**Perfect Gas Equation**

- Perfect gas equation is given by
**PV=μRT,**- WhereP,V are pressure, volume, T =absolute temperature,μ = number of moles and R =universal gas constant.
- R= k
_{B}N_{A}where k_{B}= Boltzmann constant and N_{A}= Avogadro’s Number

- This equation tells about the behaviour of gas at a particular situation.
- If a gas satisfies this equation then the gas is known as Perfect gas or an ideal gas.

__Different Forms of Perfect Gas Equation__

- PV=μRT (i)

- Where μ (no. of moles) = N/N
_{A}where N=no of molecules and N_{A}= Avogadro number(no of molecules in 1 mole of gas).Orμ = M/M_{o}where M=mass of sample of gas and M_{o}= molar mass. - PV = (N/N
_{A})RT(putting μ=N/N_{A}in equation(i)) - By simplifying PV = Nk
_{B}T - PV=Nk
_{B}T => P = (N/V) k_{B}T => P=nk_{B}T where n(number density) =N/V where N=number of molecules and V=volume. - Therefore we get
**PV=nk**_{B}T

- Substitute μ = M/M
_{o}in equation(i)

- PV=(M/M
_{o}) RT => P=(M/V)1/M_{o}RT where- ρ(mass density of the gas) = M/V

- Therefore
**P=ρRT/M**_{o}

**Ideal Gas**

- A gas that satisfies the perfectgas equation exactly at all pressures and temperatures.
- Ideal gas is atheoretical concept.
- No real gas is truly ideal.A gas which is ideal is known as real gas.
- Real gases approach the ideal gas behaviour for low pressures and high temperatures.

** Problem:-** ( Kinetic Theory Notes )

Estimate the total number of air molecules (inclusive of oxygen, nitrogen, water vapour

and other constituents) in a room of capacity 25.0 m^{3} at a temperature of 27 °C and 1 atm

pressure.

__Answer:-__

Volume of the room, V = 25.0 m^{3}

Temperature of the room, T = 27°C = 300 K

Pressure in the room, P = 1 atm = 1 × 1.013 × 10^{5} Pa

The ideal gas equation relating pressure (P), Volume (V), and absolute temperature (T)

can be written as:

PV = k_{B}NT

Where,

K_{B} is Boltzmann constant = 1.38 × 10^{–23} m^{2} kg s^{–2} K^{–1}

N is the number of air molecules in the room

N=PV/ K_{B}T

=1.013×10^{5}x25/1.38×10^{-23}x300

= 6.11 × 10^{26} molecules

Therefore, the total number of air molecules in the given room is 6.11 × 10^{26}

__Real gases deviation from ideal gas__

- Real gases approach the ideal gas behaviour for low pressures and high temperatures.
- Ideal gas equation PV=μRT, for 1 mole ,μ=1,PV=RT
- =>PV/RT=constant
- Graph should be a straight line(parallel to x-axis) for ideal gas.
- This means it has constant value at all temperature and all pressure.

But in case of real gases graph approach ideal gas behaviour at high temperature and low pressure.

- At high temperature and low pressure molecules are far apart. When temperature is increased the molecules will move randomly far from each other.
- As a result molecular interaction decreases the gas behaves as an ideal gas.
- The ideal behaviour comes into picture when the molecular present inside the gas don’t interact with each other.

Charles’s law:-Consider If P(Pressure) is constant, then