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__Gaseous State :
__

__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:
__

__Boyle’s Law:__

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**.Volume of a Given Quantity of Gas in Different Pressure, At Constant Temperature |

####
__Mathematical Representation of Boyle's Low:__

__Mathematical Representation of Boyle's Low:__

**V ∝ 1/P**

or,

**P**V = 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,

**P₁V₁ = P₂V₂**

Where,

####

**V₁**and**V₂**are the volume at**P₁**and**P₂**are the pressure respectively.####
__Graphical Representation :
__

__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.

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.

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: __

__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**

**₂**=

**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:
__

__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 :
__

__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:
__

__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

**states as,**__Charl’s Law__
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.__Graphical Representation :__

Isobars (P₁and P₂) |

###
**Absolute Temperature or Absolute Zero:**

**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

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**
∴ (

or,

or,

**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 :
__

__Gay – Lussac’s Law :__

**Pressure**of a given

**mass**of a gas at constant

**volume**is directly proportional to its

**Kelvin temperature.**

That is,

**P**∝**T**at constant Volume.###
__Combination of Boyle’s and Charl’s Law : __

__Combination of Boyle’s and Charl’s Law :__

**V**∝

**1**/

**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,

**V**∝**T**/**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.
(

**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__