Chapter 9 Other useful infomation
9.1 Physical constants
| \(R\) | 8.31446261815324 J K-1 mol-1 |
| \(N_A\) | 6.02214076 × 1023 mol-1 |
| \(c\) | 299792458 m s-1 |
| \(e\) | 1.60217653(14) × 10-19 C |
The numbers in brackets refer to the error in the previous digits
9.3 Physical constants
| Element or isotope | g mol-1 |
|---|---|
| H | 1.00794 |
| O | 15.9994 |
9.4 Maclaurin expansions of common functions
9.4.1 Maclaurin expansion of exponential functions
The function \(e^{ax}\) may be expressed as a series of polynomials:
\[\begin{equation} e^{ax} = \frac{(ax)^0}{0!} + \frac{(ax)^1}{1!} + \frac{(ax)^2}{2!} + \frac{(ax)^3}{3!} + \dots + \frac{(ax)^\infty}{\infty!} \tag{9.1} \end{equation}\]
The factoral (\(!\)) notation used in equation (9.1) is a shorthand way of expressing a series of multiplication. It is each positive integer up to that value multiplied together. For example \(5! = 5 \times 4 \times 3\times 2 \times 1\). \(0!=1\), why is explained in this video.
Knowing the Maclaurin expansion for \(e^{ax}\) it becomes obvious why the differential of this function \(\frac{\textrm{d}}{\textrm{d}x}e^{ax}=ae^{ax}\). See section 4.1.6.
9.4.2 Maclaurin expansion of trig functions
\[\begin{equation} \sin(bx) = \frac{(bx)^1}{1!} - \frac{(bx)^3}{3!} + \frac{(bx)^5}{5!} - \frac{(bx)^7}{7!} + \frac{(bx)^9}{9!} - \dots \tag{9.2} \end{equation}\]
\[\begin{equation} \cos(cx) = \frac{(cx)^0}{0!} - \frac{(cx)^2}{2!} + \frac{(cx)^4}{4!} - \frac{(cx)^6}{6!} + \frac{(cx)^8}{8!} - \dots \tag{9.3} \end{equation}\]