Charles’s law , Jolly’s law and General gas law

The gases contract by cooling and expand by heating , Equal volumes of different gases expand equally when heated through the same temperature rise , In other words they have the same volume expansion coefficient , Charles’s law expresses the effect of temperature on the volume of a gas at constant pressure .

Charles’s law

All gases have the same volume expansion coefficient , at constant pressure , so the volume expansion coefficient ( α_v ) can be determined from the following relation :

α_v = Δ ( V_ol ) / [ ( V_ol )_0°C  Δt ]

α_v = [ ( V_ol )_t − ( V_ol )_0°C ] / [ ( V_ol )_0°C  Δt ]

The measuring unit of the volume expansion coefficient is ( Kelvin )⁻¹ = ( K⁻¹ ) .

The volume expansion coefficient of gas at constant pressure ( α_v ) is the increase in volume at constant pressure per unit volume at 0°C for rise in temperature , or it is the ratio between the increase in volume to the original volume at 0° C for 1° C rise in temperature at constant pressure .

When the volume expansion coefficient of a gas under a constant pressure = ( 1/273 ) K⁻¹ , It means that the increase in volume at constant pressure per unit volume at 0° C for 1° C rise in temperature = 1/273 of the original volume .

To convert from Celsius to Kelvin , we use the following relation : T ( °K ) = t° C + 273

Where : ( T ) → Temperature on Kelvin scale , ( t ) → Temperature on Celsius scale .

Charles’s law : At constant pressure , the volume of a given mass of gas expands by 1/273 of its original volume at 0°C per each degree Kelvin rise in temperature , or at constant pressure , the volume of fixed mass of gas is directly proportional to its temperature on Kelvin scale .

Charles’s law mathematical formula : V₁ / V₂ = T₁ / T₂

( V_ol )₁ / ( V_ol )₂ = T₁ / T₂

T₁ = 273 + t₁ ,  T₂ = 273 + t₂

( V_ol )₁ / ( V_ol )₂ = ( 273 + t₁ ) / ( 273 + t₂ )

( V_ol )₁ / ( V_ol )₂ = ( 1 + 1/273 t₁ ) / ( 1 +1/273 t₂ )  , α_v = 1/273

∴ ( V_ol )₁ / ( V_ol )₂ = ( 1 + α_v t₁ ) / ( 1 + α_v t₂ )

If two gases are mixed at constant pressure , then

( V_ol ) / T ( mix. ) = [ ( V_ol )₁ / T₁ ]  + [ ( V_ol )₂ / T₂ ]

Jolly’s law or pressure law

Jolly’s law expresses the relation between gas pressure and its temperature at constant volume , The pressure of gases increases by increasing the temperature , The increase in the pressure is constant for all gases .

At constant volume , the pressure of a given mass of a gas increases by increasing temperature , At constant volume equal pressures of different gases increase equally when heated to the same temperature .

Pressure expansion coefficient is constant for all gases , Equal pressures of different gases increase equally when heated through the same rise in temperature because expansion coefficient for any gas at constant volume is constant .

At constant volume , the increase in the pressure ( Δ P ) is directly proportional to :

1. Original pressure at 0° C ( P_o ) : Δ P ∝ P_{0° C}
2. Increase in temperature ( Δ t ) : Δ P ∝ Δ t

∴ Δ P ∝ P_{0° C} Δ t

∴  Δ P = constant P_{0° C} Δ t

∴  Δ P = β_p P_{0° C} Δ t

Where ( β_p ) is the pressure expansion coefficient

β_p = Δ P / (  P_{0° C} Δ t )

β_p = ( P_t − P_{0° C} ) / ( P_{0° C} × Δ t )

The measuring unit of pressure expansion coefficient is ( Kelvin⁻¹ ) or K⁻¹ .

The pressure expansion coefficient of a gas at constant volume ( β_p ) is the increase in pressure of a gas per unit pressure at 0° C when the temperature increases 1° C at constant volume .

Or it is the ratio between the increase in gas pressure to the original pressure at 0° C when the temperature rises 1° C at constant volume .

When the pressure expansion coefficient of a gas = 1/273 K⁻¹ , It means that the increase in pressure of the gas per unit pressure at 0° C when the temperature increases 1° C at constant volume = 1/273 of the original volume .

The pressure expansion coefficient of a gas can be determined by knowing its pressure at t₁ , t₂ at constant volume from the relation :

P₁ / P₂ = ( 1+ β_pt₁ ) / ( 1+ β_pt₂ )

Jolly’s law or pressure law : The pressure of a given mass of gas , kept at constant volume , increases by 1/273 of its pressure at  0° C per each degree Kelvin rise in temperature , Or at constant volume , the pressure of fixed mass of gas is directly proportional to its temperature on Kelvin scale .

The absolute zero ( zero Kelvin ) by using Charles’s apparatus : The absolute zero is the temperature at which the volume of an ideal gas vanishes theoretically at constant pressure .

When the absolute zero is = − 273° C , It means that the temperature at which the volume of ideal gas vanishes theoretically at constant pressure = − 273° C .

The absolute zero ( zero Kelvin ) by using Jolly’s apparatus : The absolute zero is the temperature at which the pressure of an ideal gas vanishes theoretically at constant volume .

When the absolute zero is = − 273° C , It means that the temperature at which the pressure of ideal gas vanishes theoretically at constant volume = − 273° C .

The temperature on Kelvin scale is always positive value while the temperature on Celsius scale may be positive or negative value .

General gas law

General gas law studies the gas behavior when changing volume , pressure and temperature of a gas together , Also explains the relation between the three variables together .

From Boyle’s law : V_ol ∝ 1/p  , From Charles’s law :  V_ol ∝ T .

∴ V_ol ∝ T / P  ,  V_ol = constant T / P

P V_ol / T = constant

( P₁  ( V_ol )₁ ) / T₁ = ( P₂  ( V_ol )₂ ) / T₂

The general gas law : The product of the volume of a fixed mass of gas and its pressure divided by its temperature on Kelvin scale equals constant value .

ًWhen the gas at ( STP ) , So , P = 1.013 × 10⁵ N/m² , T = 273°K .

At the change of density of a gas at constant mass , So :

 P₁ / ρ₁ T₁ = P₂ / ρ₂ T₂

If two gases are mixed with each other , So :

( P V_ol / T ) ( mix. ) = ( P₁  ( V_ol )₁ ) / T₁ + ( P₂  ( V_ol )₂ ) / T₂

Gas laws , Boyle’s law and properties of gaseous materials