### Study properties of gases

If the thermal energy is much greater than the forces of attraction then we have the matter in a gaseous state.Gas molecules move very large speeds and the forces of attraction between them are not sufficient to bind the gas molecules at one place with the result the molecules move practically independent of one another.

There exist no boundary surface and therefore gas molecules tend to fill completely any available space, that is they do not possess fixed volume.

### Volume measurement or 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 decreases with the increase of pressure and increases with the decreasing pressure.

The cylinder contains 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.

#### Mathematical presentation of Boyle's law

The value of gas constant K depends on the nature of the gas and mass of the gas molecules.

For a given mass of a gas at a constant temperature,

P₁V₁ = P₂V₂

where V₁ and V₂ are the volumes at pressure P₁ and P₂ respectively.

P₁V₁ = P₂V₂

where V₁ and V₂ are the volumes at pressure P₁ and P₂ respectively.

#### Graphical presentation of Boyle's law

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

Graphical presentation of gases |

This graph shows that at a constant temperature the product of pressure and volume does not depend on its pressure.

#### Density measurement for gases

At constant temperature(T) a definite mass of gas has pressure P₁ at volume V₁ and pressure P₂ at volume V₂.Boyle’s law for gases,

P₁ V₁ = P₂ V₂

or, P₁/P₂ = V₂/V₁.

Let the mass of the gas = M and

density = D₁ at pressure P₁

density = D₂ at pressure P₂.

∴ D₁ = M/V₁ and D₂ = M/V₂

or, V₁ = M/D₁ and V₂ = M/D₂

P₁/P₂ = (M/D₂) × (D₁/M) = D₁/D₂

or, P₁/P₂ = D₁/D₂

or, P/D = constant

∴ P ∝ D

At constant temperature density of a definite mass of a gas is proportional to its pressure.P₁ V₁ = P₂ V₂

or, P₁/P₂ = V₂/V₁.

Let the mass of the gas = M and

density = D₁ at pressure P₁

density = D₂ at pressure P₂.

∴ D₁ = M/V₁ and D₂ = M/V₂

or, V₁ = M/D₁ and V₂ = M/D₂

P₁/P₂ = (M/D₂) × (D₁/M) = D₁/D₂

or, P₁/P₂ = D₁/D₂

or, P/D = constant

∴ P ∝ D

### Temperature measurement of gases

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

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 0⁰C.#### Mathematical presentation of Charles law

If V₀ is the volume of the gas at 0⁰C, then 1⁰C rise of temperature the volume of the gas rise V₀/273.5 ml.∴ 1⁰C temperature the volume of the gas, (V₀ + V₀/273) ml = V₀ (1 + 1/273) ml.

At t⁰C temperature the volume of the gas,

Vt = V₀ (1+ t/273) ml

= V₀ (273 + t°C)/273 ml

It is convenient to use the absolute temperature scale on which temperature is measured Kelvin(K). Reading on this scale is obtained by adding 273 to the Celsius scale value.At t⁰C temperature the volume of the gas,

Vt = V₀ (1+ t/273) ml

= V₀ (273 + t°C)/273 ml

The temperature on the 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,T K = 273 + t°C

∴ Vt = (V₀ × T)/273 = (V₀/273) T

Vt = K₂ T

∴ V ∝ T

Where K₂ is constant whose value depends on the nature, mass, and pressure of the gases.∴ V ∝ T

According to the above relation, Charles law states as, at constant pressure, the volume of a given mass of gas is directly proportional to its kelvin temperature.

#### Graphical presentation of Charles law

A typical variation of volume of gas with a 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 presentation of Charles law |

#### Kelvin scale for gas molecules

Since volume is directly proportional to its kelvin temperature, the volume of the gas is theoretically zero at zero kelvin or - 273⁰C.However, this is indeed hypothetical because all gases liquefy 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 = -273⁰C.

#### Density measurement from Charles law

Charles law, V₁/V₂ = T₁/T₂

Mass of the gas = M.

Density D₁ and D₂ at the volume V₁ and V₂ respectively.

V₁ = M/D₁ and V₂ = M/D₂

(M/D₁ )/(M/D₂ ) = T₁/T₂

or, D₂/D₁ = T₁/T₂

∴ D ∝ 1/T

Thus at constant pressure, the density of a given mass of gases is inversely proportional to its temperature.Mass of the gas = M.

Density D₁ and D₂ at the volume V₁ and V₂ respectively.

V₁ = M/D₁ and V₂ = M/D₂

(M/D₁ )/(M/D₂ ) = T₁/T₂

or, D₂/D₁ = T₁/T₂

∴ D ∝ 1/T

#### Charles law and Boyle's law for gas molecules

Charles law, V ∝ 1/P when T constant.

Boyle's law, V ∝ T when P constant.

When all the variables are taken into account the variation ruleBoyle's law, V ∝ T when P constant.

V ∝ T/P

or, PV/T = constant

∴ (P₁V₁)/T₁ = (P₂V₂)/T₂ = constant

∴ PV = KT

or, PV/T = constant

∴ (P₁V₁)/T₁ = (P₂V₂)/T₂ = constant

∴ PV = KT

The product of the pressure and volume of a given mass of gas is proportional to its kelvin temperature.