## Concept “mole”

November 22nd, 2022

Why do we need in concept “mole”?

Let’s consider 3 vessels with a weightless piston sliding without friction, in which there are, respectively: 2 g of hydrogen (H2 ), 18 g of water vapor ( H2O), and 44 g of carbon dioxide (CO2 ) under the same initial conditions, that is, they have: the same temperature T, the same pressure p, the same volume V (Fig. 1).
Fig. 1. Various gases under the piston.

It was experimentally established that when the gas in each vessel is heated by 1 degree, the pistons rise to the same height, that is, the gas in each vessel performs the same work. Let’s find out the physical reasons for this experimental phenomenon.
Let’s find out how many molecules are in each of the vessels: for this, we will divide the mass of gas in each of the vessels by the mass of one molecule, presenting it as the product of the relative molecular mass Mr and the values of the atomic mass unit, which is equal to 1.67.10-27kg:

This means that the same number of molecules of an ideal gas (any, regardless of its chemical composition), are able to do the same work! That is, the work of an ideal gas is determined by the number of molecules.
That is why the number of molecules of an ideal gas is a standard value of molecular physics, for which a standard is set – 1 mole: 1 mole is the amount of substance that contains as many structural elements (atoms, molecules, ions) as there are atoms in 12 g of the carbon isotope 12C. Let’s calculate how many atoms are contained in 12 g of the carbon isotope 12C:

This number of molecules is called Avogadro’s constant and has a dimension (1/mole).
1 mole of an ideal gas when heated by one degree performs 8.31 J of work, which is called the universal gas constant R.

The reason for the constancy of R lies in the constancy of the work A performed by 1 mole of gas. The work of an ideal gas, for which statistical averaging is performed, is equal to the product of the total force with which N molecules act on the piston to the lifting height of the piston h:

The force, with which each of the molecules acts on the piston, is equal to the rate of change of momentum of the molecule over time during its interaction with the piston:

It follows from the last expression that the constancy of work is determined by the constancy of the change in momentum of the molecule during interactions with the piston in each of the vessels. Molecules with a larger mass (for example, CO2) undergo a smaller change in velocity per unit of time than molecules with a smaller mass (for example, H2), and the momentum change for molecules in each vessel per unit of time will be constant. (Fig.2 and video)

Fig.2. Velocities of the gases molecules.

## Maxwell distribution law

November 11th, 2016

Maxwell  distribution law of velocities of the molecules has the form:

$f(v)=(\frac{m_0}{2\pi kT})^\frac{3}{2}exp(-\frac{m_0 v^2)}{2 kT}) 4 \pi v^2$ (1)

## Boltzmann distribution law

November 10th, 2016

Let’s consider a gas that is in the gravity field (fig.1).

Fig. 1. Ideal gas in the gravity field.

## Rigid rotor

November 9th, 2016

Let’s consider the model of diatomic molecules in two material points $m_1$ and $m_2$, attached to the ends of a weightless rigid rod (Fig. 1).

Fig.1 Rigid rotator model

## Vibrational spectra of diatomic molecules

October 25th, 2016

Let’s perform the interpretation of the vibrational spectra of diatomic molecules  with the use of the quantum harmonic oscillator model (fig.1):

Fig.1. The harmonic oscillator: A – in a static state, B – in a position of a motion

## Entropy

October 25th, 2016

Let’s construct a heat engine that converts heat into mechanical work. The heat engine consists of the next main elements: the working body (gas under the piston), heater, and cooler (fig.1).

Fig. 1. The main elements of the heat engine.