Ideal gas equation

We know that volume V of gas is

1. directly proportional to the quantity of gas i.e., number N of molecules contained in the gas. Mathematically,
   {tex}V \propto N{/tex} (if P and T are constant)
2. inversely proportional to pressure P. Mathematically,
   {tex}V \propto \frac{1}{P}{/tex} (if N and T are constant)
3. directly proportional to temperature T. Mathematically,
   {tex}V \propto T{/tex} (if N and P are constant)

Combining all these, we have {tex}V \propto \frac{N T}{P}{/tex}
{tex}\therefore{/tex} {tex}\frac{P V}{N T}{/tex} = constant = k_B = Boltzmann's constant 
Generally, N is written as N = {tex}\mu N_A{/tex}, where N_A is the number of molecules in one mole of gas and {tex}\mu{/tex} = quantity of gas in moles.
Hence, the above relation becomes
{tex}\frac{P V}{T}=N k_{B}{/tex} = {tex}\mu N_{A} k_{B}=\mu R{/tex}
Here R = {tex}\mu \cdot N_{A}{/tex} = gas constant for one mole of gas 
or {tex}P V=\mu R T{/tex} 
It is the ideal gas equation or equation of state for an ideal gas.

Ideal Gas Equation

• The gases following Boyle’s law, Charles’law and Avogadro law firmly are termed as anideal gas.
• From Boyle’s law we have  V ∝ 1/p (Constant T, n)
• From Charles’ law we have  V ∝ T (Constant p, n)
• From Avogadro’s law we have V ∝ n (Constant T, n)
• Combining them we get,  V ∝ nT/p
• or V = nRT/p 

pV = nRT ………………………… (IDEAL GAS EQUATION)

R =pV/nT

R= Gas Constant. It is same for all gases.