Ideal Gas Equation, Equation Of State, Density And Molar Mass Of Gaseous Substances, Dalton Law Of Partial Pressure, Aqueous Tension and Partial Pressure In Terms Of Mole Fraction

Ideal Gas Equation

   Combination of 3 laws (Boyles Law, Charles Law and Avogadro law) gives a single equation (PV=nRT) called as Ideal gas equation.

Mathematically,

According to Boyle’s Law; at constant T and n,

V 1/P ……….1

According to Charles Law; at constant P and n,

V T ……….2

According to Avogadro Law; at constant T and n,

V n  ……….3

From equation 1, 2 and 3; we get,

V nT/P ……….4

Or, V =R nT/P ……….5

Also, PV = nRT  …………..6

Then, R = PV/nT  ………..7

Where, R is a gas constant which is same for all gases and known as Universal Gas Constant and equation 6, PV = nRT is known as Ideal Gas Equation.

Equation Of State

                            Ideal gas equation is also known as equation of state because it gives relationship between 4 variables i.e. P, V, n and T. which describes state of any gas.

Let  if pressure, volume and temperature of fixed amount of ideal gas changes from P₁, V₁, T₁ to P₂, V₂, T₂  then,

P₁V₁/T₁ = nR …………..8

P₂V₂/T₂ = nR …………..9

So, from equation 8 & 9, we get

P₁V₁/T₁ = P₁V₁/T₁ ………..10

This above equation (eq. 10) is called Combined Gas Law.

Density And Molar Mass Of Gaseous Substances:

As per Ideal Gas Equation,

PV=nRT

Then, n/V = P/RT

On replacing n by m/M (as mole n= mass m/ molar mass M); we obtain,

m/MV=P/RT

On replacing m/V by density d; we obtain,

d/M=P/RT

Also, on rearrangement,

M = dRT/P

Where, M is molar mass, d is density, R is gas constant, T is temperature and P is pressure.

Dalton Law Of Partial Pressure:

According to Dalton law of partial pressure, “Total exerted pressure by mixture of all non-reactive gases is equal to the sum of partial pressure of all individual gases.”

At constant temperature T and Volume V

                             P_total = p₁ +p₂ +p₃…………..

Where, P_tolal = total exerted pressure of mixture of all gases.

p₁, p₂, p₃ etc. is pressure exerted by individual gases known as partial pressure.

Aqueous Tension: 

                           It is exerted by the saturated water vapors.

P_drygas = P_total – Aqueous Tension

Partial Pressure In Terms Of Mole Fraction: 

                                                    Let at T temperature, 3 gases of Volume V exert the partial pressure p1, p2, p3. Then as per ideal gas equation,

p₁=n₁RT/V

p₂=n₂RT/V

p₃=n₃RT/V

Where, n₁, n₂, n₃ are no. of moles.

 Also, according to Daltons law of partial pressure

P_total = p₁ +p₂ +p₃

Or, P_total = n₁RT/V + n₂RT/V + n₃RT/V = (n₁+n₂+n₃)RT/V

And, on dividing p₁ by P_Total , we obtain

P₁/ P_Total={n₁/(n₁+n₂+n₃)}{RTV/RTV}

P₁/ P_Total=n₁/(n₁+n₂+n₃) = n₁/n = x₁

Where, n= n₁+n₂+n₃ and x₁ is mole fraction of first gas.

So, p₁=x₁P_Total

Similarly, p₂=x₂P_Total

P₃=x₃P_Total

Then, general equation is written as-

P_i=x_iP_Total

Where, p_i is partial pressure of i^th gas.

x_i is mole fraction of i^th gas.