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), Ka the acid dissociation constant, and [H3O+] 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 Ka is constant, and thus β is dependent only on C and [H3O+] in this case. A change in the value of C and/or that of [H3O+] 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:
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 pKa value (defined as the negative logarithm of Ka) for acetic acid. The same principle applies to the concentration of hydronium ion in water ([H3O+]). Normally, the molar concentration of [H3O+] 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: