Question

An ultra-high vacuum pump can reduce the pressure from \[{\text{1}}{\text{.0atm}}\] to \[1.0 \times {10^{{\text{ - 12}}}}{\text{mm}}\]of\[{\text{Hg}}\]. Calculate the number of air molecules in a litre at this pressure and $298{\text{K}}$. Assume ideal gas behaviour.

Answer 1

Hint: To solve this question, it is required to have knowledge about the ideal gas law. We shall calculate the moles of gas present in the given pressure, volume and temperature (formula given). Then, we shall calculate the number of molecules using Avogadro's number.
The formula used:
 \[{\text{PV}} = {\text{nRT}}\]
where P is pressure, V is volume, R is the universal gas constant, n is no. of moles and T is temperature

Complete step by step answer:
According to the ideal gas law:
\[{\text{PV}} = {\text{nRT}}\]
where P is pressure, V is volume, R is the universal gas constant, n is no. of moles and T is temperature
Here, it is assumed that:
The ideal gases are made up of molecules which are in constant motion in random directions.
The molecules of an ideal gas behave as rigid spheres.
All the collisions are elastic.
The temperature of the gas is directly proportional to the average kinetic energy of the molecules.
Pressure occurs due to the collision between the molecules with the walls of the container.
Substituting this change of pressure into the ideal gas equation with appropriate values and unit conversion:$ \Rightarrow \left( {1 - \dfrac{{1.0 \times {{10}^{ - 12}}}}{{760}}} \right) \times 1 = {\text{n}} \times 0.082 \times 298$
Solving this value we will have the value of the number of moles as:
 $n = \dfrac{1}{{24.7}}$
Now multiplying this with the Avogadro number to find the number of molecules:
$N = \dfrac{1}{{24.7}} \times 6.022 \times {10^{23}} = 24.3 \times {10^{21}}$

Therefore the number of molecules in this sample of gas is $24.3 \times {10^{21}}$.

Note:
Make sure you remember that at conditions of high temperature and lower pressure, a real gas behaves like an ideal gas, because the potential energy due to intermolecular attractive forces becomes less significant compared with the particles’ kinetic energy, and the size of the molecules becomes less significant compared to the space between them. The five gas laws are:
Boyle’s Law establishes a relationship between the pressure and the volume of a gas.
Charles’s Law establishes a relationship between the volume occupied by a gas and the absolute temperature.
Gay-Lussac’s Law establishes a relationship between the pressure exerted by a gas on the walls of its container and the absolute temperature associated with the gas.
Avogadro’s Law establishes a relationship between the volume occupied by a gas and the amount of gaseous substance.