In day to day life, many times we need to measure the pressure of gases

We use instruments like barometer to measure them

We use a barometer to check the pressure of the atmosphere, or even the pressure in the tyre of our vehicle

The gas in this tyre is actually made up of many components

And when we are measuring the pressure, we are measuring the pressure exerted by all the components of the gas collectively

Like say we measure pressure for this system having 3 gases - Gas A, Gas B, Gas C

When we measure the total pressure of the system, we are actually finding the total pressure exerted by all the three gases within the system

In cases like these, the total pressure of the system is given to us by Dalton’s law

Dalton’s law is applicable in a mixture of non-reacting gases

Dalton’s law says the total pressure exerted by the mixture is the sum of the $partial pressures$ of the individual gases

To understand partial pressures, we will again consider a system consisting of three gases; A, B and C

Now let's say the same system with same temperature and volume was occupied with only Gas A

The pressure exerted by the Gas A on this system will be known as the partial pressure of Gas A. It is denoted as $P_{a}$

Similarly partial pressure of Gas B will pressure exerted if only Gas B was present in system at same conditions. It is denoted as $P_{b}$

And similarly we will get partial pressure of Gas C ( $P_{c}$ )

And according to Dalton’s law, the total pressure of the system, $P_{Total}$ would be given as $P_{Total}=P_{a}+P_{b}+P_{c}$

An important assumption we have taken is that component gases are non reactive

This is because if the components of the gas reacts within themselves, then the product can be a solid, liquid or gas. So we cannot get accurate pressure values

This law can also be expressed in terms of mole fraction of a gas

The mole fraction of a gas is defined as the ratio of the moles of the gas in the system to the total no of moles of gas present in the system

So Let us say that in our system, the three gases A, B & C gas have $n_{1},n_{2},n_{3}$ moles in the system respectively

Therefore the mole fraction of A, (denoted by $X_{a}$) in the system would be as-
$Xa=n_{a}+n_{b}+n_{c}n_{a} $

Similarly the mole fractions of Gas B and C respectively would be given as-
$X_{b}=n_{a}+n_{b}+n_{c}n_{b} $$X_{c}=n_{a}+n_{b}+n_{c}n_{c} $

Now we can substitute these mole fraction values in formula of partial pressure of gas A, B and C

From the Ideal Gas equation we know that partial pressure of Gas A

Dividing both the sides with total pressure i.e. $P_{Total}$

Taking $VRT $ common in both numerator and denominator