Why do we consider some quantities 'dimensionless'?

Question

It seems strange to define some quantities such as countable objects or the unit mol to be 'dimensionless'. For me, It leads to many sources of confusion, do we consider a dimensional quantity with respect to a dimensionless quantity to have the dimension of the numerator?.

I understand the idea of a ratio of dimensional quantities as being dimensionless. But, for example, do we consider mass/moles to have dimension of Mass? I really am struggling with this idea, and why we would arbitrarily decide certain quantities to have clear dimensions which can be followed and other quantities which cannot have this?

I've tried to consider this with the idea of a dimension $N$ which I've seen used for moles but it seems to contradict the idea that unit times pure number equals a quantity with the same unit.

Answer 1

Not all units are fundamental. In fact, one can argue that only two fundamental units are needed (see arXiv: 0711.4276 [physics.class-ph], for example). However, this does not mean that using only two units is convenient. More often than not, it is easier to introduce redundant units in order to make numbers easier to read, to exploit dimensional analysis better, and so on.

Take, for example, Thermodynamics. A fundamental constant in Thermodynamics is Boltzmann's constant, $k_B = 1.380649 × 10^{-23} \ \text{J}\text {K}^{-1}$. Notice this constant has dimensions of energy per temperature. Notice as well that every single occurrence of temperature in a formula in Thermodynamics is of the form $k_B T$ (sometimes $k_B$ is hidden inside a different constant, such as the ideal gas constant). The reason is that $k_B$ is merely a conversion factor between Joules and Kelvin: temperature is the average kinetic energy of the particles in the gas, so we can measure it in units of energy, but it is often more convenient to say the temperature is $1$ K instead of $k_B = 1.380649 × 10^{-23} \text{J}$, simply because it is more convenient to read.

The ideal gas constant is introduced in a similar way. The ideal gas law can be written as $$PV = N k_B T,$$ where $P$ stands for pressure, $V$ volume, $T$ temperature, $N$ the number of molecules of the ideal gas, $k_B$ the Boltzmann constant. However, usual volumes of gases will contain $n \sim 10^{24}$ molecules, and working with large numbers can be messy. It gets easier if we define a new dimensionless unit such that $N_A = 6.02214076 × 10^{23} = 1 \text{mol}$ because now we can write $$PV = \frac{N}{N_A} N_A k_B T = \frac{N}{N_A} R T,$$ where the last equality defined $R = N_A k_B$ ($N_A$ is Avogadro's number, by the way). Now, we can express the quantity $n = \frac{N}{N_A}$ as a number of "moles" and deal with values similar to $n \sim 10$, which can be easier to handle (especially with a computer). As a bonus, we now have the constant $R = N_A k_B = 8.31446261815324 \ \text{J}\text {K}^{-1}\text{mol}^{-1}$, i.e., $R \approx 8.31 \ \text{J}\text {K}^{-1}\text{mol}^{-1}$, which is also easier to handle.

The unit mole is not dimensional, nor fundamental. But it is convenient. In the same way we usually measure ambient temperature in Celsius or Fahrenheit instead of Joules, and in the same way we often ask for a dozen of bananas instead of saying twelve.

Answer 2

Fundamental scales such as length, mass, and time have base units in a designated unit system. Examples are SI m, kg, and s or AE ft, lbm, and s. Counting numbers are always unitless (dimensionless). Mass or moles are not dimensionless. They designate the units chosen to represent the amount of the system or substance or species.

For clarity, terms that designate amounts should be written to carry units that distinguish the substance. An example is a reaction rate, typically given simply as mols/s. The less ambiguous expression is moles substance per time. For example, we can better appreciate the equivalence in stating reaction rates for ammonia synthesis when they are expressed explicitly in one of these three ways.

+2 mols NH$_3$/s = -1 mol N$_2$/s = -3 mols H$_2$/s

Another example is the Boltzmann constant $k_B$. It is typically expressed as eV or J/K. A less ambiguous form would be eV/particle or J/K particle. This allows us to appreciate the translation to the gas law constant $R$ (J/mol K) using Avagadro's number $N_o$ (particles/mol) as $R = N_o\ k_B$. The expanded expression becomes

J/mol K = (particles/mol)(J/K particle)

A final example is found in the habit that eliminates specific designations for objects in units scaled by length. An example, we find a term in engineering that defines the exposed area of an object per volume of container $a$. The explicit units are m$^2$ packing area per m$^3$ container. The reduced units are 1/m (or m$^{-1}$).

Traditions often eliminate the exact species in mass or mole units, eliminate the unit "particle" (or per particle) on terms such as $k_B$ or $N_o$, or eliminate "object" references. Recognizing the missing expansions can be a vital part in systematic unit analysis.