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Sep 7, 2018

Gas Law's

Gaseous State :

If the thermal energy is much greater than the forces of attraction, then we have the matter in the gaseous state. Molecule in the gaseous state move with very large speeds and the forces of attraction among them are not sufficient to bind the molecules in one place, with the results that the molecules moved practically independent of one another. Because of this feature, gases are characterized by marked sensitivity of volume changes with the change of temperature and pressure. There exists no boundary surface and therefore gases tend to fill completely any available space i.e. they do not possess a fixed volume. 

Boyle’s Law:

At constant temperature, the volume of a definite mass of a gas is inversely proportional to its pressure.
That is, the volume of a given quantity of gas, at a constant temperature decrees with the increasing of pressure and increases with the decreasing pressure.
Let, a Cylinder contain 10 ml of gas, at constant temperature and 1 atm pressure. if the pressure increases to 2 atm then the volume also decrees to 5 ml.
What is the Boyle’s Law?
Volume of a Given Quantity of Gas in Different Pressure, At Constant Temperature

Mathematical Representation of Boyle's Low:

V ∝ 1/P
or, PV = K

Where, K is a constant whose value depends upon the,
a. Nature of the Gas.
b. Mass of the Gas.
For a given mass of a gas at constant temperature,
P₁V₁P₂V₂
Where, V₁ and V₂ are the volume at P₁ and P₂ are the pressure respectively.

Graphical Representation :

The relation between pressure and volume can be represented by an arm of rectangular hyperbola given below. As the value of the constant in equation will change with temperature, there will be a separate curve for each fixed temperature. These curves plotted at different fixed temperature are called isotherms.

What is the Boyle's Low?
Graphical Representation of Boyle's Low

At Constant temperature a given mass of gas the product of Pressure and Volume is always same. if the product of pressure and volume represents in Y axis and Pressure represents X axis a straight line curve is obtained with parallel to X axis.
What is the Boyle's Low?
PV vs P Graph at Constant Temperature

This Graph is shows that, at constant temperature the product of pressure and volume is does not depends on its pressure.

Relation between Pressure and density of a gas: 

At constant temperature a definite mass of gas has Pressure P₁ at Volume V₁ and Pressure P₂ at Volume V₂.
According to Boyle’s Law 
P₁ V₁ P₂ V₂
or, P₁/P₂ = V₂/V₁
Again, Let the mass of the Gas = M and Density D₁ at Pressure P₁ and the Density D₂ at Pressure P₂
Thus, D₁ = M/V₁ and D₂ = M/V₂
or, V₁ = M/D₁ and V₂ = M/D₂
Again, P₁/P₂ = (M/D₂) × (D₁/M) = D₁/D₂
P₁/P  = D₁/D₂
P/D = Constant(K)
P = K × D
P D
Thus, at constant temperature density of a definite mass of a gas is Proportional to its Pressure.

Relation between Volume and Temperature of a Gas:

At constant pressure a definite mass of gas, with the increasing of temperature, volume also increases and with decreasing temperature, volume also decreases. That is, the volume of a given mass of gas at constant pressure is directly proportional to its Kelvin temperature.

Charl’s Law :

At constant pressure, each degree rise in temperature of a definite mass of a gas, the volume of the gas expands 1/273.5 of its volume at 0°C.

Mathematical Representation:

If V₀ is the volume of the gas at 0°C, then
1℃ rise of temperature the volume of the gas rise V₀/273.5 ml
∴  1°C temperature the volume of the gas (V₀+V₀/273) ml =V₀ (1 + 1/273)ml
At t°C temperature the volume of the gas,
Vt = V₀ (1+ t/273) ml
V₀ (273+t°C)/273  ml
It is convenient to use of the absolute temperature scale on which temperature is measured Kelvin(K). A reading on this scale is obtained by adding 273 to the Celsius scale value.
Temperature on Kelvin scale is T K = 273+t°C
Vt = (V₀ × T)/273 = (V₀/273) T
Since V₀, the volume of a gas at 0°C, has a constant value at a given pressure, the above relation expressed as,
Vt = K₂ T
or, V T 
Where K₂ is constant whose value depends on the,
Nature, mass and pressure of the gases.
According to the above relation Charl’s Law states as,
At constant pressure the volume of a given mass of gas is directly proportional to its Kelvin temperature.

Graphical Representation :

A typical variation of Volume of a gas with change in its kelvin temperature a straight line plot was obtained, Called isobars. The general term isobar, which means at constant pressure, is assigned to these plots.
What is the Charl's Low?
Isobars (P₁and P₂)


Absolute Temperature or Absolute Zero:

Since volume is directly proportional to its Kelvin temperature,
the volume of the gas is theoretically zero at zero Kelvin or -273°C.
However this is indeed hypothetical because all gases liquefies and then solidity before this low temperature reached.
In fact, no substance exists as a gas at the temperature near Kelvin zero, through the straight line plots can be extra plotted to zero volume.
The temperature corresponds to zero volume is -273°C
What is the Char's Low?
Representation of Absolute Zero temperature

Relation between temperature and Density of a given gas at constant Pressure:
From the Charl’s Law,
 V₁/V₂ = T₁/T₂
Again, the mass of the gas is M and Density D₁ and D₂ at the Volume V₁ and V₂ respectively.
Then, V₁ = M/D₁ and V₂ = M/D₂ 
∴   (M/D₁ )/(M/D₂ ) = T₁/T₂
or, D₂/D₁ = T₁/T₂
or, D ∝ 1/T
Thus at constant pressure, density of a given mass of gases is inversely proportional to its temperature.

Gay – Lussac’s Law :

Pressure of a given mass of a gas at constant volume is directly proportional to its Kelvin temperature.
That is, PT at constant Volume.
P₁/T₁P₂/T₂
 What is the Gay – Lussac’s Law?
Isotherms (V₁and V₂)

Combination of Boyle’s and Charl’s Law : 

V1/P When T Constant.
From Charl's Law, 
V T When P Constant.
When all the variables taken into account the variation rule states as,
Then, VT/P 
PV/T = K(Constant)
(P₁V₁)/T₁ = (P₂V₂)/T₂=Constant
PV = KT
Thus the product of the pressure and volume of a given mass of gas is proportional to its Kelvin temperature.
At 1 atm pressure and 300 K temperature, the volume of the gas is 2000 cm³, then calculate the volume of this gas at 600 K temperature and 2 atm pressure.

From the Combination of Boyle's and Charl's Law is,
(P₁V₁)/T₁ = (P₂V₂)/T₂
Here, P₁ = 1 atm; V₁ = 2000 cm³ and T₁ = 300K and P₂= 2 atm; V₂=❓ and T₂=600 K
1×2000/300 = 2×V₂/600
or, V₂ = (1×2000×600)/(300×2)  = 2000 cm³