The narcotic cocaine is a weak base wirh \(\mathrm{p} K_{\mathrm{b}}=5.59\). Calculate the ratio of the concentration of cocaine and its conjugate acid in a solution of \(\mathrm{pH}=8.00\).

Understanding the pH and pOH relationship is crucial when dealing with acids and bases in aqueous solutions. The pH scale measures how acidic or basic a solution is, ranging typically from 0 (very acidic) to 14 (very basic), with 7 being neutral.

To make sense of the pH value, it's helpful to know that it's derived from the concentration of hydrogen ions (H+) in a solution, specifically through the negative logarithm of this concentration. The pOH, on the other hand, measures the concentration of hydroxide ions (OH-), and its value is calculated through the negative logarithm of the hydroxide ion concentration.

The key relationship between pH and pOH is simple but vital: they always add up to 14 in any aqueous solution at 25°C. This means that if you know one of these values, you can immediately determine the other by subtracting from 14. Thus, if the pH is 8, the pOH will be 14 - 8, which equals 6. This inverse relationship helps you transition between the acidic and basic perspectives of a solution.

The strength of a base can be quantified by its pKb value. The pKb value is an indication of the base's tendency to accept protons. A low pKb value corresponds to a strong base while a high pKb value indicates a weak base.

Here's why this is important: the value of pKb can be used, along with pH, in the Henderson-Hasselbalch equation to understand the balance between a base and its conjugate acid in solution. The higher the pKb, the less likely the base is to attract a hydrogen ion, which can also be described as the base needing a higher pH (less acidic environment) to be in its free, deprotonated form. In our example with cocaine, a pKb of 5.59 suggests it's a relatively weak base since it's greater than halfway on the pH scale, which peaks at 14 for strong bases.

In the realm of acids and bases, a conjugate pair consists of two species that transform into each other by the gain or loss of a proton (H+). When a base accepts a proton, it becomes its conjugate acid; conversely, when an acid donates a proton, it forms its conjugate base.

Understanding this pair is fundamental when applying the Henderson-Hasselbalch equation because this equation deals with the ratio of the concentrations of a base and its conjugate acid. The nature of this relationship informs us about the buffer capacity of a solution and how well it can maintain its pH when small amounts of acid or base are added. For instance, in the cocaine problem, we're considering cocaine as the base and its transformed version (when it gains a proton) as the conjugate acid.

The pH and pOH scales are logarithmic, which means that a change of one unit on the scale corresponds to a tenfold change in the concentration of H+ or OH- ions, respectively. Similarly, the Henderson-Hasselbalch equation uses logarithms because the concentrations of acid-base conjugate pairs in solutions can vary over several orders of magnitude.

Understanding logarithms, therefore, is key in acid-base chemistry. Specifically, you need to be comfortable with calculating the logarithm (log) of a number to find the pH or pOH and using the antilogarithm (also called the inverse log or exponentiation) to go the other way. In our exercise involving cocaine, once we figured out the difference between pOH and pKb, we took the antilogarithm of that result to find the ratio of base to conjugate acid.