`text(Definition :)` At constant temperature, the pressure of a fixed amount (i.e., number of moles `n`) of gas varies inversely with its volume. This is known as Boyle’s law.

`=>` Mathematically, it can be written as

`p prop 1/V` ( at constant T and n) ...........(1)

`=> p = k_1 1/V` ......................(2)

where `k_1` is the proportionality constant.

● The value of constant `k_1` depends upon the amount of the gas, temperature of the gas and the units in which `p` and `V` are expressed.

● On rearranging equation (2) we obtain

`pV = k_1` .............(3)

● It means that at constant temperature, product of pressure and volume of a fixed amount of gas is constant.

`=>` If a fixed amount of gas at constant temperature `T` occupying volume `V_1` at pressure `p_1` undergoes expansion, so that volume becomes `V_2` and pressure becomes `p_2`, then according to Boyle’s law :

`p_1V_1 = p_2V_2 = ` constant ...........(4)

`=> p_1/p_2 = V_2/V_1` ..............(5)

`=>` Figure shows two conventional ways of graphically presenting Boyle’s law.

`=>` Fig. is the graph of equation (3) at different temperatures.

● The value of `k_1` for each curve is different because for a given mass of gas, it varies only with temperature.

● Each curve corresponds to a different constant temperature and is known as an `text(isotherm)` (constant temperature plot).

● Higher curves correspond to higher temperature.

● It should be noted that volume of the gas doubles if pressure is halved.

● Table gives effect of pressure on volume of `0.09` mol of `CO_2` at `300 K`.

`=>` Fig above represents the graph between `p` and `1/V`.

● It is a straight line passing through origin.

● However at high pressures, gases deviate from Boyle’s law and under such conditions a straight line is not obtained in the graph.

`=>` Experiments of Boyle, in a quantitative manner prove that gases are highly compressible because when a given mass of a gas is compressed, the same number of molecules occupy a smaller space.

● This means that gases become denser at high pressure.

● A relationship can be obtained between density and pressure of a gas by using Boyle’s law :

● By definition, density `‘d’` is related to the mass `‘m’` and the volume `‘V’` by the relation `d = m/V`.

● If we put value of `V` in this equation from Boyle’s law equation, we obtain the relationship.

`d = (m/k_1) p equiv k'p`

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

`color{purple}♣ color{Violet} " Just for Curious"`
At higher altitudes, as the atmospheric pressure is low, the air is less dense. As a result, less oxygen is available for breathing. The person feels uneasiness, headache etc. This is called altitude sickness. That is why the mountaineers have to carry oxygen cylinders with them

`=>` Charles and Gay Lussac performed several experiments on gases independently to improve upon hot air balloon technology.

`text(Definition :)` For a fixed mass of a gas at constant pressure, volume of a gas increases on increasing temperature and decreases on cooling.

● They found that for each degree rise in temperature, volume of a gas increases by `1/(273.15)` of the original volume of the gas at `0 °C`.

● Thus if volumes of the gas at `0 °C` and at `t °C` are `V_0` and `V_t` respectively, then

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

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

`=> V_t = V_0 ((273.15+t)/273.15)` ..........(6)

`=>` At this stage, we define a new scale of temperature such that `t °C` on new scale is given by `T = 273.15 + t` and `0 °C` will be given by `T_0 = 273.15`.

● This new temperature scale is called the Kelvin temperature scale or Absolute temperature scale.

● Thus we add `273` (more precisely `273.15`) to the celsius temperature to obtain temperature at Kelvin scale.

`=>` If we write `T_t = 273.15 + t` and `T_0 = 273.15` in the equation (5.6) we obtain the relationship

`V_t = V_0 (T_t/T_0)`

`=> V_t/V_0 = T_t/T_0` .......(7)

● Thus, we can write a general equation as follows :

`V_2/V_1 = T_2/T_1` .......(8)

`=> V_1/T_1 = V_2/T_2`

`=> V/T =` const ........(9)

Thus `V = k_2T` ........(10)

`=>` The value of constant `k_2` is determined by the pressure of the gas, its amount and the units in which volume `V` is expressed.

`text(Statement of Charles’ Law :)` Pessure remaining constant, the volume of a fixed mass of a gas is directly proportional to its absolute temperature.

● Charles found that for all gases, at any given pressure, graph of volume vs temperature (in celsius) is a straight line and on extending to zero volume, each line intercepts the temperature axis at `-273.15 °C`.

● Slopes of lines obtained at different pressure are different but at zero volume all the lines meet the temperature axis at `-273.15 °C`.

● Each line of the volume vs temperature graph is called `text(isobar)`.

`=>` Observations of Charles can be interpreted if we put the value of `t` in equation (5.6) as `-273.15 °C`.

● We can see that the volume of the gas at `-273.15 °C` will be zero.

● This means that gas will not exist.

● In fact all the gases get liquified before this temperature is reached.

● The lowest hypothetical or imaginary temperature at which gases are supposed to occupy zero volume is called `text(Absolute zero)`.

`text(Note :)` All gases obey Charles’ law at very low pressures and high temperatures.

`color{purple}♣ color{Violet} " Just for Curious"`
Air expands on heating and hence its density decreases. Thus, hot air is lighter than the atmospheric air. This fact is used for filling hot air in the balloons which rise up for the meteorological observations.

`=>` Pressure in well inflated tyres of automobiles is almost constant, but on a hot summer day this increases considerably and tyre may burst if pressure is not adjusted properly.

`=>` During winters, on a cold morning one may find the pressure in the tyres of a vehicle decreased considerably.

`=>` The mathematical relationship between pressure and temperature was given by Joseph Gay Lussac and is known as Gay Lussac’s law.

`text(Statement :)` At constant volume, pressure of a fixed amount of a gas varies directly with the temperature.

● Mathematically,

`p prop T`

`=> p/T` = const

`=>` This relationship can be derived from Boyle’s law and Charles’ Law.

`=>` Pressure vs temperature (Kelvin) graph at constant molar volume is shown in Fig.

● Each line of this graph is called `text(isochore)`.

`=>` In 1811 Italian scientist Amedeo Avogadro tried to combine conclusions of Dalton’s atomic theory and Gay Lussac’s law of combining volumes which is now known as Avogadro law.

`text(Statement :)` Equal volumes of all gases under the same conditions of temperature and pressure contain equal number of molecules.

● This means that as long as the temperature and pressure remain constant, the volume depends upon number of molecules of the gas or in other words amount of the gas.

● Mathematically, we can write

`V prop n` where `n` is the number of moles of the gas

`=> V = k_4 n` ...........(11)

`=>` The number of molecules in one mole of a gas has been determined to be `6.022 ×10^(23)` and is known as Avogadro constant.

`=>` Since volume of a gas is directly proportional to the number of moles; one mole of each gas at standard temperature and pressure (STP) will have same volume.

● Standard temperature and pressure means `273.15 K (0°C)` temperature and `1` bar (i.e., exactly `10^5` pascal) pressure.

● These values approximate freezing temperature of water and atmospheric pressure at sea level.

● At STP molar volume of an ideal gas or a combination of ideal gases is `22.71098 L mol^(–1)`.

`=>` Molar volume of some gases is given in (Table ).

`=>` Number of moles of a gas can be calculated as follows

`n = m/M` .......(5.12)

Where `m` = mass of the gas under investigation and `M` = molar mass. Thus,

`V = k_4 m/M` ............(13)

Equation (5.13) can be rearranged as follows :

`M = k_4 m/V = k_4 d` ............(14)

Here ‘d’ is the density of the gas.

● We can conclude from equation (14) that the density of a gas is directly proportional to its molar mass.

`=>` A gas that follows Boyle’s law, Charles’ law and Avogadro law strictly is called an ideal gas.

● Such a gas is hypothetical.

● It is assumed that intermolecular forces are not present between the molecules of an ideal gas.

● Real gases follow these laws only under certain specific conditions when forces of interaction are practically negligible.

● In all other situations these deviate from ideal behaviour.