## What is an ideal gas law?

Ideal gas law states that the product of pressure and volume of a gas directly varies with the product of the number of moles, the universal gas constant, and the temperature of the gas.

The mathematical equation for the given law is:

$PV=nRT$

where,

- the pressure of the gas is represented by P,
- the volume of the gas is represented by V,
- the number of moles of the gas is represented by n,
- the universal gas constant is represented by R,
- the temperature of the gas is represented by T.

## Postulates of the kinetic theory of gases

An ideal gas is a supposition gas. Kinetic theory of gases proposed the rules for Ideal gas are as follows:

- Small particles or molecules make up the gas. The volume occupied by the gas particles is negligible to the volume of the container.
- The molecules in kinetic theory are assumed to be perfectly hard spheres and undergo collision amongst themselves and the walls of the container elastically. It means that total kinetic energy before the collision and after the collision is the same.
- The molecules are moving in the container continuously in different directions with different velocities.
- The distance between the molecules of a gas is large such that interactions between molecules are negligible.
- The gravitational force on the molecules of a gas is negligible.
- To simplify calculations, it is usually assumed that they have identical masses.

### Units of physical quantities in ideal gas equation

- When the value of R = 8.314 J K
^{-1}mol^{-1}, then the units of pressure should be in pascals (Pa) or kilopascal, volume should be in liters (L) and temperature should be in Kelvin (K). - When the universal gas constant is R = 0.0821 L.atm K
^{-1}mol^{-1}, then the units of pressure or gauge pressure should be in the atmosphere (atm), volume should be in liters (L) and temperature should be in Kelvin (K).

## Simple gas laws

There are four different simple gas laws. The combination of these four laws makes an ideal gas equation. They are:

- Boyle's law
- Charles' law
- Gay-Lussac law
- Avogadro's law

### Boyle's law

At a fixed temperature, the absolute pressure of the gas varies inversely with its volume.

Mathematically,

$P\propto \frac{1}{V}\phantom{\rule{0ex}{0ex}}PV=K\left(cons\mathrm{tan}t\right)$

At constant temperature, *V _{1}* is the Volume at pressure P

_{1}, then the volume of the same gas changes to V

_{2}when the new pressure of the gas is P

_{2}. According to Boyle's law, the relation between P

_{1},

*V*, and P

_{1}_{2, }V

_{2}is:

${P}_{1}.{V}_{1}={P}_{2}.{V}_{2}$

The graphical representation of Boyle's law:

This graph represents the variation of pressure with the volume of the gas at a constant temperature.

### Charles' law

At a fixed pressure, the volume of the gas varies directly with its temperature.

Mathematically,

$V\propto T\phantom{\rule{0ex}{0ex}}\frac{V}{T}=K\left(cons\mathrm{tan}t\right)$

At constant Pressure, V_{1} is the Volume of the gas when the temperature of the gas is T_{1} , the volume of the same gas changes to V_{2} when the new temperature of the gas is T_{2}. According to Charles' law, the relation between V_{1, }T_{1,} and V_{2, }T_{2} is:

$\frac{{V}_{1}}{{T}_{1}}=\frac{{V}_{2}}{{T}_{2}}$

The graphical representation of Charles' Law:

This graph represents the variation between volume and temperature at constant pressure.

### Gay-Lussac law

At a fixed volume, the absolute pressure of the gas varies directly with its temperature.

Mathematically,

$P\propto T\phantom{\rule{0ex}{0ex}}\frac{P}{T}=K\left(cons\mathrm{tan}t\right)$

At constant volume, P_{1} is the pressure of the gas when the temperature of the gas is T_{1} , the pressure of the same gas changes to P_{2} when the new temperature of the gas is T_{2}. According to Gay-Lussac's Law, the relation between P_{1, }T_{1,} and P_{2, }T_{2} is :

$\frac{{P}_{1}}{{T}_{1}}=\frac{{P}_{2}}{{T}_{2}}$

The graphical representation of Gay-Lussac law:

This graph represents the variation between the pressure, temperature of the gas at constant volume.

### Avogadro's law

The graphical representation of Avogadro's Law;

At fixed pressure and temperature, an equal volume of different gases have an equal number of molecules.

Mathematically,

V∝n

$\frac{V}{n}=cons\mathrm{tan}t$

At constant pressure and temperature, *V _{1}* is the volume of the gas and

*n*is the number of molecules of gas when the volume of gas changes to

_{1}*V*,then the number of molecules of the gas is

_{2}*n*. According to Avogadro's law, the relation between

_{2}*V*and

_{1, }n_{1,}*V*is:

_{2, }n_{2}$\frac{{V}_{1}}{{n}_{1}}=\frac{{V}_{2}}{{n}_{2}}$

This graph represents the variation of volume and the molecules of a gas at constant pressure and temperature. It tells that the volume of gas increases, then the number of molecules also increases.

## Importance of an ideal gas law

Ideal gas law relates four different variables in one equation. If three variables are known, then by substitution method, one can calculate the fourth variable.

For example:

Find the value of the temperature of the gas in kelvin, if at 2-atmosphere pressure of the gas, the volume of the gas is 5 liters and number of moles of the gas is 2 mol.

Substitute all the known values in the ideal gas Law, that is PV =*nRT* to get the value of an unknown variable.

$\begin{array}{rcl}PV& =& nRT\\ & & \\ 2atm\times 5L& =& 2mol\times 0.0821L.atm/K.mol\times T\\ T& =& \frac{2atm\times 5L}{2mol\times 0.0821L.atm/K.mol}\\ T& =& 60.9K\end{array}$

## Uses of ideal gas law in daily life

- Ideal gas law is applicable in airbags in vehicles. As the airbags deploy, the bag is filled with nitrogen gas. The airbag is inflated but not overfilled.
- Ideal gas law is used in the helium gas balloons to calculate the maximum mass the balloon can lift.

## Limitations of ideal gas law

Ideal gas law is only applicable to the ideal gases, but the fact is that in nature, there is no ideal gas in the atmosphere. All gases violate the rules of kinetic theory of ideal gas.

- Real gas has intermolecular attraction between the molecules of the gas.
- The nature of the collision of molecules is inelastic.
- The volume of gas molecules cannot be negligible. It occupies comparable volume to the volume of the container.

Gases can behave alike ideal gas only in the condition of low pressure and high temperature, not at high pressure and low temperature.

## Real Gas equation

Real gases or non-ideal gas do not obey an ideal gas law because, in real gases, molecules can interact with each other, and they experience attraction and repulsion between the molecules of the gas.

The equation of real gas is:

$\left(P+\frac{{n}^{2}a}{V}\right)\left(V-b\right)=nRT$

where a is the attraction factor and b is the repulsion factor.

## Context and Applications

Ideal gas law has several applications and is studied in courses like-

- Bachelors in Science (Physics, Chemistry)
- Masters in Science (Physics, Chemistry)

## Practice Problems

1. What is the unit of R if the gas constant R = 8.31?

- L.atm/K.mol
- J/K.mol
- J/K
- J/mol

Answer- b

Explanation: When R = 8.31, then the unit of R is J/K mol.

2. Calculate atmospheric pressure if 8 liter is the volume of the gas and number of moles is 6 mol and temperature of the gas is 230 K?

- 15 atm
- 20 atm
- 14.16 atm
- 25 atm

Answer- c

Explanation: Substitute all the known values in the ideal gas Law, that is PV = nRT , to get the value of an unknown variable.

$PV=nRT\phantom{\rule{0ex}{0ex}}P\times 8L=6mol\times 0.0821L.atm/K.mol\times 230K\phantom{\rule{0ex}{0ex}}P=\frac{6mol\times 0.0821L.atm/K.mol\times 230K}{8L}\phantom{\rule{0ex}{0ex}}P=14.16$

3. Who derived the ideal gas law?

- Benoit Paul Emile Clapeyron
- Robert Boyle
- Jacques Charles
- None of the above

Answer- a

Explanation: Benoit Paul Emile Clapeyron derives the ideal gas law.

4. If the gas constant R = 0.082, then what is the unit of R?

- L.atm/K.mol
- J/K.mol
- J/K
- J/mol

Answer- a

Explanation: When R = 0.082, then the unit of R is L.atm/K.mol.

5. Which term is not involved in an ideal gas equation?

- Time
- Volume
- Temperature
- Atmospheric Pressure

Answer-a

Explanation: Time is not involved in an ideal gas equation because the equation is PV = nRT where, T is not time, it is the temperature.

## Common Mistake

It is important to concentrate on the units of the variables. Sometimes questions gave different units, then convert it into given units and then, substitute into the formula. Firstly, convert it into SI units.

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