Chemical equilibrium and analysis of a mixture. (Warning! This is a long problem.) A remote optical sensor for CO₂ in the ocean was designed to operate without the need for calibration.

The sensor compartment is separated from seawater by a silicone membrane through which CO₂, but not dissolved ions, can diffuse. Inside the sensor, CO₂ equilibrates with HCO^-₃ and CO₃^2-. For each measurement, the sensor is flushed with fresh solution containing 50.0 M bromothymol blue indicator (NaHIn) and 42.0 M NaOH. All indicator is in the form HIn^- or In^2- near neutral pH, so we can write two mass balances: (1) [HIn^-]  [In2^-]  FIn  50.0 M and (2) [Na⁺]  F_Na = 50.0μM + 42.0 μM + 92.0 μM. HIn^- has an absorbance maximum at 434 nm and In^2- has a maximum at 620 nm. The sensor measures the absorbance ratio R_A = A₆₂₀/A₄₃₄ reproducibly without need for calibration. From this ratio, we can find [CO₂(aq)] in the seawater as outlined here:

(a) From Beer’s law for the mixture, write equations for [HIn^-] and [In^2-] in terms of the absorbances at 620 and 434 nm (A₆₂₀ and A₄₃₄). Then show that 

(b) From the mass balance (1) and the acid dissociation constant KIn, show that 

(c) Show that

(d) From the carbonic acid dissociation equilibria, show that 

(e) Write the charge balance for the solution in the sensor compartment. Substitute in expressions B, C, E, and F for [HIn^-], [In^2-], [HCO^3-], and [CO₃ ^2-]. 

(f) Suppose that the various constants have the following values: 

From the measured absorbance ratio R_A = A₆₂₀/A₄₃₄  2.84, find [CO₂(aq)] in the seawater. 

(g) Approximately what is the ionic strength inside the sensor compartment? Were we justified in neglecting activity coefficients in working this problem?

Transcribed Image Text:

CO2(g) CO2(aq)Co2laq) K, CO2(aq) + H2O 근 HCOg + H+ Hln In 620 60 Hln In 434 ε 434 Seawater Inside sensor compartment Silicone membrane allows CO2 but not other species, to diffuse across