Ideal Gas Equation – Derivation, Relation with Density

A theoretical gas made up of a collection of randomly moving point particles that only interact through elastic collisions is known as an ideal gas. We can change water’s solid, liquid, gaseous states by altering their temperature, pressure, and volume. The ideal gas equation is written as PV = nRT.

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The effect of pressure on the volume of a gas at constant pressure and the effect of temperature on the volume of gas at constant pressure is studied with the help of Boyle’s law and Charles’s law, respectively. Now, what is the effect of both pressure and temperature on the volume of the gas? To this question, you will find the answer in the Ideal Gas Equation article.

What is the Ideal Gas Equation?

The equation which gives the simultaneous effect of pressure and temperature on the volume of a gas is called the ideal gas equation. It is also known as combined gas law. The ideal gas equation is also defined as the equation which gives the simultaneous effect of pressure and temperature on the volume of a gas. 

The ideal gas equation is \({\rm{PV = nRT}}\)

The ideal gas equation can be derived directly by combining Boyle’s law, Charles’ law, and Avogadro’s law.

What is Boyles Law?

Boyles law states that “The volume of a given mass of a gas is inversely proportional to its pressure provided the temperature remains constant”.

Mathematically, Boyles law is expressed as: 

\({\rm{V\alpha }}\frac{{\rm{1}}}{{\rm{P}}}\) , at constant temperature and for a given mass of the gas.

Or \({\rm{V = k}}\frac{{\rm{1}}}{{\rm{P}}}\), where k is the constant of proportionality, \({\text{V}}\) is the volume, and \({\text{P}}\) is the pressure of the gas. The value of constant k depends upon the mass and the temperature of the gas.

By rearranging the equation, we get \({\rm{PV = k}}\).

This implies that the product of the pressure and volume of a given mass is constant at a constant temperature.

Which Law Relates to Volume and Pressure at Constant Temperature?

Charles law states that the volume of a given mass of gas increases or decreases by \(\frac{1}{{273}}\) of its volume at \({{\rm{0}}^{\rm{o}}}{\rm{C}}\) for every one-degree rise or fall in temperature provided pressure is kept constant.

It can also be stated as the volume of a given mass of gas is directly proportional to the absolute or Kelvin temperature at constant pressure.

It is given by the equation, \(\frac{{\rm{V}}}{{\rm{T}}}\) Constant, or \({\rm{V\alpha T}}\) at constant pressure.

What is Avogadro’s Law?

Avogadro’s Law states that equal volume of all the gases under similar conditions of temperature and pressure contain an equal number of molecules.

No. of moles \(\left( {\rm{n}} \right)\) \({\rm{\alpha }}\) No. of molecules \(\left( {\rm{N}} \right)\)

Or Volume of gas \(\left( {\rm{V}} \right){\rm{\alpha n}}\)

The 1 mole of gas contains Avogadro’s number of molecules which is equal to \(6.022 \times {10^{23}}\). 

Derivation of Ideal Gas Equation

According to Boyle’s law,  \({\rm{V\alpha }}\frac{{\rm{1}}}{{\rm{P}}}\) at constant \({\text{T}} –  –  –  –  > (1)\)

According to Charles’ law, \({\rm{V\alpha T}}\) at constant \({\text{P}} –  –  –  –  > (2)\)

According to Avogadro’s law, \({\rm{V\alpha n}}\) at constant \({\text{T}}\) and \({\text{P}} –  –  –  –  > (3)\)

Where \({\text{V}}\) is the volume, \({\text{T}}\) is the temperature, \({\text{P}}\) is pressure, and \({\text{n}}\) is the number of moles of the gas.

On combining equation \((1), (2),\) and \((3),\) we get 

\({\rm{V\alpha }}\frac{{\rm{1}}}{{\rm{P}}}{\rm{ \times T \times n}}\)

\({\rm{V = R \times }}\frac{{\rm{1}}}{{\rm{P}}}{\rm{ \times T \times n}}\)

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

Where \({\text{R}}\) is the molar gas constant or Universal Gas Constant and the gas which follows the ideal gas equation strictly is called an ideal gas.

What is Universal Gas Constant (R)?

The constant \({\text{R}}\) is the ideal gas equation \({\rm{PV = nRT}}\) is called the universal gas constant.

From the ideal gas equation, \({\rm{PV = nRT}}\)

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

\({\rm{R = }}\frac{{{\rm{ Pressure \times Volume }}}}{{{\rm{ Moles \times Temperature }}}}\)

But, pressure is force per unit area, therefore

\({\rm{R = }}\frac{{{\rm{ }}\frac{{{\rm{Force}}}}{{{\rm{Area}}}}{\rm{ \times Volume }}}}{{{\rm{ Moles \times Temperature }}}}\)

\({\rm{R = }}\frac{{\frac{{{\rm{ Force }}}}{{{{{\rm{( Length )}}}^{\rm{2}}}}}{\rm{ \times ( Length }}{{\rm{)}}^{\rm{3}}}}}{{{\rm{ Moles \times Temperature }}}}\)

\({\rm{R = }}\frac{{{\rm{ Force \times Length }}}}{{{\rm{ Moles \times Temperature }}}}\)

\({\rm{R = }}\frac{{{\rm{ Work }}}}{{{\rm{ Moles \times Temperature }}}}\)

Thus, \({\text{R}}\) represents work done per degree per mole of the gas. \({\text{R}}\) can be expressed in different units depending upon the units of work. The magnitude of the unit of \({\text{R}}\) depends upon the units in which pressure, volume, and temperature are expressed.

The values of \({\text{R}}\) in different units of measurement:

How to Express Ideal Gas Equation in Terms of Density?

From the ideal gas equation, \({\rm{PV = nRT}}\) \(————-> (1)\)

If m is the mass of the  gas in grams and \(M\) is the molecular mass of a gas, then

\({\rm{n = }}\frac{{\rm{m}}}{{\rm{M}}}\) \(———–> (2)\)

On substituting equation \((2)\) in equation \((1)\), we get

\({\rm{PV = }}\frac{{\rm{m}}}{{\rm{M}}}{\rm{RT}}\)

\({\rm{P = }}\frac{{\rm{m}}}{{\rm{V}}}\frac{{{\rm{RT}}}}{{\rm{M}}}\)

\({\rm{P = d}}\frac{{{\rm{RT}}}}{{\rm{M}}}\)

\({\rm{d = }}\frac{{{\rm{MP}}}}{{{\rm{RT}}}}\)

\({\rm{M = d}}\frac{{{\rm{RT}}}}{{\rm{P}}}\) \(——> (3)\)

This equation gives the relation between the molar mass and the density of the gas. From equation \((3),\) it may be concluded that under similar conditions of temperature and pressure, densities of different gases are all directly proportional to their molar masses.

What is the Formula of Combined Gas Law?

The combined gas equation is,

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

This is the most convenient form of the ideal gas equation for calculation when any five variables are given, and the 6th is to be calculated.

Equation \((1)\) implies that  \(\frac{{{\rm{PV}}}}{{\rm{T}}}{\rm{ = K}}\)\(——–(2)\)

The value of constant \({\text{K}}\) depends only upon the amount of gas taken. If \(n\) is the number of moles of gas taken, then it is found that,

\({\rm{K}}\,{\rm{\alpha }}\,{\rm{n}}\) or \({\rm{K = nR}}\)\(———(3)\)

On substituting equation \((3)\) in \((2)\), we get

\(\frac{{{\rm{PV}}}}{{\rm{T}}}{\rm{ = nR}}\) or \({\rm{PV = nRT}}\)

This is the most common form of the ideal gas equation.

Solved Questions

1. Calculate the temperature of \(4.0\) mole of a gas occupying \({\rm{5d}}{{\rm{m}}^{\rm{3}}}\) at \(3.32\) bar \({\text{R=0}}{\text{.083}}\) bar \({\rm{d}}{{\rm{m}}^{\rm{3}}}{{\rm{K}}^{{\rm{ – 1}}}}{\rm{mo}}{{\rm{l}}^{{\rm{ – 1}}}}\)

According to the ideal gas equation \({\rm{pV = nRT}}\)

\({\rm{T = }}\frac{{{\rm{PV}}}}{{{\rm{RT}}}}\)

\({\rm{T = }}\frac{{{\rm{3}}{\rm{.32 \times 5}}}}{{{\rm{4}}{\rm{.0 \times 0}}{\rm{.083}}}}{\rm{ = 50K}}\)

2. Calculate the volume occupied by \({\rm{8}}{\rm{.8\;g}}\) of \({\rm{C}}{{\rm{O}}_{\rm{2}}}\) at \({\rm{31}}{\rm{.}}{{\rm{1}}^{\rm{o}}}{\rm{C}}\) and \(1\) bar pressure. \({\text{R=0}}{\text{.083}}\) bar \({\rm{d}}{{\rm{m}}^{\rm{3}}}{{\rm{K}}^{{\rm{ – 1}}}}{\rm{mo}}{{\rm{l}}^{{\rm{ – 1}}}}\)

The Molecular mass of \({\rm{C}}{{\rm{O}}_{\rm{2}}}{\rm{,M = 12 \times 1 + 16 \times 2 = 44}}\)

The temperature in Kelvin, \({\rm{T = 31}}{\rm{.}}{{\rm{1}}^{\rm{^\circ }}}{\rm{C + 273 = 304}}{\rm{.1\;K}}\)

According to the ideal gas equation \({\rm{PV = nRT = }}\frac{{\rm{m}}}{{\rm{M}}}{\rm{RT}}\)

\({\rm{1 \times V = }}\frac{{{\rm{88}}}}{{{\rm{44}}}}{\rm{ \times 0}}{\rm{.083 \times 304}}{\rm{.1}}\)

\({\rm{V = 5}}{\rm{.05d}}{{\rm{m}}^{\rm{3}}}\)

Summary

In this article, you have understood the meaning and how to derive the Ideal Gas Equation and how to relate the ideal gas equation to density. This idea will help in understanding ideal and non-ideal gas.

FAQs

Q.1. What is the ideal gas equation?
Ans: The equation which gives the simultaneous effect of pressure and temperature on the volume of a gas is called the ideal gas equation.

The ideal gas equation is \({\rm{pV = nRT}}\)

Q.2. How to derive the ideal gas equation?
Ans: According to Boyle’s law,  \({\rm{V\alpha }}\frac{{\rm{1}}}{{\rm{P}}}\) at constant \({\text{T}} – – – – > (1)\)

According to Charles’ law, \({\rm{V\alpha }}\,{\rm{T}}\) at constant \({\text{P}} –  –  –  –  > (2)\)

According to Avogadro’s law, \({\rm{V\alpha }}\,{\rm{n}}\) at constant \({\text{T}}\) and \({\text{P}} –  –  –  –  > (3)\)

Where \({\text{V}}\) is the volume, \({\text{V}}\) is the temperature, \({\text{P}}\) is pressure and \(n\) is the number of moles of the gas.

On combining equation \((1), (2)\), and \((3)\), we get 

\({\rm{V\alpha }}\frac{{\rm{1}}}{{\rm{P}}}{\rm{ \times T \times n}}\)

\({\rm{V = R \times }}\frac{{\rm{1}}}{{\rm{P}}}{\rm{ \times T \times n}}\)

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

Q.3. What is \({\text{R}}\) in \({\rm{PV = nRT}}\)?
Ans: In the ideal gas equation, \({\rm{PV = nRT}}\), \({\text{R}}\) is the universal gas constant. The value of \({\text{R}}\) varies according to the units, and the \({\text{SI}}\) unit value of \({\text{R}}\) is \(8.314\;{\rm{J}}\;{{\rm{K}}^{ – 1}}\;{\rm{mo}}{{\rm{l}}^{ – 1}}.\)

Q.4. What are the units for the ideal equation \({\rm{pV = nRT}}\)?
Ans: In \({\rm{pV = nRT}}\), units are chosen based on the value of \({\text{R}}\). The volume is measured in liters, and temperature is measured in Kelvin.

Q.5. What is the relationship between the density of a gas and the ideal gas equation?
Ans: According to the ideal gas equation, \({\rm{pV = nRT}}\) 
\({\rm{PV = }}\frac{{\rm{m}}}{{\rm{M}}}{\rm{RT}}\) (since  \({\rm{n = }}\frac{{\rm{m}}}{{\rm{M}}}\))
\({\rm{P = }}\frac{{\rm{m}}}{{\rm{V}}}\frac{{{\rm{RT}}}}{{\rm{M}}}\)
\({\rm{P = d}}\frac{{{\rm{RT}}}}{{\rm{M}}}\) or d=MPRT