Priyam Study Centre

A Man would do nothing, if he waited until he could do it so well that no one would find fault with what he has done.

Sep 5, 2018

Ideal Gas Equation

Ideal Gas Equation:

A gas at equilibrium has definite value of Pressure(P), Volume(V), Temperature(T) and Composition(n).
These are called state Variables and are determined experimentally. The state of the gas can be defined by these variables. Boyle's(1662), Charles's(1787) and Avogadro laws gives the birth of an equation of state for Ideal Gas. 
According to,
Boyle's Low V 1/P When n and T are constant for a gas.
Charl's Low VT When n and P are constant for a gas.
Avogadro's Low V n When P and T are constant for a gas.
When all the variables are taken into account, The variation rule states that,
V ∝ (1/P) × T × n
or, V = R × (1/P) × T × n
or, PV = nRT
Where R is the Universal Gas Constant. This is called ideal gas equation of state for ideal gas. This equation is found to hold most satisfactory when P tense to zero. At ordinary temperature and pressure, the equation is found to deviated about 5%.

Value Of Universal Gas constant (R) at NTP : 

At NTP 1 mole gas at 1 atm Pressure occupied 22.4 lit of gas.
Thus, R = (PV)/(nT)
Putting the values above equation,We have, R = ( 1 atm × 22.4 lit)/(1 mol × 273 K)
= 0.082 lit atm mol⁻¹ K⁻¹

Value of R in C.G.S. and S.I. system:

1 atm = 76 cm Hg = 76 cm × 13.6 gm cm⁻² × 981 cm sec⁻² = 76 × 13.6 × 981 dyne cm⁻²
Thus, R = (76 × 13.6× 981 dyne cm⁻² ×22.4 × 10³ cm³)/(1 mol × 273 K)
= 8.314 × 10⁷ dyne cm² mol⁻¹ K⁻¹
Again, Work (W) = Force(F) × Displacement(d), So, erg = dyne cm².
Thus, R = 8.314 × 10⁷ erg mol⁻¹ K⁻¹
We Know That, 1 J = 10⁷ erg, thus the vale of R in S.I. Unit = 8.314 J mol⁻¹ K⁻¹ 
Again, 4.18 J = 1 Cal, hence, R = 8.314 / 4.18 Cal mol⁻¹ K⁻¹
= 1.987 Cal mol⁻¹ K⁻¹2 Cal mol⁻¹ K⁻¹

Physical Significance of Gas Constant R:

The universal gas constant R = PV/nT
Thus, it has the units of (Pressure × Volume) divided by (amount of gas × temperature). Now the dimension of pressure and volume are,
Pressure = (force/area) = (force/ length²) = force × length⁻² and Volume = length³.
Thus, R = (force × length⁻² × length³)/(amount of gas × Kelvin
= (force × length)/(amount of gas × kelvin) 
= (Work or Energy)/(amount of gas × kelvin).
Thus, the dimensions of R are energy per mole per kelvin and hence it represents the amount of work or energy) that can be obtained from one mole of a gas when its temperature is raised by one kelvin.
Determine the value of gas constant R when pressure is expressed in Torr and Volume in dm³.

 61.54 Torr dm³ mol⁻¹ K⁻¹
See the Solution to Problem 6(Properties of Gases). 

Derive the value of R when, (a) pressure is expressed in atom, and volume in cm³and (b) Pressure in dyne m⁻² and volume mm³.
(a) 82.05 atm cm³ mol⁻¹K⁻¹
(b) 8.314 × 10¹⁴ dyne m⁻² mm³ mol⁻¹ K⁻¹
See the Solution to Problem 7 (Properties of Gases).

Determination of Molar mass from Ideal Gas Equation:

The Ideal Gas Equation is,
PV = nRT 
or, PV = (g/M)RT
Where g = weight of the gas in gm and M = Molar mass of the gas.
Again, P = ( g/V) (RT/M)
We know that, Density (d) = Weight (g)/Volume (V).
Thus, P = dRT/M
Find the Molar mass of ammonia at 5 atm pressure and 30°C temperature (Density of ammonia = 3.42 gm lit⁻¹).
17 gm mol⁻¹
See the Solution to Problem 8 (Properties of Gases).
What is the molecular weight of a gas, 12.8 gms of which occupy 10 liters at a pressure of 750 mm and at 27°C ?
31.91 gm mol⁻¹

Determination of Number of Molecule Present in Ideal Gas From Ideal Gas Equation:

The Ideal gas equation for n mole gas is PV = nRT.
Again, PV = (N/N₀) RT, 
Where N = Number of molecules present in the gas and N₀ = Avogadro Number.
Thus, P = (N/V) × (R/N₀) × T
or, P = N′ k T
Where N′ = number of molecules present per unit Volume and k = Boltzmann Constant = R/N₀ = 1.38 × 10⁻¹⁶ erg molecule⁻¹ K⁻¹
Calculate the number of molecules present per ml of an ideal gas maintained at pressure of 7.6 × 10⁻³ mm of Hg at 0°C.
 2.69 × 10⁻¹⁴