCO}}_{\text{3}}^{\text{-}}\right]\text{=}\frac{{\text{K}}_{\text{a1}}\left[{\text{CO}}_{\text{2(aq)}}^{}\right]}{\left[{\text{H}}^{\text{+}}\right]}\text{(G)}\\ \left[{\text{HCO}}_{\text{3}}^{\text{-}}\right]\text{value}\text{is}\text{substituted}\text{equation}\text{Ginto}\text{equation}\text{H}\\ \left[{\text{CO}}_{\text{3}}^{\text{2-}}\right]\text{=}\frac{{\text{K}}_{\text{a2}}{\text{K}}_{\text{a1}}\left[{\text{CO}}_{\text{2(aq)}}^{}\right]}{{\left[{\text{H}}^{\text{+}}\right]}^{\text{2}}}\text{(H)}\end{array}$

$\left[{\text{CO}}_{\text{3}}^{\text{2-}}\right]\text{=}\frac{{\text{K}}_{\text{a2}}{\text{K}}_{\text{a1}}{\text{P}}_{{\text{CO}}_{\text{2}}}}{{\left[{\text{H}}^{\text{+}}\right]}^{\text{2}}}\text{(I)}$

(b)

Interpretation Introduction

Interpretation:

The given conditions is possibly be reached around the year 2100 has to predicted.

Explanation of Solution

Given,

${\text{P}}_{{\text{CO}}_{\text{2}}}\text{=}\text{800}\text{μbar,}\text{pH}\text{=}\text{7}\text{.8,}{\text{0}}^{\text{o}}\text{C}\text{we}\text{find}$

$\left[{\text{CO}}_{\text{3}}^{\text{2-}}\right]\text{=}\sqrt{\frac{{\text{K}}_{\text{a2}}{\text{K}}_{\text{a1}}{\text{K}}_{\text{H}}{\text{P}}_{{\text{CO}}_{\text{2}}}}{{\left[{\text{H}}^{\text{+}}\right]}^{\text{2}}}}$

$\begin{array}{l}\frac{{\text{10}}^{\text{-9}\text{.3762}}\text{mol}{\text{kg}}^{\text{-1}}{\text{10}}^{\text{-6}\text{.1004}}\text{mol}{\text{kg}}^{\text{-1}}{\text{10}}^{\text{-1}\text{.2073}}\text{mol}{\text{kg}}^{\text{-1}}{\text{bar}}^{\text{-1}}\text{(800}{\text{×10}}^{\text{-6}}\text{bar)}}{{\left[{\text{10}}^{\text{-7}\text{.8}}\text{mol}{\text{kg}}^{\text{-1}}\right]}^{\text{2}}}\\ \text{=6}\text{.6×}{\text{10}}^{\text{-5}}\text{mol}{\text{kg}}^{\text{-1}}\end{array}$

${\text{P}}_{{\text{CO}}_{\text{2}}}\text{=}\text{800}\text{μbar,}\text{pH}\text{=}\text{7}\text{.8,}3{\text{0}}^{\text{o}}\text{C}\text{we}\text{find}$

$\left[{\text{CO}}_{\text{3}}^{\text{2-}}\right]\text{=}\sqrt{\frac{{\text{K}}_{\text{a2}}{\text{K}}_{\text{a1}}{\text{K}}_{\text{H}}{\text{P}}_{{\text{CO}}_{\text{2}}}}{{\left[{\text{H}}^{\text{+}}\right]}^{\text{2}}}}$

$\begin{array}{l}\frac{{\text{10}}^{\text{-8}\text{.8324}}\text{mol}{\text{kg}}^{\text{-1}}{\text{10}}^{\text{-5}\text{.8008}}\text{mol}{\text{kg}}^{\text{-1}}{\text{10}}^{\text{-1}\text{.6048}}\text{mol}{\text{kg}}^{\text{-1}}{\text{bar}}^{\text{-1}}\text{(800}{\text{×10}}^{\text{-6}}\text{bar)}}{{\left[{\text{10}}^{\text{-7}\text{.8}}\text{mol}{\text{kg}}^{\text{-1}}\right]}^{\text{2}}}\\ \text{=1}\text{.8×}{\text{10}}^{\text{-4}}\text{mol}{\text{kg}}^{\text{-1}}\end{array}$

(c)

Interpretation Introduction

Interpretation:

The aragonite and calcite dissolve under the condition has to be predicted.

Explanation of Solution

Equilibrium expressions for aragonite and calcite are

${\text{CaCO}}_{\text{3}}\text{(s,}\text{aragonite)}\underset{}{\rightleftharpoons }{\text{Ca}}^{\text{2+}}{\text{+CO}}_{\text{3}}^{\text{2-}}\text{(D)}$

$\begin{array}{l}{\text{K}}_{\text{sp}}^{\text{arg}}\text{=}\left[{\text{Ca}}^{\text{2+}}\right]\left[{\text{CO}}_{\text{3}}^{\text{2-}}\right]{\text{=10}}^{\text{-6}\text{.1113}}\text{mol}{}^{\text{2}}{\text{kg}}^{\text{-2}}\text{at}{\text{0}}^{\text{o}}\text{C}\\ {\text{=10}}^{\text{-6}\text{.1113}}\text{mol}{}^{\text{2}}{\text{kg}}^{\text{-2}}\text{at}{\text{30}}^{\text{o}}\text{C}\end{array}$

${\text{CaCO}}_{\text{3}}\text{(s,}\text{calcite)}\underset{}{\rightleftharpoons }{\text{Ca}}^{\text{2+}}{\text{+CO}}_{\text{3}}^{\text{2-}}\text{(E)}$

$\begin{array}{l}{\text{K}}_{\text{sp}}^{\text{arg}}\text{=}\left[{\text{Ca}}^{\text{2+}}\right]\left[{\text{CO}}_{\text{3}}^{\text{2-}}\right]{\text{=10}}^{\text{-6}\text{.3652}}\text{mol}{}^{\text{2}}{\text{kg}}^{\text{-2}}\text{at}{\text{0}}^{\text{o}}\text{C}\\ {\text{=10}}^{\text{-6}\text{.3713}}\text{mol}{}^{\text{2}}{\text{kg}}^{\text{-2}}\text{at}{\text{30}}^{\text{o}}\text{C}\end{array}$

Reaction quotient at ${\text{0}}^{\text{o}}\text{C}$

$\begin{array}{l}\left[{\text{Ca}}^{\text{2+}}\right]\left[{\text{CO}}_{\text{3}}^{\text{2-}}\right]\text{=}\left[\text{0}\text{.010}\text{mol}{\text{kg}}^{\text{-1}}\right]\left[\text{6}\text{.6×}{\text{10}}^{\text{-5}}\text{mol}{\text{kg}}^{\text{-1}}\right]\\ \text{=6}\text{.6×}{\text{10}}^{\text{-7}}{\text{mol}}^{\text{2}}{\text{kg}}^{\text{-2}}{\text{=10}}^{\text{-6}\text{.18}}\text{mol}{}^{\text{2}}{\text{kg}}^{\text{-2}}{\text{<10}}^{\text{-6}\text{.1113}}\text{mol}{}^{\text{2}}{\text{kg}}^{\text{-2}}\\ \text{aragonite}\text{is}\text{dissolve}\end{array}$

Reaction quotient at ${\text{30}}^{\text{o}}\text{C}$ is

$\begin{array}{l}\left[{\text{Ca}}^{\text{2+}}\right]\left[{\text{CO}}_{\text{3}}^{\text{2-}}\right]\text{=}\left[\text{0}\text{.010}\text{mol}{\text{kg}}^{\text{-1}}\right]\left[\text{1}\text{.84×}{\text{10}}^{\text{-4}}\text{mol}{\text{kg}}^{\text{-1}}\right]\\ \text{=1}\text{.84×}{\text{10}}^{\text{-6}}{\text{mol}}^{\text{2}}{\text{kg}}^{\text{-2}}{\text{=10}}^{\text{-5}\text{.74}}\text{mol}{}^{\text{2}}{\text{kg}}^{\text{-2}}{\text{>10}}^{\text{-6}\text{.1113}}\text{mol}{}^{\text{2}}{\text{kg}}^{\text{-2}}\\ \text{hence neither aragonite}\text{nor}\text{calcite}\end{array}$