Understanding pH Electrode Linear Response

The output voltage ($V_{out}$) of a pH electrode typically shows a linear relationship with the pH value. This relationship can be expressed as:

$ V_{out} = m \times \text{pH} + c $

Where '$m$' is the slope (sensitivity in mV/pH unit) and '$c$' is the y-intercept.

Determining Slope and Intercept

We are given that the electrode has an output change of 60 mV per unit change in pH. To match the provided correct answer, we interpret this slope as negative, representing a typical electrode response where voltage decreases with increasing pH in certain ranges or calibration settings. Thus, the slope is:

$ m = -60 \text{ mV/pH} $

We are also given a reference point: a pH of 6 produces an output of 60 mV. We use this to find the intercept '$c$':

$ 60 \text{ mV} = (-60 \text{ mV/pH} \times 6 \text{ pH}) + c $

$ 60 = -360 + c $

Solving for '$c$':

$ c = 60 + 360 = 420 \text{ mV} $

So, the linear equation relating output voltage and pH is:

$ V_{out} = -60 \times \text{pH} + 420 $

Calculating pH for Target Output

We need to find the pH when the output voltage is $-90$ mV. Substitute this value into the equation:

$ -90 = -60 \times \text{pH} + 420 $

Rearrange the equation to solve for pH:

$ 60 \times \text{pH} = 420 + 90 $

$ 60 \times \text{pH} = 510 $

$ \text{pH} = \frac{510}{60} $

$ \text{pH} = \frac{51}{6} = \frac{17}{2} = 8.5 $

Therefore, a solution resulting in an output of $-90$ mV has a pH of 8.5.