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 (*H _{2}* ), 18 g of water vapor (

*H*), and 44 g of carbon dioxide (

_{2}O*CO*) under the same initial conditions, that is, they have: the same temperature

_{2}*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 *M _{r}* and the values of the atomic mass unit, which is equal to 1.67

^{.}10

^{-27}kg:

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*. Let’s calculate how many atoms are contained in 12 g of the carbon isotope

^{12}C*:*

^{12}CThis 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, *CO _{2}*) undergo a smaller change in velocity per unit of time than molecules with a smaller mass (for example,

*H*), and the

_{2}*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.