Ideal Gas Equation: Derivation of Ideal Gas Law, R Constant Meaning, Absolute Temperature, Other Gas Law

The ideal gas equation describes the state of an ideal gas. It's a theoretical model where gas particles have negligible volume and no intermolecular forces except during elastic collisions. The equation is: $PV=nRT$ where $P$ is pressure $\left(\mathrm{P}\mathrm{a}\right),V$ is volume $\left({\mathrm{m}}^3\right),n$ is moles, $R=8.314\mathrm{J}/(\mathrm{m}\mathrm{o}\mathrm{l}\cdot \mathrm{K})$ is the universal gas constant, and $T$ is absolute temperature (K). Absolute temperature is measured on the Kelvin scale, where 0 K (absolute zero) is the point of zero molecular kinetic energy.

Why Should You Learn the Ideal Gas Equation and Absolute Temperature? 

The ideal gas equation and absolute temperature are ideal scenarios or theoretical models, discussed in the "Thermal Properties of Matter" chapter, that help you progress to advanced concepts in Physics. You can say it's a simplification that becomes a springboard to more realistic models (like van der Waals). When you are dealing with thermodynamic state variables and equation of state, you need to know this aspect. 

As this is an interrelated subject, you can expect it to be included in various entrance tests after your CBSE boards. For instance, it's in the IAT Physics syllabus where you can expect questions that combine all the gas laws (explained below) and concepts like their limitations. Have a look at the previous year's questions for the IISER Entrance exam.

Even while preparing for your upcoming NEET exam, you should look into these concepts, as earlier questions have tests the candidate's knowledge on interpreting graphs on the behaviour of gases in the ideal state, and comparing properties of gases, or similar.  

The ideal gas model typically assumes several key conditions. 

1. Molecules are point masses with negligible volume.
2. No intermolecular forces except during elastic collisions.
3. Molecules move randomly, obeying Newton's laws.
4. Collisions with container walls cause pressure.

Real gases approximate ideal behaviour at low pressure and high temperature. 

The ideal gas equation unifies empirical gas laws. 

• Boyle's Law: $PV=$ constant (at constant $T$ ).
• Charles' Law: ${\bar{V}}_T=$ constant (at constant $P$ ).
• Gay-Lussac's Law: ${\bar{P}}_T=$ constant (at constant $V$ ).

These laws are derived from $PV=nRT$ by holding variables constant.

Absolute temperature relates to molecular kinetic energy.

The average kinetic energy of a gas molecule is: $KE=\frac{3}{2}kT$ where ${}^k=\frac{R}{N_A}=1.38\times {10}^{-23}\mathrm{J}/\mathrm{K}$ is Boltzmann's constant, and $N_A$ is Avogadro's number. At absolute zero ( $T=0\mathrm{K}$ ), molecular motion theoretically stops.

The ideal gas equation is derived from kinetic theory $P=\frac{1}{3}\frac{mN}{V}⟨v^2⟩$ where $m$ is molecular mass, $N$ is number of molecules, and $⟨v^2⟩$ is mean square velocity. Equating kinetic energy to thermal energy yields: $PV=\frac{NkT}{n}=nRT$

The ideal gas equation and absolute temperature are applied in several applications. 

• Weather Forecasting: Balloon volume changes with temperature
• Engineering: Gas behaviour in engines and turbines.
• Chemistry: Calculating molar quantities in reactions.

Based on previous year questions for JEE Main, here are some numerical problems to look at.  

1. Problem: A gas occupies 2 L at ${27}^{\circ }\mathrm{C}$ and 1 atm . Find its volume at ${127}^{\circ }\mathrm{C}$ and 2 atm

$\frac{P_1{V}_1}{T_1}=\frac{P_2{V}_2}{T_2}⟹\frac{1\cdot 2}{300}=\frac{2\cdot V_2}{400}⟹V_2=1.33$

2. Problem: Calculate the temperature when 1 mol of gas expands from $0.5{\mathrm{m}}^3$ to $1{\mathrm{m}}^3$ at constant pressure of 10 Pa

$\begin{array}{c}\frac{V_1}{T_1}=\frac{V_2}{T_2}⟹T_2=T_1\frac{V_2}{V_1},\mathrm{}P{V}_1=nR{T}_1⟹T_1=\frac{{10}^5\cdot 0.5}{1\cdot 8.314}\approx 6015\mathrm{K}\\ T_2=6015\cdot \frac{1}{0.5}\approx 1203{}_{\mathrm{K}}\end{array}$

3. Problem: Find the number of moles in 11.2 L of gas at STP ( $0{}^{\circ }\mathrm{C},1\mathrm{a}\mathrm{t}\mathrm{m}$ )

$PV=nRT⟹n=\frac{1\cdot {10}^5\cdot 11.2\cdot {10}^{-3}}{8.314\cdot 273}\approx {0.5}_{\mathrm{m}\mathrm{o}\mathrm{l}}$