# Appendix: Basic Mathematical Concepts

This appendix covers briefly the basic mathematical procedures that are important for carrying out applications in physical pharmacy. Refer to calculus and mathematical textbooks for a more in-depth discussion.

In any equation in physical pharmacy, two main components must be considered first: the dependent variable and the independent variable(s). Consider for instance the following equation for the calculation of the buffer capacity (see Chapter 4):

To identify the dependent variable, first look at what you are solving for. In the equation, (*β*) is the independent variable. However, *β* *depends* on *C* (the total molar concentration of the buffer), *K _{a}* the acid dissociation constant, and [H

_{3}O

^{+}] the hydronium ion concentration in the buffer solution. Therefore, three

*independent*variables can influence the value of

*β*. For a given buffer system, the value of

*K*is constant, and thus

_{a}*β*is dependent only on

*C*and [H

_{3}O

^{+}] in this case. A change in the value of

*C*and/or that of [H

_{3}O

^{+}] results in a change in the value of

*β*.

In any equation, two sides are clearly identified, usually separated by an equals sign. Whenever a number is added or subtracted from either side of the equation, the other side should be modified by the same value. For example, consider the following equation:

*Y* = 50 *X* + 5

Adding +10 to the left side of the equation requires adding +10 to the right side, so that the equation remains balanced. The same rule applies to dividing or multiplying by a number. Both sides of the equation must be divided or multiplied by the same number. However, remember that division by zero is not allowed.

The logarithmic function is a useful arithmetic transformation in physical pharmacy. Often pharmacists deal with variables that have small numeric values. For example, the acid dissociation constant for acetic acid is 1.74 × 10^{−5}. Converting this value to a logarithmic number results in −4.76; multiplying this number by −1 yields 4.76, which is the p*K _{a}* value (defined as the negative logarithm of

*K*) for acetic acid. The same principle applies to the concentration of hydronium ion in water ([H

_{a}_{3}O

^{+}]). Normally, the molar concentration of [H

_{3}O

^{+}] is a small number. Taking its log value and multiplying it by −1 results in the pH of the solution. Two types of logarithmic expressions exist: the logarithm of base 10 (log) and the natural logarithm (ln). The relationship between these two forms is: