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Ideal Gas Equation - (Concept)

Ideal Gas Equation
Ideal gas equation is an equation which is followed by the ideal gases. A gas that would obey Boyle's and Charles Law under all the conditions of temperature and pressure is called an ideal gas.
As discussed the behaviour of gases is described by certain laws as Avogadro's Law, Boyle's Law, Charles' Law.

\begin{array}{l}{\text { According to Avogadro's Law ; } \mathrm{V} \propto \mathrm{n}(\mathrm{P} \text { and } \mathrm{T} \text { constant) }} \\ {\text { According to Boyle's Law ; } \mathrm{V} \propto \frac{1}{P}(\mathrm{T} \text { and n constant) }} \\ {\text { According to Charles' Law; } \mathrm{V} \propto \mathrm{T}(\mathrm{P} \text { and n constant) }} \\ {\text { Combining the three laws; we get: }}\end{array}

\begin{array}{l}{V \propto \frac{n T}{P}} \\\\ {V=R \frac{n T}{P}}\end{array}

\begin{array}{l}{\text { 'R' is the proportionality constant. On rearranging the above equation we get: }} \\ {\text { PV = nRT }} \\ {\text { This is the ideal gas equation as it is obeyed by the hypothetical gases called ideal gases under all conditions }}\end{array}

Universal Gas Constant or Ideal Gas Constant
\begin{array}{l}{\text { Ror S: Molar gas constant or universal gas constant }} \\ {\text { Values of } \mathrm{R}=0.0821 \text { lit, atm, } \mathrm{K}^{-1}, \mathrm{mol}^{-1}} \\ {\qquad \begin{array}{l}{=8.314 \text { joule } \mathrm{K}^{-1} \mathrm{mol}^{-1}} \\ {=8.314 \times 10^{7} \operatorname{erg} \mathrm{K}^{-1} \mathrm{mol}^{-1}} \\ {=2 \mathrm{\, cal} \mathrm{\, K}^{-1} \mathrm{\, mol}^{-1}}\end{array}}\end{array}

\begin{array}{l}{\text {For a single molecule, gas constant is known as }} \\ {\text {Boltzmann constant }(\mathrm{k}) \text { . }} \\ {\mathrm{k}=\mathrm{R} \mathrm{N}_{0}} \\ {\: \: \: \, =1.38 \times 10^{-3} \mathrm{J} / \mathrm{deg-abs} / \mathrm{molecule}} \\ {\: \: \: \, =1.38 \times 10^{-16} \text { erg/deg-abs/molecule }}\end{array}

Combined gas law:
The Boyle's and Charles' Law can be combined to give a relationship between the three variables P, V and T. The initial temperature, pressure and volume of a gas are . With the change in either of the variables, all the three variables change to . Then we can write:


Combining the both above equation we get


The above relation is called the combined gas law.

Density and Molar Mass of a Gaseous Substance

\\\mathrm{Ideal \: gas\: equation\: is \: PV = nRT \quad \quad \quad.......(i)}\\\mathrm{\text {On rearranging the above equation, we get }}\\\\\mathrm{\frac{\mathrm{n}}{\mathrm{V}}=\frac{\mathrm{P}}{\mathrm{RT}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad\: \: \: .......(ii)}\\\\\mathrm{n\left(N_{0} . \text { of moles }\right)=\frac{\text { Given mass }(m)}{\text { Molar mass }(M)}\quad....(iii)}\\\\\text {Putting the value of 'n' from equation (iii) in equation (ii), we get: }\\\\\frac{\mathrm{m}}{\mathrm{MV}}=\frac{\mathrm{P}}{\mathrm{RT}}\quad \quad \quad \quad \quad \quad \quad \quad \quad\quad\quad\: \: \: \: .........(iv)\\\\\text {We know that density (d) is mass (m) per unit volume (V) }\\\\\mathrm{d}=\frac{\mathrm{m}}{\mathrm{V}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad .........(v)\\\\\text {Replacing } \frac{\mathrm{m}}{\mathrm{V}} \text { in eq. (iv) with d(density), then equation (iv) becomes:}\\\\

\frac{\mathrm{d}}{\mathrm{M}}=\frac{\mathrm{P}}{\mathrm{RT}}

\text { Rearanging the above equation, we get } \mathrm{M}=\frac{\mathrm{dRT}}{\mathrm{P}}

\text {The above equation gives the relation between the density and molar mass of a gaseous substance. }

Exam Chapter
MHT-CET States of Matter : Gases and liquids
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