Calculate the pH of a water sample from a playa lake in the western US that has the evaporite mineral potassium bicarbonate (KHCO3) at a concentration of 0.009 mol/kg. Assume the water is in equilibrium with the atmosphere (PCO2 = 10-3.4 bar) and the temperature is 25 ̊C. Note that the pH will be greater than 7. State any other assumptions that you make to simplify the problem. Calculate the pH using the proton condition method.
To calculate the pH of the water sample using the proton condition method, we need to consider the equilibrium reactions involving potassium bicarbonate (KHCO3) and the dissolved carbon dioxide (CO2) in the water. The proton condition method assumes that the proton concentration can be used as an independent basis for calculating the pH.
Assumptions:
1. The water sample is in equilibrium with the atmosphere, meaning it has reached equilibrium with gaseous CO2.
2. The temperature of the water sample is 25°C.
3. The concentration of KHCO3 is given as 0.009 mol/kg.
4. The pCO2 (partial pressure of CO2) in the atmosphere is given as 10^-3.4 bar.
Steps to calculate the pH using the proton condition method:
1. Write the relevant equilibrium reactions:
a) Dissolved CO2 reacts with water to form carbonic acid (H2CO3):
CO2 + H2O ⇌ H2CO3
b) Carbonic acid (H2CO3) dissociates to bicarbonate (HCO3-) and a proton (H+):
H2CO3 ⇌ HCO3- + H+
c) Bicarbonate (HCO3-) further dissociates to carbonate (CO3^2-) and another proton (H+):
HCO3- ⇌ CO3^2- + H+
2. Use the proton concentrations to construct the proton condition equations:
a) For the carbonic acid equilibrium:
[CO2] / [H2CO3] = K1
b) For the bicarbonate equilibrium:
[HCO3-] / [H2CO3] = K2
c) For the carbonate equilibrium:
[CO3^2-] / [HCO3-] = K3
d) For the water autoionization equilibrium (pH equation):
[H+] [OH-] = Kw, where Kw is the autoprotolysis constant of water.
3. Now, substitute the equilibrium expressions and concentrations into the proton condition equations:
a) For the carbonic acid equilibrium:
[CO2] / [H2CO3] = (pCO2) / (K1[H2O])
b) For the bicarbonate equilibrium:
[HCO3-] / [H2CO3] = (K2[H2O]) / (K1 + [H+])
c) For the carbonate equilibrium:
[CO3^2-] / [HCO3-] = [H+] / (K2 + [H+])
4. The concentration of H2O is approximately constant, so we can combine the equations obtained in steps 3a, 3b, and 3c to form a quadratic equation:
pCO2 / K1 = ([H+]^2) / (K3 + K2[H+] + [H+]^2)
5. Rearrange the equation to solve for [H+]:
[H+]^3 + (K3 + K2pCO2 / K1)[H+]^2 + (K2pCO2 / K1)[H+] - pCO2 = 0
6. Solve the quadratic equation to find the concentration of [H+].
7. Calculate the pH using the equation: pH = -log10[H+].
It should be noted that solving this equation might be cumbersome without a calculator or computer program due to the complexity of the quadratic equation.