##
__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.

According to,**Boyle's(1662**),**Charles's(1787)**and**Avogadro laws**gives the birth of an equation of state for Ideal Gas.__Boyle's Low__

**V**∝

**1/P**When

**n**and

**T**are constant for a gas.

__Charl's Low__

**V**∝

**T**When

**n**and

**P**are constant for a gas.

__Avogadro's Low__

**V**∝

**n**When

**P**and

**T**are constant for a gas.

**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 :**** **

**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__Value of R in C.G.S. and S.I. system:__

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**

**W**) = Force(

**F**) × Displacement(

**d**), So, erg = dyne cm².

Thus,

We Know That, 1 J = 10⁷ erg, thus the vale of R in S.I. Unit = **R = 8.314 × 10⁷ erg mol⁻¹ K⁻¹****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⁻¹**≃**2 Cal mol⁻¹ K⁻¹**###
__Physical Significance of Gas Constant R:__

__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.

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.

(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,__Determination of Molar mass from Ideal Gas Equation:__

**PV = nRT**

or,

Where g = weight of the gas in gm and M = Molar mass of the gas.**PV = (g/M)RT**
Again,

We know that, **P = ( g/V) (RT/M)****= Weight (g)/Volume (V).**

__Density (d)__
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).

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 ?

####
__Determination of Number of Molecule Present in Ideal Gas From Ideal Gas Equation:__

__Determination of Number of Molecule Present in Ideal Gas From Ideal Gas Equation:__

**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.