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States of Matter: gaseous states, gas laws, ideal gas equation

 Introduction 

Matter is made up of atoms or molecules. The arrangement of these molecules determines the state of matter. There are three recognised state of matter : Solid, Liquid and Gas. Matter can change between states when the temperature or pressure is changed. State changes of matter are physical rather than chemical.

Difference between states of matter 

GASEOUS STATE 

Important properties of gases : 
( i ) Mass : Mass in gm = Moles × Molecular mass. 
(i i) Volume : Volume of the gas is the volume of container in which they are filled in. 
(iii) Temperature : Temperature of a gas is the measure of kinetic energy of gas.
                     Kinetic energy ∝ Temperature
( i v) Pressure : Pressure of gas is defined as the force exerted by the gas on the walls of its container. It is often assumed that pressure is isotropic, i.e. it is the same in all the three directions.
 Pressure = Force/ Area 

 P = Mg/ A = (v× d× g )/A  = (A ×h× d× g)/ A 
 P = hdg 
Where.... h = height of the mercury column supported by the barometer.   
d = density of mercury. 
g = acceleration due to gravity.

GAS LAW

The gas laws are a set of laws that describe the relationship between thermodynamic temperature (T), pressure (P) and volume (V) of gases.

Boyle's law

 It states "at a constant temperature (T), the pressure (P) of a given mass (or moles, n) of any gas varies inversely with the volume (V)". Pressure (P) and volume (V) of gases. 
i.e. P ∝1 /V (for given n and T)
 or, PV = K = constant
 For two or more gases at constant temperature 
P1V1 = P2V2 = ...... = K 
Also, density d∝ 1 /V 
Hence P∝ d 
 or p1/d1 = p2/d2= .............= k

Graphical representation :

Charles' law :

It states "at constant pressure, the volume of a given mass of a gas, increases or decreases by 1 273.15 th of its volume at 0ºC for every rise or fall of one degree in temperature".

V /Vo = 1 + 1 /273.15 t(at constant n and P)

or Vt = Vo [1+(t/273.15)]

or Vt = [Vo(273.15+ t)]/273.15

0ºC on the Celsius scale is equal to 273.15 K at the Kelvin or absolute scale. i.e. Tt (Temperature in Kelvin scale) = 273.15 + t 

Therefore, From the above equation we get Vt/Vo = Tt/To

or Vt/Tt =Vo /To

i.e. The volume of a given gas is proportional to the absolute temperature. 

V1/T1 =V2/T2 (at constant P)

Graphical representation :

Gay-Lussac's law : Dependence of Pressure on Temperature :

It states "at constant volume, the pressure of a given mass of a gas is directly proportional to the absolute temperature of the gas". 

P ∝ T or P = KT

or P1/T1 = P2/T2

Graphical representation :

The combined gas Law :

It states "for a fixed mass of gas, the volume is directly proportional to absolute temperature and inversely proportional to the pressure".

Boyle's Law, V∝ 1 /P (at constant n, T) 

Charle's Law, V ∝T (at constant n, P)

Therefore, V∝ T/P    or V = K (T/P)  or PV/T  = K  or P1V1/T1 = P2V2/T2

Volume coefficient (𝛂v) of a gas : 

The ratio of increase in volume of a gas at constant pressure per degree rise of temperature to its volume at 0ºC is the volume coefficient of the gas.

𝛂v = (Vt- Vo)/(Vo x t)      or    Vt = Vo(1 +𝛂vt )

For all gases, 𝛂v  = 1/ 273

Pressure coefficient (𝛂p) of a gas : 

The ratio of increase in pressure of the gas at constant volume per degree rise of temperature to its pressure at 0ºC is the pressure coefficient of the gas.

𝛂p = (Pt -Po) /Po x t                 or Pt =Po(1 + 𝛂pt)

For all gases , 𝛂p = 1/273 

Avogadro's Law :

 It states "equal volumes of any two gases at the same temperature and pressure contain the same number of molecules".

 V 𝛂 n (At constant P and T )

or V1/n1 = V2/n2

THE IDEAL GAS EQUATION 

An ideal gas is defined to be a system in which there are no inter molecular/interatomic forces. Such a system can only exist as a gas. Any real system will approach ideal gas behaviour in the limit that the pressure is extremely low and the temperature is high enough to overcome attractive intermolecular forces. An ideal gas is a gas to go which the laws of Boyle and Charles are strictly applicable under all conditions of temperatures and pressures. 

From Boyle's law we get, V 𝛂 1/P (at constant n and T)

 From charles law we get, V 𝛂T (at constant n and P) 

From Avogadro's law we get, V 𝛂 n (at constant T and P)

 Combining the above three equations we get 

V 𝛂 nT/ P or V = R (nT/ P) [Where R = ideal gas constant]

 or PV = nRT 

Ideal gas equation is a relation between four variables and it describes the state of any gas. For this reason, it is also called Equation of State.

Dimension of R :

Physical significance of R : 

The dimension  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 1K. 

Units of R 

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