# Constantly changing

## Problem

Physical constants can only be determined by experiment and can never be known exactly, even if in principle an exact value does exists. As a result, physical quantities are given as a probable range of values with an uncertainty registered in the last two digits, as follows:

$1.234\, 5678(32) \rightarrow 1.234\, 5678 \pm 0.000\, 0032$

$245.234\, 789\, 123(45) \rightarrow 245.234\, 789\, 123\pm 0.000\, 000\, 045$

The following table contains the best currently known measurements for various physical quantities:

Name | Value |

Avogadro constant | $6.022\, 141\, 79(30) \times 10 ^{23}$ mol $^{-1}$ |

Atomic mass constant | $1.660 \,538\, 782(83) \times 10^{-27}$ kg |

Electron mass | $9.109\, 382 \,15(45)\times 10^{-31}$ kg |

Proton-electron mass ratio | $1836.152\, 672\, 4718(80)$ |

Proton mass | $1.672\, 621\, 637(83) \times 10^{-27}$ kg |

Neutron mass | $1.674\, 927 \,211(84) \times 10{-27}$ kg |

Speed of light in vacuum | $299\, 792\, 458$ m s$^{-1}$ exactly |

National Institute of Standards and Technology Reference on Constants, Units and Uncertainty provides detailed information on the bounds of measurements of physical constants.

See http://physics.nist.gov/cuu/Constants/index.html for more details .

For a list of all constants, see http://physics.nist.gov/cuu/Constants/Table/allascii.txt

Interestingly, there is a strong element of statistics used to determine the probable values of constants. Key to this idea are the concepts of error and uncertainty in measurement. Cleverly designed experiments based on a strong understanding of statistics can be used to minimise this uncertainty.

To read about the essentials of expressing measurement uncertainty see http://physics.nist.gov/cuu/Uncertainty/index.html

Note that the speed of light given is a numerically exact quantity because the length of a metre has now been defined in terms of the speed of light!

## Student Solutions

The error $\Delta Z$ of the quantity $Z=\frac{A}{B}$ where $A$ and $B$ are independent satisfies $\left(\frac{\Delta Z}{Z}\right)^2 = \left(\frac{\Delta A}{A}\right)^2 + \left(\frac{\Delta B}{B}\right)^2$.

The error $\Delta Z$ of the quantity $Z=A+B$ where $A$ and $B$ are independent satisfies $(\Delta Z)^2 = (\Delta A)^2 + (\Delta B)^2$.

The error $\Delta Z$ of the quantity $Z=kA$ satisfies $(\Delta Z)^2 = (|k|\Delta A)^2 $.

$$\begin{align*}\left(\frac{\Delta r'}{r'}\right)^2 &= \left(\frac{8.3\times10^{-35}}{1.67...\times10^{-27}}\right)^2 + \left(\frac{4.5\times10^{-38}}{9.10...\times10^{-31}}\right)^2 \\&= 4.90\times^{-15}.\\\Rightarrow \Delta r' &= 1.29\times10^{-4}\end{align*}$$

Therefore, $r' = 1836.15267(13)$.

The proton/electron mass ratio, $r$, is $1836.152\, 672\, 4718(80)$. These values are consistent, as the given value for $r$ is within the error of $r'$. It appears the mass ratio is known to much greater accuracy than the individual masses.

The atomic mass of oxygen-16 is 15.99491461956(16)u, and the atomic mass of hydrogen-1 is 1.00782503207(10)u. The atomic mass of a water molecule is therefore $18.0104536837(26)u = 2.99072411(15)\times10^{-26}kg$. Therefore 1 mole of water weighs (mulitplying by Avogadro's constant) $1.80105647(13)\times10^{-2}kg$.