Ideal Gas Equation

The Ideal Gas Equation is a fundamental equation in physical chemistry that relates the pressure, volume, temperature, and amount of an ideal gas. An ideal gas is a hypothetical gas that perfectly follows the kinetic molecular theory of gases. The equation is an approximation of the behavior of many real gases under a wide variety of conditions, although it has its limitations.

The Equation

The Ideal Gas Equation is expressed as:

[ PV = nRT ]

where:

( P ) is the pressure of the gas,
( V ) is the volume of the gas,
( n ) is the number of moles of the gas,
( R ) is the ideal gas constant,
( T ) is the temperature of the gas in Kelvin.

Ideal Gas Constant (R)

The ideal gas constant is a proportionality factor that relates the energy scale in physics to the temperature scale. Its value depends on the units used for pressure and volume. Some common values for ( R ) are:

( 8.314 \, \text{J} \cdot \text{mol}^{-1} \cdot \text{K}^{-1} ) (SI units)
( 0.0821 \, \text{L} \cdot \text{atm} \cdot \text{mol}^{-1} \cdot \text{K}^{-1} )
( 62.364 \, \text{L} \cdot \text{Torr} \cdot \text{mol}^{-1} \cdot \text{K}^{-1} )

Derivation from Empirical Gas Laws

The Ideal Gas Equation is derived from a combination of empirical gas laws, which are:

Boyle's Law: ( PV = \text{constant} ) (at constant ( n ) and ( T ))
Charles's Law: ( V/T = \text{constant} ) (at constant ( n ) and ( P ))
Avogadro's Law: ( V/n = \text{constant} ) (at constant ( P ) and ( T ))

Combining these laws gives us the Ideal Gas Equation.

Assumptions of an Ideal Gas

For a gas to be considered ideal, it must adhere to the following assumptions:

Gas particles are point masses with no volume.
There are no intermolecular forces between the particles.
Collisions between gas particles and with the walls of the container are perfectly elastic.
The average kinetic energy of gas particles is directly proportional to the temperature in Kelvin.

Differences Between Ideal and Real Gases

Aspect	Ideal Gas	Real Gas
Volume of Particles	Negligible compared to the container	Not negligible
Intermolecular Forces	None	Exist, especially under high pressure
Behavior	Follows Ideal Gas Equation at all conditions	Deviates from Ideal Gas Equation under certain conditions
Conditions	Ideal at high temperature and low pressure	Deviates at low temperature and high pressure

Applications of the Ideal Gas Equation

The Ideal Gas Equation is used in various applications, such as:

Calculating the molar mass of a gas.
Determining the density of a gas.
Finding the yield of a reaction involving gases.

Example Problems

Example 1: Calculating the Volume of a Gas

Problem: Calculate the volume occupied by 2 moles of an ideal gas at a pressure of 1 atm and a temperature of 273 K.

Solution:

Given:

( n = 2 \, \text{moles} )
( P = 1 \, \text{atm} )
( T = 273 \, \text{K} )
( R = 0.0821 \, \text{L} \cdot \text{atm} \cdot \text{mol}^{-1} \cdot \text{K}^{-1} ) (since pressure is in atm)

Using the Ideal Gas Equation:

[ V = \frac{nRT}{P} = \frac{2 \cdot 0.0821 \cdot 273}{1} = 44.7 \, \text{L} ]

Example 2: Finding the Pressure of a Gas

Problem: What is the pressure exerted by 0.5 moles of an ideal gas in a 10 L container at 298 K?

Solution:

Given:

( n = 0.5 \, \text{moles} )
( V = 10 \, \text{L} )
( T = 298 \, \text{K} )
( R = 0.0821 \, \text{L} \cdot \text{atm} \cdot \text{mol}^{-1} \cdot \text{K}^{-1} )

Using the Ideal Gas Equation:

[ P = \frac{nRT}{V} = \frac{0.5 \cdot 0.0821 \cdot 298}{10} = 1.22 \, \text{atm} ]

The Ideal Gas Equation provides a simple way to relate the macroscopic properties of gases. However, it's important to remember that it is an approximation and that real gases may deviate from this behavior under certain conditions, such as high pressures and low temperatures.