States of Matter

Gaseous State

Characteristics of Gases

Gases are highly compressible.
Gases exert pressure equally in all directions.
Compared to solids and liquids, gases have much lower density.
A gas does not have fixed volume or shape.
Gases evenly and completely mix in all proportions without a mechanical aid.

Boyle’s Law
(Pressure-Volume Relationship)

“At constant temperature, the pressure of a fixed amount of gas (number of moles) varies inversely with its volume.”

If p is pressure and V is volume then at constant T and n

`p∝1/V`

Or, `p=k1/V`

Or, `pV=k` ……………..(1)

Where, k is the constant of proportionality. This equation means that at constant temperature, the product of pressure and volume of a fixed amount of gas is constant.

Let us assume, for a given amount of gas at constant temperature T the volume is V₁ and pressure is p₁. After expansion, the volume becomes V₂ and pressure becomes p₂. Then, according to Boyle’s Law

`p_1V_1=p_2V_2` = constant

Or, `(p_1)/(p_2)=(V_2)/(V_1)`

We know that density is related to mass and volume by this relation.

`d=m/V`

So, `V=m/d`

Putting this value in equation (1), we get

`p\m/d=k`

Or, `p\m=kd`

This shows that at a constant temperature, pressure is directly proportional to the density of a fixed mass of gas.

Charles’ Law
(Temperature-Volume Relationship)

“At constant pressure, the volume of a fixed mass of a gas is directly proportional to its absolute temperature.”

`V∝T`

Or, `V/T=k` ……………(2)

Where, k is constant of proportionality.

Charles observed that for all gases, at any given pressure, graph of volume vs temperature is a straight line. When the line is extended to zero volume, each line intercepts the temperature axis at -273.15° C. Volume of gases will become zero at this temperature. In other words, a gas will cease to exist at this temperature. In fact, all the gases get liquefied before this temperature is reached. This temperature is called Absolute Zero. The lowest hypothetical temperature at which gases are assumed to occupy zero volume is called absolute zero. All gases obey Charles’ Law at very low pressure and high temperatures.

Absolute Temperature Scale

During their experiments, Charles and Guy Lussac found that for each degree rise in temperature, volume of a gs increases by `1/(273.15)` of the original volume of the gas at 0° C. Thus, if volume of a gas at 0° C is V₀ and at t° C is V_t then,

`V_t=V_0+1/(273.15)V_0`

Or, `V_t=V_0(1+t/(273.15))`

Or, `V_t=V_0((273.15+t)/(273.15))`

Here, a new scale of temperature is defined. On this scale, t° C is T = 273.15 + t and 0° C is T₀ = 273.15. This new temperature scale is called Kelvin Temperature Scale or Absolute Temperature Scale.

Guy Lussac’s Law
(Pressure-Temperature Relationship)

“At constant volume, pressure of a fixed amount of gas varies with the temperature.

`p∝T`

Or, `P/T=k` ………..(3)

Avogadro’s Law
(Volume-Amount Relationship)

“Equal volume of all gases under the same conditions of temperature and pressure contain equal number of molecules.

`V∝n`

Or, `V=kn` …………………(4)

We know that number of molecules in 1 mole of a gas is 6.022 × 10²³. This is also called Avogadro’s Constant.

Standard temperature and pressure mean 273.15 K (0° C0 and 1 bar (or 10⁵ Pa). An STP molar volume of an ideal gas or a combination of ideal gases is 22.70198 L mol^-1.

Number of moles of a gas can be calculated as follows:

`n=m/M`

Substituting this value in equation (4) we get

Or, `V=km/M`

Or, `M=km/V=kd`

Here, d is the density. So, it can be concluded that the density of a gas is directly proportional to its molar mass.

Ideal Gas Equation

A gas that follows Boyle’s Law, Charles’ Law and Avogadro’s Law strictly is called an ideal gas. Such a gas is hypothetical.

The three laws can be combined together in a single equation, and the equation is called ideal gas equation.

Boyle’s Law: At constant T and n: `V∝t/p`

Charles’ Law: At constant p and n: `V∝T`

Avogadro’s Law: At constant p and T: `V∝n`

So, `V∝(nT)/p`

Or, `V=R(nT)/p`

Or, `pV=nR\T`

Or, `R=(pV)/(nT)`

R is called the gas constant or Universal Gas Constant, because it is same for all the gases.

At constant temperature and pressure, n moles of any gas will have the same volume because:

`V=(nR\T)/p`

Volume of 1 M of an ideal gas under STP (273.15 K and 1 bar) is 22.710981 L mol^-1. Value of R for one mole of an ideal gas can be calculated as follows:

`R=((10^5Pa)(22.71xx10^(-3)m^3))/((1M)(273.15K))`

= 8.314 Pa m³ K^-1 mol^-1

= 8.314 × 10^-2 bar L K^-1 mol^-1

= 8.314 J K^-1 mol^-1

Combined Gas Law

If temperature, volume and pressure of a fixed amount of gas vary from T₁,V₁ and p₁ to T₂, V₂ and p₂ then ideal gas equation can be written as follows:

`(p_1V_1)/(T_1)=nR`

And `(p_2V_2)/(T_2)=nR`

Or, `(p_1V_1)/(T_1)=(p_2V_2)/(T_2)`

This equation is called the combined gas law.

Density and Molar Mass of a Gas

Ideal gas equation can be written as follows:

`n/V=p/(RT)`

We know that `n=m/M`. Substituting this in above equation we get

`m/(MV)=p/(RT)`

Or, `d/M=p/(RT)`

Or, `M=(dR\T)/p`