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MVSAT Dec 2023

What is the formula of partial pressure?

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Hint :Formula can be derived using Dalton's law of partial pressure which is defined as, The total pressure of a mixture of non-reacting gases in a system is the sum of their partial pressure.

Complete Step By Step Answer:
Partial Pressure of a gas is defined when the gas is present in a gaseous mixture. It is the pressure that the gas could have exerted when it is allowed to occupy the entire volume of the gas mixture alone at the same temperature.
Let us assume there is a container of volume V with two gases in it, say nitrogen and oxygen. According to ideal gas equation,
 $ {{P}_{t}}V=\left( {{n}_{{{O}_{2}}}}+{{n}_{{{N}_{2}}}} \right)RT $ ---Equation $ \left( 1 \right) $
According to the definition of partial pressure, Partial pressure because of oxygen (if present alone in the container)
Using Ideal gas equation,
 $ {{P}_{{{O}_{2}}}}V={{n}_{{{O}_{2}}}}RT $ ---Equation $ \left( 2 \right) $
Similarly, Partial pressure because of nitrogen (if present alone in the container)
Using Ideal Gas Equation,
 $ {{P}_{{{N}_{2}}}}V={{n}_{{{N}_{2}}}}RT $ ---Equation $ \left( 3 \right) $
Adding equations $ 2 $ and $ 3 $ , we get
 $ \left( {{P}_{{{O}_{2}}}}+{{P}_{{{N}_{2}}}} \right)V=\left( {{n}_{{{O}_{2}}}}+{{n}_{{{N}_{2}}}} \right)RT $ ---Equation $ \left( 4 \right) $
Compare equation $ 4 $ with equation $ 1 $ , we get
 $ {{P}_{t}}={{P}_{{{O}_{2}}}}+{{P}_{{{N}_{2}}}} $
It is called dalton’s law of partial pressure.
Divide equation $ 2 $ by equation $ 1 $ , we get
 $ \frac{{{P}_{{{O}_{2}}}}}{{{P}_{t}}}=\left( \frac{{{n}_{{{O}_{2}}}}}{{{n}_{{{O}_{2}}}}+{{n}_{{{n}_{2}}}}} \right) $
 $ {{P}_{{{O}_{2}}}}={{X}_{{{O}_{2}}}}{{P}_{t}} $
 $ {{X}_{{{O}_{2}}}}= $ Mole fraction of oxygen $ =\left( \frac{{{n}_{{{O}_{2}}}}}{{{n}_{{{O}_{2}}}}+{{n}_{{{n}_{2}}}}} \right) $
Therefore, Partial pressure of oxygen $ ={{P}_{{{O}_{2}}}}={{X}_{{{O}_{2}}}}{{P}_{t}} $
Similarly, Partial pressure of nitrogen can be obtained by dividing equation $ 3 $ by $ 1 $ , we get
 $ {{P}_{{{N}_{2}}}}={{X}_{{{N}_{2}}}}{{P}_{t}} $

Note :
Through this derivation, we came to know dalton’s law of partial pressure and the partial pressure formula. Remember that Dalton's law is applicable to non-reacting gases only such as oxygen and nitrogen. If HCL and $ N{{H}_{3}} $ are present, then the reaction would take place as one is acid and the other is base.