Making sense of unit and dimension

Developing a qualitative appreciation of the scale of units and dimension helps to rule out what is clearly impossible to generate intuitive pictures. Here we go over the basic discussion on length, energy, and time to illustrate how this can be done for typical liquids.

density

The density of most organic liquids is of order $\rm 1\,g/cm^3$. This is roughly the density of atoms. What do we mean? The mass of atom is dominated by that of protons and neutrons, i.e., of order $\rm 10^{-27}\,kg = 10^{-24}\,g$. The volume is of order $\rm \unicode{x212B}^3 = 10^{-24}\,cm^3$. Therefore the density is about $\rm 1\,g/cm^3$. This density is held by hard-core repulsion, making most liquids and solides nearly incompressible. As a result, the thermaldynamic properties of the condensed phases are more sensitive to temperature than pressure normally.

mole, liquid volume

The atom size is of order $\unicode{x212B}$, so the volume of one mole of atoms is about $\rm 6{\times}10^{-7}m^3 \simeq 1\,cm^3$. The molar mass of organics is of order $\rm 10\,g/mol$. Combining this with the liqudi density, we say that the molar volume of typical liquid is of order $\rm 10\,cm^3$, about the size of a smaller piece of apple. For liquids, we may loosely say that, $\rm 1\,g \sim 1\,cm^3 \sim 1\,mole$.

gas volume

In contrast, the molar volume of ideal gas at standard condition is $\rm 22.4L$, which is about 1000 timesthat of molar volume of liquid. So the typical volume change factor upon liquifaction or vaporization is about 1000, which sets the scale of the gas compression.

energy scale

The average kinetic energy is $3 RT/2$. At room temperature with $T \rm = 300\,K$, this gives $\rm 3.7\,kJ/mol$. So the reference energy for liquid is of order $\rm kJ/mol$. The Trouton's rule (Frederick Trouton, Phil. Mag., 18:110, 54–57, 1884) says that at the melting point, the latent heat is about 6 times this value, i.e., $\rm 22\,kJ/mol$. On the other hand, the reference value for entropy is $3 R / 2 = \rm 12\,J/(mol\cdot K)$. The Lindeman criterion (Frederick Lindemann, Z. Phys., 11, 609, 1910) plays an analogous rule as the Trouton's rule for melting transition.

relaxation time

The molecular collision time is a fraction of pico-seconds, $\rm 10^{-13}$—$\rm 10^{-12} s$. Sufficient collisions relaxes the velocity-correlation, so the time scale for atoms to lose the memory of velocity is picosecond. When the average velocity and atom size is considered, the mass diffusivity of small organics can be estimated to be of order $\rm 1\, nm^2/ns$. In contrast, the diffusivity for momentum is slower by about 1000 times, which is set by the scale of viscosity.

viscosity

The viscosity is approximately the product between modulus and relaxation time. The modulus of liquids is about $\rm k_{\rm B}T/\unicode{x212B}^3 \sim 1\, GPa$. Multiplying it by the relaxation time, $\rm 1\,ps$, gives $\rm 10^{-3}\, Pa\cdot s$ for viscosity, a typical order of magnitude.