Ideal Gas: Overview, Equations, Units

Ideal Gas: Chemists worldwide dreamt of having an equation describing a gas molecule’s relation to its environmental parameters such as temperature or pressure. However, they encountered many difficulties to do so because other factors were affecting the state of the gas, such as intermolecular forces. While putting a blind eye to minor factors, they came up with an equation known as the ideal gas equation. But we already know that anything ideal does not exist. An ideal gas is hypothetical. Real gases approach ideal gas behaviour only when the following assumptions are made.

What is Ideal Gas?

An ideal gas is a hypothetical gas proposed to simplify calculations of a gas molecule with respect to its temperature and pressure. It does not have any real existence.

1. An ideal gas comprises molecules that can randomly move in all directions. The collision between the particles is considered to be perfectly elastic, i.e. no loss in kinetic energy of the particles due to collision.
2. In reality, no gas is ideal. All gases are real and tend to approach ideal gas behaviour when the density is low. This happens because when the density is low, the gas molecules are pretty far apart to interact with each other. Thus, the ideal gas concept helps us in studying real gases.

Derivation of the Ideal Gas Law

Gas is a collection of particles in which a significant distance exists between the molecules. It is due to this distance; colourless gases are invisible to the human eye. They are studied using four measurable parameters: pressure \((\text {P})\), volume \((\text {V})\), number of moles \((\text {n})\), and temperature \((\text {T})\). The ideal gas law is a mathematical expression that relates the pressure and volume of a gas to its moles and temperature. Moreover, the ideal gas law is a combination of several other gas laws that describe the behaviour of gases.

Boyle’s Law

According to Boyle’s law, if the temperature and number of moles of the gas are held constant, then the pressure of the gas is inversely proportional to its volume.

\({\rm{P}} \propto \frac{{\rm{1}}}{{\rm{V}}}\)

Charle’s Law

According to Charle’s law, if the pressure and number of moles of the gas are held constant, then the volume of the gas is directly proportional to its temperature.

\({\rm{V}} \propto {\rm{T}}\)

Gay-Lussac’s law

According to Gay-Lussac’s law, at a constant volume and number of moles, the pressure of an enclosed gas is directly proportional to its temperature.

\({\rm{P}} \propto {\rm{T}}\)

Avogadro’s Law

According to Avogadro’s law, the volume of a gas is directly proportional to the number of moles present.

\({\rm{V}} \propto {\rm{n}}\)

The law describes how equal volumes of two gases contain an equal number of molecules with the same temperature and pressure.

A gas that follows the above laws strictly is known as an ideal gas. Emile Clapeyron first proposed the ideal gas law in \(1834\), in which all these laws were combined to form the ideal gas equation.

What is Ideal Gas Equation?

The three laws (Charle’s law, Boyle’s law, Avogadro’s law) can be combined in a single equation to form an ideal gas equation. 

At constant \(\text {T}\) and \(\text {n}\); \({\rm{V}} \propto \frac{{\rm{1}}}{{\rm{P}}}\)……………. Boyle’s Law 

At constant \(\text {P}\) and \(\text {n}\); \({\rm{V}} \propto {\rm{T}}\)……………. Charles’ Law 

At constant \(\text {p}\) and \(\text {T}\); \({\rm{V}} \propto {\rm{n}}\)……………. Avogadro Law 

Thus,

\({\rm{V}} \propto \frac{{\rm{nT}}}{{\rm{P}}}\)

\({\rm{V}} = {\rm{R}}\frac{{{\rm{nT}}}}{{\rm{P}}}\)

Where,

\(\text {P}\) is the pressure of the ideal gas.

\(\text {V}\) is the volume of the ideal gas.

\(\text {n}\) is the amount of ideal gas measured in terms of moles.

\(\text {R}\) is the proportionality constant or the gas constant.

\(\text {T}\) is the temperature.

\({\rm{PV}} = {\rm{nRT}}\)…………..Eqn(1)

\( \Rightarrow {\rm{R}} = \frac{{{\rm{PV}}}}{{{\rm{nT}}}}\)

The Gas Constant, R

The constant of proportionality \(\text {R}\) is called the gas constant and is the same for all gases. It is also known as Universal Gas Constant. Equation \((1)\) is called the ideal gas equation. The value of \(\text {R}\) depends on the units in which \(\text {p}, \text {V}\) and \(\text {T}\) are measured.

The value of \(\text {R}\) for one mole of an ideal gas can be calculated under these conditions as follows :

\({\rm{R}} = \frac{{\left( {{{10}^5}\,{\rm{Pa}}} \right)\left( {22.71 \times {{10}^{ – 3}}{{\rm{m}}^3}} \right)}}{{\left( {1\,{\rm{mol}}} \right)\left( {273.15\,{\rm{K}}} \right)}}\)

\( = 8.314\,{\rm{Pa}}\,{{\rm{m}}^3}{{\rm{K}}^{ – 1}}{\rm{mo}}{{\rm{l}}^{ – 1}}\)

\( = 8.314\,{10^{ – 2}}{\rm{bar}}\,{\rm{L}}{{\rm{K}}^{ – 1}}{\rm{mo}}{{\rm{l}}^{ – 1}}\)

\( = 8.314\,{\rm{J}}{{\rm{K}}^{ – 1}}{\rm{mo}}{{\rm{l}}^{ – 1}}\)

Where the volume of one mole of an ideal gas under STP conditions (\(273.15 \,\text {K}\) and \(1\) bar pressure) is \(22.71{\mkern 1mu} {\rm{L}}{\mkern 1mu} {\rm{mo}}{{\rm{l}}^{ – 1}}\).

The ideal gas equation describes the state of any gas; therefore, it is also called the Equation of state.

Ideal Gas Equation Units

The Ideal gas equation consists of four parameters and one constant of proportionality. The units of these quantities are-

Combined Gas Law

For a fixed amount of gas, if temperature, volume and pressure vary from \({{\rm{T}}_1},\,{{\rm{V}}_1}\) and \({{\rm{P}}_1}\) to \({{\rm{T}}_2},\,{{\rm{V}}_2}\) and \({{\rm{P}}_2}\) then we can write-

\(\frac{{{{\rm{P}}_{\rm{1}}}{{\rm{V}}_{\rm{1}}}}}{{{{\rm{T}}_{\rm{1}}}}} = {\rm{nR}}\) and \(\frac{{{{\rm{P}}_{\rm{2}}}{{\rm{V}}_{\rm{2}}}}}{{{{\rm{T}}_{\rm{2}}}}} = {\rm{nR}}\)

\(\frac{{{{\rm{P}}_{\rm{1}}}{{\rm{V}}_{\rm{1}}}}}{{{{\rm{T}}_{\rm{1}}}}} = \frac{{{{\rm{P}}_{\rm{2}}}{{\rm{V}}_{\rm{2}}}}}{{{{\rm{T}}_{\rm{2}}}}}\)

The above Equation is also known as Combined gas law.

Density and Molar Mass of a Gaseous Substance 

The ideal gas equation can be rearranged as follows-

\(\frac{{\rm{n}}}{{\rm{V}}} = \frac{{\rm{P}}}{{{\rm{RT}}}}\)

Replacing \(\text {n}\) by \(\frac{{\rm{m}}}{{\rm{M}}}\), we get-

\(\frac{{\rm{m}}}{{{\rm{MV}}}} = \frac{{\rm{P}}}{{{\rm{RT}}}}\)

\(\frac{{\rm{d}}}{{\rm{M}}} = \frac{{\rm{P}}}{{{\rm{RT}}}}\) (where d is the density)

\({\rm{M}} = \frac{{{\rm{dRT}}}}{{\rm{P}}}\)

Assumptions of the Ideal Gas Law

The ideal gas law assumes that gases adhere to the following characteristics: 

(1) the collisions suffered by the molecules are elastic, and their motion is frictionless, i.e. there is no loss in the kinetic energy of the molecules suffering collision; 

(2) the volume that the gas occupies is more than the total volume of the individual molecules; 

(3) no intermolecular forces are acting between the molecules or their surroundings; 

(4) the molecules are in constant random motion, and the distance between two molecules is larger than the atomic size of the individual molecule. 

Considering all the above assumptions, an ideal gas will not change to a liquid state at room temperature.

Behaviour of Real Gases: Deviation from Ideal Gas Behaviour

Real gases do not obey Boyle’s law, Charles law and Avogadro law strictly under all conditions. This is because, in real gases, molecules interact with each other. At high pressures, molecules of gases are very close to each other and do not strike the walls of the container with full impact. This is because molecules are dragged back by other molecules due to attractive intermolecular forces. Thus, the pressure exerted by the real gas is lower than the pressure exerted by the ideal gas.

\({{\rm{P}}_{{\rm{ideal}}}} = {{\rm{P}}_{{\rm{real}}}}\left( {{\rm{observed}}\,{\rm{pressure}}} \right) + \frac{{{\rm{a}}{{\rm{n}}^2}}}{{{{\rm{V}}^2}}}\left( {{\rm{correction}}\,{\rm{term}}} \right)\)

Repulsive forces also become significant at high pressure. These forces cause the molecules to behave as small impenetrable spheres. The volume occupied by the molecules reduces from \(\text {V}\) to \((\text {V}–\text {nb})\), where \(\text {nb}\) is approximately the total volume occupied by the molecules themselves. Having considered the corrections for pressure and volume, the ideal gas equation can be written as-

\(\left( {{\rm{P}} + \frac{{{\rm{a}}{{\rm{n}}^2}}}{{{{\rm{V}}^2}}}} \right)\left( {{\rm{V}} – {\rm{nb}}} \right) = {\rm{nRT}}\)

The above Equation is known as the Van der Waals equation.

Where \(\text {n}\) is the number of moles of the gas.

\(\text {a}\) and \(\text {b}\) are van der waals constants

\(‘\text {a}’\) represents the magnitude of attractive intermolecular forces within the gas and is independent of temperature and pressure.

\(\text {b}\) is the volume correction term. 

Real gases exhibit ideal behaviour when pressure approaches zero and temperature and pressure are such that the intermolecular forces are practically negligible. 

Compressibility factor, Z

The deviation of real gas from ideal behaviour can be measured in terms of compressibility factor \(\text {Z}\). It is defined as the ratio of \(\text {PV}\) to \(\text {nRT}\). Mathematically-

\({\rm{Z}} = \frac{{{\rm{PV}}}}{{{\rm{nRT}}}}\)

For an ideal gas, \(\text {Z} = 1\), at all temperatures and pressures. This is because for an ideal gas, \(\text {PV} = \text {nRT}\). The graph of \(\text {Z}\) vs \(\text {p}\) will be a straight line parallel to the pressure axis.

For real gases, the value of \(\text {Z}\) deviates from unity. 

At very low pressures, all gases behave like an ideal gas and have \(\text {Z} \approx 1\). 

At high pressure, all the gases have \(\text {Z} > 1\) as they are difficult to compress. 

At intermediate pressures, most gases have \(\text {Z} < 1\).

Variation of compressibility factor for some gases

Thus, real gases exhibit ideal behaviour when the volume occupied by the gas is so large that the individual volume of the molecules can be neglected, i.e. at low pressure, the behaviour of the real gas becomes more ideal. 

Boyle Temperature or Boyle Point 

The temperature at which a real gas obeys ideal gas law over an appreciable pressure range is called Boyle temperature or Boyle point.

Above Boyle’s temperature, \(\text {Z} > 1\), real gases show positive deviations from ideality due to feeble intermolecular forces of attraction. 

Below Boyle temperature, with increasing pressure, the \(\text {Z}\) value first decreases and reaches a minimum value and then increases continuously. Hence, at low pressure and high-temperature real gases exhibit ideal behaviour. However, these conditions are different for different gases.

\({\rm{Z}} = \frac{{{\rm{P}}{{\rm{V}}_{{\rm{real}}}}}}{{{\rm{nRT}}}}\)

If the gas exhibits ideal behaviour, then-

\({{\rm{V}}_{{\rm{ideal}}}} = \frac{{{\rm{nRT}}}}{{\rm{P}}}\)

Substituting the value of \(\frac{{{\rm{nRT}}}}{{\rm{P}}}\) in \(\text {Z}\), we get-

\({\rm{Z = }}\frac{{{{\rm{V}}_{{\rm{real}}}}}}{{{{\rm{V}}_{{\rm{ideal}}}}}}\)

Hence, the compressibility factor is the ratio of actual molar volume to the molar volume of it if it were an ideal gas at that temperature and pressure. 

Limitations of Ideal Gas

Although the ideal gas equation helps solve numericals, it has many limitations. These are-

The ideal gas model fails at lower temperatures or higher pressures when intermolecular forces and molecular size become important. It also fails for heavy gases, such as many refrigerants and gases with strong intermolecular forces, notably water vapour. 

Summary

An ideal gas is hypothetical and does not have any real existence. Almost all gases are real and approach ideal gas behaviour only under certain specific conditions. This page explains the concept of an ideal gas and develops the ideal gas law and ideal gas equation. It also describes how a real gas deviates from ideal gas and the use of the compressibility factor.

Frequently Asked Questions (FAQs)

Q.1. Why is the ideal gas equation known as the Equation of state?
Ans: The ideal gas equation describes the state of any gas; therefore, it is also called the Equation of state.

Q.2. What are the two most ideal gases?
Ans: Hydrogen and helium are the two gases that behave closest to ideal gases. These gases have the weakest intermolecular attractions, and molar volume is close to that of ideal gases.

Q.3. Do ideal gases exist?
Ans: An ideal gas is a theoretical concept and has no real existence. However, many gases approach the ideal gas behaviour under certain conditions. 

Q.4. What is the compressibility factor of an ideal gas?
Ans: For an ideal gas, \({\rm{Z = 1}}\), at all temperatures and pressures. This is because for an ideal gas, \({\rm{PV = nRT}}\).