# 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 V_{1} and pressure is p_{1}. After expansion, the volume becomes V_{2} and pressure becomes p_{2}. 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 V_{0} 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 T_{0} = 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 × 10^{23}. This is also called Avogadro’s Constant.

Standard temperature and pressure mean 273.15 K (0° C0 and 1 bar (or 10^{5} 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 m^{3} 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 T_{1},V_{1} and p_{1} to T_{2}, V_{2} and p_{2} 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`