Partial Pressure - Dalton’s Law, Application, Formula, Units, FAQs

Dalton’s Law of Partial Pressure

Dalton’s Law of Partial Pressure states that the total pressure exerted by a mixture of non-reactive gases is equal to the sum of the partial pressures of all the individual gases present in the mixture.

The pressure exerted by an individual gas in a mixture is called its partial pressure.

$P_{\text {total }}=P_1+P_2+P_3+\cdots$

where:

• $P_{\text {total }}=$ total pressure of the gaseous mixture
• $P_1, P_2, P_3=$ partial pressures of individual gases

Partial pressure is defined as the pressure exerted by an individual gas present in a mixture of non-reactive gases. The partial pressure of a gas can be calculated using the formula:

$P_n=X_n \times P_{\text {total }}$

where:

• $P_n=$ partial pressure of the gas
• $X_n=$ mole fraction of the gas
• $P_{\text {total }}=$ total pressure of the mixture

This law is applicable only to mixtures of non-reactive gases and is based on the ideal gas assumption. It is widely used in atmospheric chemistry, respiration, and gas collection over water.

Application of Dalton’s Law of Partial Pressure of Gas

• The pressure of gases over the surface of the liquid can be calculated by using Dalton’s Law of partial pressure. The total pressure of a mixture of gas and water will be equal to atmospheric pressure if the water level inside and outside the vessel are equal.

• The gas present over water exerts combined pressure due to its vapor pressure and due to the pull of gravity. This gas contains water vapors. The pressure of dry gas can be calculated easily using the following equation.

When the level of water gets equal on the inside and outside-

$\begin{gathered}P_{\text {atm }}=P_{\text {total }} \\ P_{\text {total }}=P_{\text {gas }}+P_{\text {water }} \\ P_{\text {gas }}=P_{\text {atm }}-P_{\text {water }} \\ P_{\text {water }}=\text { Aqueous tension }\end{gathered}$

The vapor pressure of a gas is different at different temperatures; therefore, P_water is known.

Aqueous tension is the pressure exerted by water vapors. This Aqueous tension is the vapor pressure. Vapor pressure is the pressure exerted by the vapors of the liquid. Above the surface of a liquid, there always exist vapors of that liquid which exist in equilibrium with the water.

Derivation of Partial Pressure Formula

Consider a container containing two non-reactive gases, Gas A and Gas B. Let their partial pressures be $p_A$ and $p_B$, respectively.

According to the ideal gas equation:

$p_A=\frac{n_1 R T}{V}$

and

$p_B=\frac{n_2 R T}{V}$

where:

• $n_1$ and $n_2$ are the number of moles of gases A and B ,
• $R$ is the universal gas constant,
• $T$ is the temperature,
• $V$ is the volume of the container.

According to Dalton's Law, the total pressure of the gaseous mixture is equal to the sum of the partial pressures of the individual gases.

$P_{\text {total }}=p_A+p_B$

Substituting the values of $p_A$ and $p_B$ :

$P_{t o t a l}=\frac{n_1 R T}{V}+\frac{n_2 R T}{V}$

Taking $\frac{R T}{V}$ common:

$P_{\text {total }}=\frac{\left(n_1+n_2\right) R T}{V}$

Now, dividing the partial pressure of Gas A by the total pressure:

$\frac{p_A}{P_{\text {total }}}=\frac{n_1}{n_1+n_2}$

The ratio:

$X_1=\frac{n_1}{n_1+n_2}$

It is called the mole fraction of Gas A.

Therefore,

$p_A=X_1 P_{\text {total }}$

Similarly, for any gas in a gaseous mixture, the general equation is:

$p_n=X_n P_{\text {total }}$

where:

• $p_n=$ partial pressure of the $n^{\text {th }}$ gas
• $X_n=$ mole fraction of the $n^{\text {th }}$ gas
• $P_{\text {total }}=$ total pressure of the gaseous mixture.

Units of Partial Pressure

Since partial pressure is a type of pressure, it is measured in the same units as ordinary pressure.
Common units of partial pressure are:

• Atmosphere (atm)
• Pascal (Pa)
• Kilopascal (kPa)
• Millimeter of mercury (mmHg)
• Torr
• Bar

Relationship between common units:

$1 \mathrm{~atm}=760 \mathrm{mmHg}=760 \mathrm{Torr}=1.013 \times 10^5 \mathrm{~Pa}$
The Sl unit of partial pressure is Pascal (Pa).

Also Read

Some Solved Examples

Example 1. Equal weights of ethane and hydrogen are mixed in an empty container at 25∘C The fraction of the total pressure exerted by Hydrogen is -

a) 1:2

b) 1:1

c) 1:16

d) 15:16

Answer 1

let Xg of each gas is mixed.

$
\begin{gathered}
\text { mole of ethane }=\frac{x}{30} \\
\text { mole of hydrogen }=\frac{x}{2}
\end{gathered}
$
$
\begin{aligned}
\therefore \text { Mole fraction of hydrogen } & =\frac{x / 2}{x / 2+x / 30}=\frac{15}{16} \\
\Rightarrow \frac{\text { partial pressure of } H_2}{\text { Total pressure }} & =\text { mole fraction of Hydrogen } \\
= & 15: 16
\end{aligned}
$

Example 2.

At definite temperature, the tootal pressure of a gas mixture consisting of three gases A,B and C are 2,4,and 6 , then the increasingorder of their partial pressure is ___

a) $P_A=P_B=P_C$
b) P $_{\text {A }}>$ P $_{\mathrm{B}}>$ P $_{\mathrm{C}}$
c) P $_{\mathrm{A}}<$ P $_{\mathrm{B}}<$ P $_{\mathrm{C}}$
d) $P_A \neq P_B \neq P_C$

Answers (a)

The mole fractions of $A, B$, and $C$ in the mixture,

$
\begin{aligned}
& x_A=\frac{2}{12}=\frac{1}{6} \\
& x_B=\frac{4}{12}=\frac{1}{3} \\
& x_C=\frac{6}{12}=\frac{1}{2}
\end{aligned}
$
Therefore, the partial pressure of gas A,

$
\begin{aligned}
\underline{P_A} & =\frac{P}{6} \\
P_B & =\frac{P}{3} \\
\hline P_c & =\frac{P}{2} \\
\therefore \mathrm{P}_{\mathrm{A}}<\mathrm{P}_{\mathrm{B}}<\mathrm{C} &
\end{aligned}
$

Question 3: The partial pressure of a gas is given by:

$p_n=X_n P_{\text {total }}$
Here, $X_n$ represents:
(A) Mass fraction
(B) Mole fraction
(C) Volume of gas
(D) Density of gas

Solution:

The formula for partial pressure is:

$p_n=X_n P_{\text {total }}$

where:

• $p_n=$ partial pressure of the gas
• $P_{\text {total }}=$ total pressure of the gaseous mixture
• $X_n=$ mole fraction of the gas

The mole fraction of a gas is defined as:

$X_n=\frac{\text { moles of the gas }}{\text { total moles of all gases }}$

Thus, $X_n$ represents the mole fraction of the gas in the mixture.

Hence, the correct answer is option (B)

Practice more questions from the link given below

For more questions to practice, the following MCQs will help in the preparation for competitive examinations