**Definition**

It is a quantitative relationship between temperature and volume of a gas. It was given by French scientist J. Charles in 1787.

According to this law, the volume of the given mass of a gas is directly proportional to the absolute temperature when the pressure is kept constant.

V α T (when pressure and number of moles are constant)

V = kT

V/T = k

If the temperature is
changed from T_{1} to T_{2} and volume changes from V_{1}
to V_{2}, then

V _{1}/ T_{1} = k and V_{2} / T_{2} = k

So, V _{1}/ T_{1} = V_{2}/ T_{2}

V_{1} = initial
volume

T_{1 }= initial
temperature

V_{2}= final
volume

T_{2} = final
temperature

It can be written as

V_{2} = V_{1}
(T_{2} / T_{1})

V_{2}= final
volume

T_{2} = final
temperature

V_{1} = initial
volume

T_{1 }= initial
temperature

The ratio of volume to temperature remains constant for same amount of gas at same pressure.

The law determines the linear relationship between volume and temperature. The temperatures are measured in Kelvin, which is SI unit of temperature.

**Explanation
**

James Clerk Maxwell claimed that a space is occupied by a gas is dependent upon the motion of its particles. The particles continuously collide with the walls of the container. This rapid collision exerts force on the container surface, that force creates pressure.

The impact of this force is inconsequential but collectively the collision exerts pressure on the surface of the container. Gas pressure is proportional to the magnitude of collision and the force. So, the more collision results in higher pressure. However, it is also realized that the pressure is increasing with the increase in temperature provided the volume of the container kept constant. This behaviour is evident in the air pumps that churn out hot air when their piston is periodically pushed and pulled.

Volume increases in heated gas because volume isn’t bounded as ball expands pressure is also increases. The rubber expands as more hot gas is pumped in and the exhilarated particles bounce and push on the inside of the surface, pushing it outward. They obey Charles’s law [1].

**Experimental
Verification**

Let us consider a certain
amount of a gas enclosed in a cylinder fitted with a moveable piston. The
volume of the gas is V_{1} and its temperature is T_{1}. When
the gas in the cylinder is heated both volume and the temperature of the gas
increase. The new values of volume and temperature are V_{2} and T_{2}
respectively.

**Example
**

If 250 cm^{3} of
hydrogen is cooled from 127^{o}C to -27^{o}C by maintaining the
pressure constant. Calculate the new volume of the gas at low temperature.

**Solution
**

Pressure has been kept constant so this gas is obeying the Charles’s law

Initial
volume (V_{1}) =250cm_{3} =0.25 dm^{3}

_{ }Initial temperature (T_{1}) = 127^{O}C +273 K =400K

Final temperature (T_{2}) = -27^{O}C +273 K = 246 K

Final volume (V_{2}) = Ɂ

According to Charles’s law

V _{1}/ T1 = V_{2}/ T_{2}

V_{2}= V_{1} x T_{2} / T1

_{ } V_{2} = 0.25 dm^{3} x 246K /400K = 0.153 dm^{3} = 153 cm^{3} Answer

So, by decreasing the temperature the volume of the gas has decreased at constant temperature.

**Graphical Explanation **

If we plot a graph between temperature on x-axis and the volume of one mole of an ideal gas on the y-axis, we get a straight line which cuts the temperature axis at -273.16^{o} C. this can be possible only if we extrapolate the graph upto -273.16^{o}C. This temperature is the lowest possible temperature which would have been achieved if the substance remains in the gaseous state. Actually, all the gases are converted into liquids above this temperature.

Charles‘s law is obeyed when the temperature is taken on Kelvin scale. For example, at 238 K (10^{O} C) the volume is 566cm^{3} while at 373 K (100^{o}C) the volume is 746 cm^{3. }According to Charles’s law

V _{1}/ T_{1} = V_{2}/T_{2 } = k

566/283 = 746/373 =2= k

Greater the mass of gas taken, greater will be the slope of straight line. The reason is that greater the number of moles greater the volume occupied. All these straight lines when extrapolated meet at a single point of -273.16^{o}C (0 Kelvin). It is apparent that this temperature of -273.16^{o}C will be attained when the volume becomes zero. But for real gas the zero volume is impossible which shows that this temperature cannot be attained for a real gas. This is how we recognized that -273.16^{o} C must represent the coldest temperature.

**Relation to Absolute zero**

According to Charles’s law gas will descend to zero at certain temperature. Gay- Lussac said that this law is not applicable at low temperature:

He said that compressed vapours remain entirely in their elastic state and they require temperature to enable them to resist the pressure.

At absolute zero temperature gas has zero energy so molecule restrict their motion Gay-Lussac also demonstrate that the Charles’s law doesn’t applicable just above the boiling point of the liquid. For example, ether is only a little above of its boiling point.

William Thomson was the first who mention that volume of a gas might descend to zero. He describe absolute zero in terms of the second law of thermodynamics.

**Derivation of Absolute
Zero**

In order to derive absolute zero of temperature, consider the following quantitative definition of Charles’s law.

At constant pressure the volume of
the given mass of the gas increases or decreases by 1\273 of its original
volume at 0^{o}C for every 1^{o}C rise or fall in temperature
respectively.

The general equation to know the volume of the gas at various temperature is

V_{t}= V_{0}
(1+t\273)

V_{t}
= volume of gas at temperature T

V_{0}
= volume of gas at 0^{o} C

t = temperature on centigrade or Celsius scale

if
a gas is warmed by 1^{o}C, it expands by 1\273 of its original volume
at 0^{0} C. Since original volume is 546cm^{3}, so for 1^{0}
C rise in temperature 2cm^{3 }increase in volume will take place. 2cm^{3}
is the 1\273 of 546cm^{3}. Similarly for 100^{0}C rise in
temperature, a change of 200cm^{3}will take place.

For
example, the increase in temperature from 10^{0}C to 100^{0} C,
increase in volume from 566cm^{3 }to 746cm^{3}.

**Relation to kinetic
theory**

The kinetic theory of gases is related to the macroscopic properties of gases like pressure and volume. It is also related to the microscopic properties of molecules like speed and mass of the molecules which make up the gas. In this theory temperature is taken as proportional to the average kinetic energy of the gas molecule.

T α *E*_{k}

**Application **

**Hot air balloons**

`An object floats in a fluid when it weigh less than the fluid it displaces. According to Charles’s law, balloon is filled with heated gas, so volume expand. At elevated volume, the balloon occupies a greater volume in the same weight. Its density is lesser than the cold air so balloon begins to rise.

**Bloated tires**

According to Charles law, the bloated tubes protruding out from a tire when we left in summer heat. The heat outside steadily enter into tubes and gradually causes the tire to expand. It is highly recommended to check your tires daily in summers. Because if subjected to further expansion the tires can burst at any second.

**Automobiles**

In automobile engine series of pistons are lined up they are periodically bob up and down when there is presence or absence of fluid. The one end of piston is attached with crankshaft so rise or fall rotate the shaft and other end is attached with rear wheel of automobile. According to Charles’s law pistons are pushed by gas which is produced by fuel combustion. The combustion produce great amount of heat. So temperature rise and converted gas expands. They push them with force and thrust the vehicle forward.

**Properties of Gases**

- Gases don’t have a definite volume and occupy all the available space. The volume of a gas is the volume of the container.
- They don’t have a definite shape and take the shape of the container just like liquids.
- Due to low densities of gases, as compared to those of liquids and solids, the bubble through liquids and tend to rise up.
- Gases can diffuse and effuse.
- Gases can be compressed by applying a pressure because there are large empty spaces between their molecules.
- Gases can expands on heating.
- When sudden expansion of gases occurs cooling takes place, it is called Joule Thomson effect.
- Molecules of gases are in a constant state of random motion. They can exert a certain pressure on the walls of the container and this pressure is due to the number of collisions.
- The intermolecular forces in gases are very weak [3].
- Gases temperature is proportional to the average kinetic energy of the molecules.

**The Ideal Gas Equation **

According to Boyle’s law:

Vα 1/P

According to Charles’s law:

Vα T

It is well known fact that volume of the given gas at constant temperature and pressure is directly proportional to the number of moles (Avogadro’s law)

Vα n

So by these equations

Vα nT/P

V= constant nT/P

The constant is R which is called general gas constant.

V= R nT/P

**Reference **

- https://www.scienceabc.com/nature/universe/charles-law-definition-formula-equation.html
- https://en.wikipedia.org/wiki/Charles%27s_law
- https://www.chemguide.co.uk/physical/kt/idealgases.html