Charle's Law Formula

The law of volumes, or Charles law, is an experimental gas law that describes how gases expand when heated.

Charles law states that "When the pressure on a sample of a dry gas is held constant, the Kelvin temperature and the volume will be in direct proportion".

What is the Formula of Charles Law?

According to Charles law equation:

V ∝ T

The Charles law mathematical expression will be 

V = kT

V/T = k

Where V is the volume of the gas.

T is the temperature of the gas in kelvin.

k is the non zero constant.

This law illustrates how a gas expands as the temperature rises, whereas a fall in temperature causes the volume to decrease. The mathematical expression of Charles law can be expressed as follows for comparing the same material under two different sets of conditions:

\[\frac{V_{1}}{T_{1}}=\frac{V_{2}}{T_{2}}\] or 

\[\frac{V_{1}}{V_{2}}=\frac{T_{1}}{T_{2}}\] or

V1T2 = V2T1

Where V1 is the initial volume of the system.

V2 is the final volume of the system.

T1 is the initial temperature of the system in Kelvin.

T2 is the final temperature of the system in Kelvin.

This Charles law equation indicates that as the absolute temperature of the gas rises, so does the volume of the gas.

Charles Law Derivation

In this section, we will discuss step by step derivation of the Charles law formula.

Consider a system whose initial volume is V1 and initial temperature is T1. 

Now according to Charles law, the volume will be directly proportional to temperature at constant pressure. So we can write the relationship between volume and temperature as follows:

V1 ∝ T1……………….(1)

Since volume is directly proportional to temperature, if the temperature of the system is increased to T2 the volume will also increase proportionally to V2. Now the mathematical expression of Charles law at temperature T2 will be as follows:

V2 ∝ T2……………….(2)

Now removing the proportionality sign by introducing the non zero constant k we can write equations (1) and (2) as follows:

V1 = kT1

\[\frac{V_{1}}{T_{1}}=k\]…………………(3)

V2 = kT2

\[\frac{V_{2}}{T_{2}}=k\]……………………………(4)

The non zero constant in equations (3) and (4) are the same, so by comparing equations (3) and (4) we will get the mathematical expression of Charles law as follows:

\[\frac{V_{1}}{T_{1}}=\frac{V_{2}}{T_{2}}\] or 

\[\frac{V_{1}}{V_{2}}=\frac{T_{1}}{T_{2}}\] or

V1T2 = V2T1

This expression gives the Charles law when the same material is under two different sets of conditions.

Problems on Charles Law Formula

1. A sample of gas under constant pressure has an initial temperature of 230 C and an initial volume of 25.6 cm3. What will be the temperature of the gas if the final volume is 15.2 cm3?

Ans: Here pressure is constant P = Constant

Initial temperature is T1 = 230 C = 273 + 23 = 296 K

Initial volume is V1 = 25.6 cm3

Final volume is V2 = 15.2 cm3

Final temperature is T2 = ?

Since the process is happening under constant pressure we can apply the Charles law formula to find the final temperature.

The Charles law equation is given as 

\[\frac{V_{1}}{V_{2}}=\frac{T_{1}}{T_{2}}\]

Substituting the values we get

\[\frac{25.6}{15.2}=\frac{296}{T_{2}}\]

\[T_{2}=\frac{296\times15.2}{25.6}\]

\[T_{2}=\frac{4499.2}{25.6}\]

T₂ = 175.76 K = -97.25⁰ C

Therefore the final temperature of the gas sample is -97.25⁰ C.

2. A gas occupies the volume of 80 cm3 at a temperature of 700 C. If the temperature is doubled by keeping the pressure constant, what will be the final volume of the gas?

Ans: Here the initial volume occupied by the gas is V1 = 80 cm3

Initial temperature of the gas is T1 = 700 C = 273 + 70 = 343 K.

Now the gas is heated such that the temperature is doubled.

So the final temperature of the gas will be T2 = 1400 C = 273 + 140 = 413 K.

Since the process is happening at constant pressure, to find the final volume of the gas we will use the Charles law formula.

The Charles law equation is given as 

\[\frac{V_{1}}{V_{2}}=\frac{T_{1}}{T_{2}}\]

Substituting the values we get

\[\frac{80}{V_{2}}=\frac{343}{413}\]

\[V_{2}=\frac{413\times 80}{343}\]

\[V_{2}=\frac{33040}{343}\]

V₂ = 96.4 cm³

Therefore the final volume of the gas is 96.4 cm3.

Conclusion

Charles Law is a gas law that defines how gases expand when heated in an experimental setting. The rule asserts that if an amount of gas is held at constant pressure, the volume and temperature, measured in degrees Kelvin, have a direct relationship. The molecules in any particular gas begin to move around more swiftly as the temperature increases. The molecules hit with each other and the walls of their container more frequently and with greater force as they move faster. These more frequent and severe impacts will result in higher pressure if the gas container is rigid. However, if the container is flexible, such as a balloon, the pressure will remain constant as the gas volume expands.