# Physical properties of gases and gas laws

## Properties of gases and gas laws

Gases are physical properties of mark sensitivity of volume change with the change of temperature and pressure. Thus the gases are generally concerned with the relations among four properties, namely mass, pressure, volume, and temperature.

The relationship between these physical properties of gases describes by Boyles, Charles, and Avogadro gas laws.

Gas molecules move very large speeds because of the forces of attraction between them very low. Thus the gas molecules do not possess fixed volume, they move partially independent of one another.

### Boyle’s law of gases

Boyle’s law states as, at a constant temperature, the volume of a definite mass of a gas is inversely proportional to its pressure.

Thus the volume of a given quantity of gas, at constant temperature decreases with the increase of pressure of gases.

At constant temperature and 1 atm pressure, a cylinder contains 10 ml of methane gas.  If the pressure increases to 2 atm then according to this law volume decreases to 5 ml.

#### Mathematical derivation of Boyle’s law

According to the mathematical definition of Boyle’s law at a constant temperature

At constant T, V ∝ 1/P
∴ PV = K = constant for a gas

Thus the value of gas constant depends on the nature and mass of the gases.

For a given mass of gas, the volume V1 at pressure P1 and V2 at pressure P2. Thus according to Boyle’s Law

P1V1 = P2V2

#### Graphical representation of Boyle’s law

The relation between pressure and volume of gas can be represented by an arm of a rectangular hyperbola given below.

1. The value of gas constant change with temperature. Thus there will be a separate curve for each fixed temperature. These curves plotted at different fixed temperatures are called isotherms.
2. At constant temperature, a given mass of gas is the product of pressure and volume. If the product of pressure and volume represents in Y-axis and pressure represents X-axis a straight line curve obtained parallel to X-axis.

#### The relation between pressure and density of gas

Let at a constant temperature, M mass of gas has pressure P1 at volume V2 and pressure P2 at volume V2.

Thus according to Boyle’s law
P1V1 = P2V2
or, P1/P2 = V2/V1

Let the density of the gas D1 at pressure  P1 and D2 at pressure P2.

$\therefore&space;D_{1}=\frac{M}{V_{1}}\,&space;and\,&space;D_{2}=\frac{M}{V_{2}}$

$or,&space;V_{1}=\frac{M}{D_{1}}\,&space;and\,&space;V_{2}=\frac{M}{D_{2}}$

$\therefore&space;\frac{P_{1}}{P_{2}}=\frac{V_{2}}{V_{1}}=\left&space;(&space;\frac{M}{D_{2}}&space;\right&space;)\times&space;\left&space;(&space;\frac{D_{1}}{M}&space;\right&space;)=\frac{D_{1}}{D_{2}}$

or, P/D = constant
∴ P ∝ D

Thus at a constant temperature, the density of a definite mass of a gas proportional to its pressure.

#### Relationship between pressure and volume

At constant pressure a given mass of gas, volume increases with the increasing temperature. Thus the volume of a given mass of gas at constant pressure is directly proportional to its kelvin temperature.

### Statement of Charles law

Charles made the measurement of the volume of a fixed mass of gases at the various temperature at constant pressure and states the gas law as

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.

#### Charles law formula for ideal gases

Let V0 = volume at 0⁰C, then 1⁰C rise of temperature the volume of the gas rise V0/273.5 ml.

10C rise of temperature the volume

$V_{1}=\left&space;(&space;V_{0}+\frac{V_{0}}{273}&space;\right&space;)=V_{0}\left&space;(&space;1+\frac{1}{273}&space;\right&space;)$

Thus at t0C temperature the volume

$V_{t}=V_{0}\left&space;(&space;1+\frac{t}{273}&space;\right&space;)=V_{0}\left&space;(&space;\frac{273+t^{0}C}{273}&space;\right&space;)$

It is convenient to use the absolute temperature scale on which temperature is measured in Kelvin. Thus the reading on this scale obtained by adding 273 to the celsius value.

TK = 273 + t0C

$\therefore&space;V_{t}=\frac{V_{0}\times&space;T}{273}=\left&space;(&space;\frac{V_{0}}{273}&space;\right&space;)\times&space;T$

Since V0 = initial volume = constant at a given pressure, thus the above relation expressed as,

Vt = K2 T
where K2 = constant for a gas
∴ V ∝ T

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

#### Graphical representation of Charles law

A typical variation of volume of gas with a change in its kelvin temperature a straight line plot is obtained. These plots are known as isobars. Thus the general term isobar, which means at constant pressure assigned to these plots.

#### Absolute zero temperature gases

Since volume is directly proportional to its kelvin temperature. Thus the volume is theoretically zero at zero kelvin or – 2730C.

This is hypothetical because the gases from liquid and then solid before this low temperature reached. In reality, no substance exists as gases at the temperature near kelvin zero.

#### Temperature density relationship

Let at constant pressure, for M mass has volume V1 at temp T1 and V2 at temp T2.

Charles law, V1/V2 = T1/T2

Let density D1 and D2 at the volume V2 and V1.

$\therefore&space;\frac{T_{1}}{T_{2}}=\frac{V_{1}}{V_{2}}=\left&space;(&space;\frac{M}{D_{1}}&space;\right&space;)\times&space;\left&space;(&space;\frac{D_{2}}{M}&space;\right&space;)=\frac{D_{2}}{D_{1}}$

D ∝ 1/T

Thus at constant pressure, the density of a given mass of gases inversely proportional to its temperature.

#### Derivation of combined gas laws equation

Charles law, V ∝ 1/P when T constant.
Boyle’s law, V ∝ T when P constant.

Thus when all the variables are taken into account the variation rule states as

V ∝ T/P
or, PV/T = constant

$\therefore&space;\frac{P_{1}V_{1}}{T_{1}}=\frac{P_{2}V_{2}}{T_{2}}$

or, PV = KT

Thus this ideal gas law state the physical properties as the product of the pressure and volume of a given mass of gases is proportional to its kelvin temperature.