Kaplan MCAT General Chemistry Review

pH and pOH are specific calculations of the more generic “p-scale.” A p-scale is defined as the negative logarithm of the number of items. There’s no reason why this logarithmic calculation cannot be applied to the population values of all the world’s countries [pPop = -log(population)] or to the number of hairs on the head of every living human [pHair = -log(hairs)]. The log scale system could be applied to anything, but let’s be real: pPop and pHair aren’t going to earn us any Test Day points. It’s much more valuable for us to understand the significance of the p-scale expression for the concentrations of hydrogen ions and hydroxide ions in acid and base aqueous solutions.

The pH of a solution is given by

pH = -log[H⁺] = log(1/[H⁺])

Likewise, the pOH of a solution is given by

pOH = -log[OH^-] = log(1/[OH^-])

For pure water at equilibrium at 298 K, the concentration of the hydrogen ion equals the concentration of the hydroxide ion and is 10^-7 mol/L. Therefore, pure water at 298 K has a pH of 7 and a pOH of 7. (Note: -log 10^-7 = 7.)

From the water dissociation constant expression (K_w = [H₃O⁺][OH^-] = 10^-14), we find that

pH + pOH = 14 (for aqueous solutions at 298 K)

Key Concept

This equation demonstrates a fundamental property of logarithms: The log of a product is equal to the sum of the logs; that is, log(xy) = log x + log y.

For any aqueous solution at 298 K, a pH less than 7 (or pOH greater than 7) indicates a relative excess of hydrogen ions, and the solution is acidic; a pH greater than 7 (or pOH less than 7) indicates a relative excess of hydroxide ions, and the solution is basic. A pH (or pOH) equal to 7 indicates equal concentrations of hydrogen and hydroxide ions, resulting in a neutral solution.

Estimating p-Scale Values

An essential skill that you must hone for Test Day, applicable to many problems involving acids and bases, is the ability to convert pH, pOH, pK_a, and pK_b values quickly into nonlogarithmic form and vice versa.

When the original value is a power of 10, the operation is relatively straightforward: Changing the sign on the exponent gives the corresponding p-scale value directly. For example,

If [H+] = 0.001 or 10^-3, then the pH = 3 and pOH = 11. If K_b = 1.0 × 10^-7, then pK_b = 7.

MCAT Expertise

Other important properties of logarithms include these: Log x ⁿ = n Log x and Log 10 ^x = x. From these two properties, one can derive the particularly useful relationship that will be seen on Test Day (and we can see in the example):-Log 10^-x=x.

More difficulty arises (in the absence of a calculator or a superhuman ability to calculate logarithms in your head) when the original value is not an exact power of 10. The MCAT is not a math test, and it is not primarily interested in determining your ability to perform mathematical calculations. Exact calculation of the logarithmic value of a number that is not an integer power of 10 will be excessively onerous, if not outright impossible. The test writers are interested, however, in testing your ability to apply mathematical concepts appropriately in solving certain problems. Fortunately, there is a simple method of approximation that will be foolproof for Test Day.

If the nonlogarithmic value is written in proper scientific notation, it will look like n × 10^-m, where n is a number between 1 and 10. Using the basic log rule that the log (xy) = log x + log y, we can express the negative log of this product, -log (n × 10^-m), as -log (n) - log (10^-m). Since log refers to the common log with base of 10, we can simplify -log (n) - log (10^-m) to -log (n) - (-m), or m - log (n). Now, since n is a number between 1 and 10, its logarithm will be a fraction between 0 and 1 (note: log 1 = 0 and log 10 = 1). Thus, m - log (n) will be between (m - 1) and (m - 0). Furthermore, the larger n is (that is, the closer to 10), the larger the fraction, log (n) will be; consequently, the closer to (m - 1) our answer will be. If this is too much to remember—and given the amount of information you need to remember for Test Day, it might be—all you need to remember is this: When the nonlogarithmic value is n × 10^-m, the logarithmic value will be between (m - 1) and m. It’s that simple!

Example: If K_a = 1.8 × 10^-5, then pK_a = 5 - log 1.8. Because 1.8 is small, its log will be small, and the answer will be closer to 5 than to 4. (The actual answer is 4.74.)

MCAT Expertise

Learning how to estimate when using logarithms is an important skill that can save us a lot of time on Test Day.

STRONG ACIDS AND BASES

Strong acids and bases are those that completely dissociate (or nearly so) into their component ions in aqueous solution. For example, when sodium hydroxide, NaOH, is added to water, the ionic compound dissociates, for all intents and purposes, completely according to the net ionic equation:

NaOH (s) = Na⁺ (aq) + OH^- (aq)