Subsection 10.5.3 Law of Equipartition of Energy
According to Maxwell, total kinetic energy of a dynamical system consisting of large number of particles is equally distributed among its various degrees of freedom. The average kinetic energy of a monoatomic gas molecule is given by
\begin{equation*}
\frac{1}{2}m\overline{v^{2}} = \frac{3}{2}k_{B}T
\end{equation*}
\begin{equation*}
\text{or,}\quad \frac{1}{2}m\overline{v_{x}^{2}}=\frac{1}{2}m\overline{v_{y}^{2}} =\frac{1}{2}m\overline{v_{z}^{2}}
\end{equation*}
\begin{equation*}
=\frac{1}{3}\cdot \frac{1}{2}m\overline{v^{2}} =\frac{1}{3}\cdot \frac{3}{2}k_{B}T=\frac{1}{2}k_{B}T
\end{equation*}
Hence, the average kinetic energy per molecule in each degree of freedom is \(\frac{1}{2}k_{B}T.\)
Subsubsection 10.5.3.1 Avogadro’s Number, \(N_{A}\)
It is the number of molecules in one gram of hydrogen gas or the number of atoms in 12 grams of the carbon-12 isotope. The value of \(N_{A} = 6.022 \times 10^{23}\) molecules in one mole of carbon-12 atoms. The mole is a base unit in the SI system for an amount of substance. The symbol of mole is [mol]. The mole is defined as the amount of any substance (entity) that contains as many atoms as there are in 12 grams of \(C-12\) atom. At standard temperature and pressure (STP, T = 273 K, p = 1 atm) 1 mole of any substance occupies Avogadro’s number of molecules. At STP, volume of an ideal gas V=22.4 liter. If we weigh one mole of any substance its comes out to be the molecular weight of that substance in gram Subsubsection 10.5.3.2. One mole of water \((H_{2}O)\) weighs 18.015 g. One mole of ethyl alcohol, \((C_{2}H_{6}O)\text{,}\) weighs 46.069 g. If we weigh whether one mole of iron or one mole of sulfur they contain the same number of atoms equal to Avogadro’s number. This is an important aspect of the mole.
1
Molecular weight and molar weight are related terms that are often used interchangeably, but they have slightly different meanings. Molecular weight refers to the mass of a single molecule of a substance in atomic mass units (amu) or unified atomic mass units (u). Molar weight, also known as molar mass, refers to the mass of one mole of a substance in unit of grams per mole (g/mol). The molar mass of a compound is numerically equal to its molecular weight but has units of grams per mole. For example, the molar mass of water (\(H_2O\)) is approximately 18.015 g/mol, which is the same as its molecular weight (18.015 amu).
Subsubsection 10.5.3.2 Atomic Mass Unit, u
It is the mass of one hydrogen atom. It is normally a sum of number of protons and neutrons in a nucleus. The weight of 1 atom of \(_{6}^{12}C\) weighs 12.00 amu. A 12 g of carbon contains \(N_{A}\) atoms, each weighing exactly 12 amu. Therefore,
\begin{equation*}
N_{A}\,amu = 1 g
\end{equation*}
or,
\begin{equation*}
1 u = (1/N_{A}) \,g = 1.660 \times 10^{-24} \,g.
\end{equation*}
