Ideal Gas or Perfect Gas

1. Statement and Concept Overview

An Ideal Gas or Perfect Gas is a hypothetical gas that strictly obeys all gas laws (Boyle’s law, Charles’s law, and Gay-Lussac’s law) at all temperatures and pressures.
It is assumed that the molecules of an ideal gas are point masses having no intermolecular forces between them.

Real gases such as oxygen, hydrogen, and nitrogen behave nearly as ideal gases under low pressure and high temperature conditions.

2. Explanation and Mathematical Formulation

An ideal gas satisfies the perfect gas equation:

[
pV = nRT
]

where:

[p] → pressure of the gas
[V] → volume of the gas
[n] → number of moles of gas
[R] → universal gas constant
[T] → temperature of the gas (in Kelvin)

If the mass of gas is [m] and its molar mass is [M], then:

[
n = \dfrac{m}{M}
]

Substituting in the perfect gas equation:

[
pV = \dfrac{m}{M}RT
]

or,

[\dfrac{pV}{T} = \dfrac{R}{M}m] [= \text{constant for a given gas mass}]

This shows that for a fixed mass of gas, the ratio [\dfrac{pV}{T}] remains constant.

3. Dimensions and Units

Quantity	Symbol	Dimensions	SI Unit
Pressure	[p]	[M L^{-1} T^{-2}]	Pascal (Pa)
Volume	[V]	[L^{3}]	m³
Temperature	[T]	[Θ]	Kelvin (K)
Gas Constant	[R]	[M L^{2} T^{-2} mol^{-1} K^{-1}]	J mol⁻¹ K⁻¹

4. Key Features of an Ideal Gas

Molecules are point-sized particles with negligible volume.
No intermolecular forces exist between molecules.
Elastic collisions occur between molecules and the container walls.
The internal energy depends only on temperature, not on volume or pressure.
It follows the relation [pV = nRT] at all temperatures and pressures.

5. Important Formulas to Remember

Formula	Description
[pV = nRT]	Ideal Gas Equation
[R = \dfrac{pV}{nT}]	Universal Gas Constant
[p_1V_1/T_1 = p_2V_2/T_2]	Combined Gas Law
[n = \dfrac{m}{M}]	Relation between mass and moles
[pV = \dfrac{mRT}{M}]	Ideal gas equation in terms of mass

6. Conceptual Questions with Solutions

1. Does an ideal gas actually exist?

No, an ideal gas is a hypothetical concept. However, real gases such as oxygen or nitrogen behave nearly ideally at low pressures and high temperatures.

2. Why do real gases deviate from ideal behavior?

Real gases have finite molecular sizes and intermolecular forces, which are ignored in the ideal gas model.

3. Under what conditions does a real gas approach ideal behavior?

At **low pressure** and **high temperature**, intermolecular forces are minimal, and real gases behave ideally.

4. How is the number of moles related to mass and molar mass?

The relation is given by [n = \dfrac{m}{M}], where *m* is the mass and *M* is the molar mass of the gas.

5. What does the constant [R] represent in the ideal gas equation?

It is the **universal gas constant**, equal to [8.314 \, J \, mol^{-1} \, K^{-1}].

7. FAQ / Common Misconceptions

1. Does the ideal gas law apply to liquids or solids?

No, it only applies to gases. Liquids and solids have strong intermolecular forces, which the law doesn’t consider.

2. Is air an ideal gas?

Air is a **mixture of gases** that behaves approximately like an ideal gas under normal conditions.

3. Can pressure and temperature be independent in an ideal gas?

No, they are related by [pV = nRT]; a change in one affects the other if volume is constant.

4. Why is the internal energy of an ideal gas dependent only on temperature?

Because there are **no intermolecular forces**, only kinetic energy (which depends on temperature) contributes to internal energy.

5. Does [R] vary from gas to gas?

No, [R] is a **universal constant**; it remains the same for all ideal gases.

8. Practice Questions (with Step-by-Step Solutions)

Q1. A gas occupies 3 L at 2 atm and 300 K. Find its volume at 1 atm and 400 K.
Solution:
Using [\dfrac{p_1V_1}{T_1}] [= \dfrac{p_2V_2}{T_2}]
[V_2] [= \dfrac{p_1V_1T_2}{p_2T_1}] [= \dfrac{2 \times 3 \times 400}{1 \times 300}] [= 8 L]

Q2. Find the pressure exerted by 2 moles of an ideal gas at 300 K occupying a volume of 10 L.
[p = \dfrac{nRT}{V}] [= \dfrac{2 \times 8.314 \times 300}{10 \times 10^{-3}}] [= 498.84 , kPa]

Q3. A gas has 5 g of hydrogen ([M = 2]). If it occupies 50 L at 300 K, find its pressure.
[n] [= \dfrac{5}{2} = 2.5], [\quad] [p= \dfrac{nRT}{V}] [= \dfrac{2.5 \times 8.314 \times 300}{50 \times 10^{-3}}] [= 124.71 kPa]

Q4. Why does the ideal gas law fail at high pressures?
Because molecular volume and forces become significant, which are neglected in the ideal model.

Q5. What is the value of [R] in calories?
[R = 8.314 J mol^{-1} K^{-1}] [= 1.987 \, cal \,mol^{-1} \, K^{-1}]