Charle’s Law

Charles' Law is a key principle in thermodynamics that explains how gases expand when heated and contract when cooled, at constant pressure. Named after French scientist Jacques Charles, the law states that the volume of a gas is directly related to its temperature. As the gas temperature increases, its molecules gain energy, move faster, and occupy more space. Similarly, cooling the gas causes its volume to decrease.

1.0Statements of Charle’s Law

• This law indicates that for a specified mass of an ideal gas at unchanging pressure, the volume is directly related to its absolute temperature.
• It states that if the pressure remains constant,then the volume of given mass of a gas increase or decrease by $\frac{1}{273.15}$ of its volume at 0°Cfor each $1^{\circ }$ rise or fall of temperature.
• Let $V_0$ be the volume of given mass of a gas at $0^{\circ }$.According to Charle’s Law,its volume at $1^{\circ }$ is,

$V_1=\left(V_0+\frac{V_0}{273.15}\right)=V_0\left(1+\frac{1}{273.15}\right)$

Volume of  gas at $2^{\circ }C$ is

$V_2=V_0\left(1+\frac{2}{273.15}\right)$

Volume of the gas at  $t^{\circ }C$ is

$V_t=V_0\left(1+\frac{t}{273.15}\right)=V_0\left(\frac{273.15+t}{273.15}\right)$

If $T_0$ and T are temperatures on Kelvin scale corresponding to  $0^{\circ }Cand{t}^{\circ }C$ then,

$T_0=273.15+0=273.15\text{ and }T=273.15+t$

$\therefore V_t=V_0\frac{T}{T_0}\text{ or }\frac{V_t}{T}=\frac{V_0}{T_0}$

$\frac{V_t}{T}=\text{Constant}\Rightarrow V\propto T$

With pressure constant, the volume of a given gas is directly related to its absolute temperature.

$\frac{V_1}{V_2}=\frac{T_1}{T_2}$

2.0Molecular Explanation

• Raising the temperature of a gas boosts the kinetic energy of its molecules.
• These faster-moving molecules collide more frequently and with greater force against the container walls.  
• To maintain constant pressure, the volume of the container must increase, allowing the molecules to spread out and reduce the frequency of collisions per unit area.
• Conversely, decreasing the temperature reduces molecular kinetic energy, leading to a decrease in volume.

3.0Charles' Law Experiment

• Inflate the Balloon: Inflate the balloon and knot it securely.
• Initial Observation: Observe the balloon’s size at room temperature.
• Hot Water Setup: Place the balloon in hot water (not boiling) and observe it expand as the gas inside heats up.
• Cold Water Setup: Move the balloon to ice water and watch it shrink as the gas cools.
• Conclusion: The balloon expands in hot water and contracts in cold, demonstrating Charles' Law: gas volume is directly related to temperature at constant pressure.

4.0Derivation of Charle’s Law

Applying Ideal Gas Equations, $PV=nRT$

$P\to Pressure,V\to Volume,n\to no.ofmoles,R\to GasConstant,T\to TemperatureinKelvin$

Keeping the pressure P and the number of moles n constant. So, if both pressure and the number of moles are constant, we can express the Ideal Gas Law as,

$V=\frac{nRT}{P},k=\frac{nR}{P}$

$V=kT$

$V\propto T\Rightarrow PressureisConstant$

5.0Graphs for Charle’s Law

6.0Concept of "Absolute Zero"

• A key takeaway from Charles's Law is the concept of absolute zero, the lowest temperature at which a gas theoretically has zero volume. As temperature decreases, so does the gas volume. At absolute zero (0 K or -273.15°C), the gas volume reaches zero, and molecular motion ceases. Temperatures below this point are not possible.

7.0Applications of Charles’s Law

• Hot Air Balloons: It operates on Charles's Law: heating the air inside the balloon expands its volume, decreasing its density and allowing the balloon to rise in cooler air.
• Internal Combustion Engines: The expansion of hot gases in the cylinders of an engine is related to Charles's Law.
• Industrial Processes: Many industrial processes involving gases, such as drying and heating, rely on the principles of Charles's Law.

8.0Crucial Consideration About Charle’s Law

• Absolute Temperature (Kelvin):Charles's Law is only applicable when temperature is measured on the Kelvin scale.  Kelvin scale is an absolute temperature grading scale, with 0 K representing absolute zero, the point where all molecular motion theoretically stops. Using Celsius or Fahrenheit would result in inaccurate outcomes.
• Constant Pressure: The pressure of the gas must remain constant for Charles's Law to apply.  
• Ideal Gas Behavior: Charles's Law, like other gas laws, is strictly applicable to ideal gases.Real gases depart from ideal behavior under conditions of high pressure and low temperature.
• Amount of Gas: The amount of gas (number of moles) must remain constant.

9.0Solved Example: Charle’s Law

Q-1.A gas occupies a volume of 8.0 L at 273 K. What will be its volume at 0°C (273 K), assuming pressure remains constant?

Solution:Since the temperature doesn't change $(T_1=T_2=273\text{ K})$, the volume also remains constant.So $V_1=V_2=V.\text{ So, the volume remains 8L.}$

Q-2.A gas occupies 12.0 L at 320 K. The gas is then heated to 480 K. If the volume of the gas becomes 15.0 L, find the change in pressure. Assume the amount of gas is constant.

Solution:

$\frac{P_1{V}_1}{T_1}=\frac{P_2{V}_2}{T_2}$

$\frac{P_1\times 12}{320}=\frac{P_2\times 15}{480}$

$P_2=P_1\times \frac{12}{320}\times \frac{480}{15}=P_1\times \frac{5760}{4800}=P_1\times 1.2$

Thus, the pressure increases by a factor of 1.2.if the initial pressure was $P_1$ the new pressure $P_2\text{ is }1.2\times P_1$

Q-3.A 2.0 L gas is heated from 350 K to 400 K. The gas is then expanded from 2.0 L to 4.0 L. What is the final temperature of the gas?

Solution:

$\frac{V_1}{V_2}=\frac{T_1}{T_2}$

$\frac{V_1}{T_1}=\frac{V_2}{T_2}\Rightarrow \frac{2}{350}=\frac{4}{T_2}\Rightarrow T_2=\frac{4\times 350}{2}=700\text{ K}$

Q-4.The rms velocity of hydrogen at S.T.P is $u\text{ m/s}$. If the gas is heated at constant pressure till its volume is three fold,calculate its final temperature?

Solution:

$\frac{V_2}{V_1}=\frac{T_2}{T_1}$

$T_2=\frac{V_2}{V_1}\times T_1=\frac{3V}{V}\times 273=819\text{ K}$