# Charles law

Solids and liquids have a fixed volume. This makes it easy for us to calculate their properties. Gases on the other hand have no fixed volume. They occupy up the volume of the container they're held in. If that was not enough, the volume of a gas can be affected by changes in temperature and pressure. French scientist, Jacques Alexandre César Charles, formulated a law we now know as Charles' law which defined the relationship between temperature and the volume of a gas. With the help of this law, Jacques Charles designed and launched the world's first-ever hot air balloon! Let us have a look at how this law expanded the boundaries of mankind and how it literally took us to infinity and beyond.

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## Charles law' definition

Charles law also called the law of volumes is used to describe the effect that a change in temperature has on the volume of a gas. It states the following: At constant pressure, the volume of a gas is proportional to its temperature.

$$V\;\propto\;T$$

Or written as an equation:

$$V=kT$$.

This means that the quantity $$V/T$$ is a constant value.

What this means is that gases expand when their temperature increases and contract when their temperature decreases. But why does this happen? and why does it not affect the volume of solids or liquids? what makes gases special? In order to explain this we will need to scale down to the subatomic level.

The temperature here is always measured in degrees Kelvin. To convert between degrees Celcius and degrees Kelvin, we add 273 to the temperature in degrees Celsius to obtain the temperature in degrees Kelvin. $$T_K\;=\;273\;+\;T_{^\circ C}$$

## Why does Temperature affect the Volume of a Gas?

We already know that gases do not have a distinct shape or volume. The molecules are spread out and move randomly, this property allows them to expand and compress as the size of the container changes. When a gas is compressed its volume decreases (density increases) as the molecules become closely packed. If a gas expands, the volume increases, and the density decreases. The volume of a gas is usually measured in $$\mathrm{m^3}$$, $$\mathrm{cm^3}$$ or $$\mathrm{dm^3}$$. But why is this important? We spoke about how gas molecules move randomly in the container they're held in. This motion gives each of these particles its own kinetic energy.

Due to the random motion of the gas molecules, they collide with each other and with the walls of the container. These collisions are the reason why gases exert pressure.

As the temperature of the gas increases, the average kinetic energy of the molecules increases. This increases the average speed of their random motion. Put simply, the higher the temperature, the greater the speed and kinetic energy of the molecules. Charles' law makes an important assumption: the gas must be contained at a constant pressure. When the pressure is constant, the increase in the kinetic energy of the molecules will cause the gases to expand. This is because of the increase in the rate of collision of the gas molecules.

Effect of temperature on volume. Charles' law states that the volume is directly proportional to the temperature of a gas, Florida State University

## Equations for Charles' Law

The equations for Charles' law can also be used while comparing the same gas under different conditions. Because the ratio of volume and gas is constant, we can equate the ratio of volume and temperature of a gas under different temperatures.

$$\frac{V_{\mathit1}}{T_{\mathit1}}\mathit=\frac{V_{\mathit2}}{T_{\mathit2}}$$

$$\frac{\text{Initial}\;\text{volume}}{\text{Initial}\;\text{temperature}}=\frac{\mathrm{Final}\;\text{volume}}{\text{Final}\;\text{temperature}}$$

For a fixed quantity of gas at constant pressure, the ratio of volume and temperature is constant.

When the temperature decreases the speed of the gas molecules reduces. After a point the speed reaches zero i.e. the gas molecules stop moving, this temperature is called absolute zero, $$-273.15\;^\circ\mathrm C$$. And because the speed cannot decrease below zero there is no temperature below absolute zero. Quantum mechanics becomes a more appropriate theory when we want to describe systems at super low temperatures.

## Application of Charles' Law

One of the most famous applications of Charles' law is the hot air balloon!

Let's try and figure out how Charles' law explains the working of a hot air balloon.

The working of a hot air balloon is explained by Charles law, as the temperature of the gas increases it becomes less dense, causing it to rise and fill the balloon above it, Chemistry God

A hot air balloon works by burning fuel like propane to heat the air under an open balloon. Two things happen once the propane starts to heat up: The temperature of the gas under the balloon increases, and it starts two expand. As the volume of the gas increases the density decreases. This makes it lighter and it pushes the balloon upward, making it more buoyant. i.e the lighter air tries to rise up but is held by the inflated balloon. At a certain temperature, the pressure exerted by the light heated air pushing upwards will be enough to overcome the weight of the balloon and its passengers and lift them upwards into the air.

Charles' law can be used as a simple model to describe certain weather phenomena. When the air in the atmosphere is cold it has a lower volume. This makes the air denser. This is why it's difficult to perform physical activities outdoors during the winter. Our lungs have to exert more while breathing denser air. Another example of Charles's law is when car tires deflate during the winter and over-inflate during the summer.

## Charles' Law Examples

Here we will go through some examples of Charles' law which will allow us to test our understanding of the relevant equations.

A $$600\,\mathrm{ml}\;$$ sample of nitrogen is heated from $$10\;^\circ\mathrm C$$ to $$57\;^\circ\mathrm C$$ at constant pressure. What is the final volume?

Step 1 - Write down the given quantities:

$$V_1=600\,\mathrm{ml},\;V_2\;=?,\;T_1=10^\circ\mathrm C,\;T_2=57^\circ\mathrm C$$

Step 2 - Convert the temperature to Kelvin:

$$T_K=\;^\circ C+273\\$$ $${T_1}_K=10^\circ C+273\\$$ $${T_1}_K =283\;K\\$$ $${T_2}_K=57^\circ C+273\\$$ $${T_2}_K=330\;K$$

Step 3 Since the temperature of the gas is increasing under constant pressure. We can use the Charles law to find the final volume:

$$\frac{V_{\mathit1}}{T_{\mathit1}}\mathit=\frac{V_{\mathit2}}{T_{\mathit2}}$$ $$V_{\mathit2}\mathit\;=\;\frac{T_{\mathit2}}{T_{\mathit1}}V_{\mathit1}$$ $$V_{\mathit2}\mathit\;=\;\frac{330\;\mathrm K}{283\;\mathrm K}\times600\,\mathrm{ml}$$ $$V_{\mathit2}\mathit\;=\;700\,\mathrm{ml}$$

The final volume of the gas after the expansion is $$700\;\mathrm{ml}$$.

Always make sure that your answer makes sense at the end. for instance in the above case the temperature increases. this means that the final volume should be greater than the initial volume.

Let's look at another example.

Calculate the change in temperature when $$2\,\mathrm{l}$$ at $$21\;^\circ\mathrm C$$ is compressed to $$1\,\mathrm{l}$$.

Step 1 Write down the given quantities

$$V_{\mathbf1}=2\;\mathrm L,\;V_2\;=1\;\mathrm L,\;T_1=21\;^\circ\mathrm C,\;T_2=?$$

Step 2 Convert the temperature to Kelvins

$$T_K=^\circ C+273\\{T_K}_1=21^\circ C+273\\{T_K}_1=294\;K$$

Step 3 Since the volume of the gas is decreasing the final temperature must be lower. We can use Charles' law to find it

$$\frac{V_{\mathit1}}{T_{\mathit1}}\mathit=\frac{V_{\mathit2}}{T_{\mathit2}}$$ $$T_{\mathit2}\mathit=\frac{V_{\mathit2}}{V_{\mathit1}}T_{\mathit1}$$ $$T_{\mathit2}\mathit=\frac{1\;\mathrm{ml}}{2\;\mathrm{ml}}\times293\;\mathrm K$$ $$T_{\mathit2}\mathit=146.5\;\mathrm K$$

The final temperature of the gas after the compression is $$146.5\;\mathrm K$$ or $$-126.5\;^\circ\mathrm C\;$$

This brings us to the end of the article. Let us go through what we've learned so far.

## Charles' Law - Key takeaways

• Jacques Charles formulated a law that defined the relationship between temperature and the volume of a gas.
• The law states that at constant pressure, the volume of a fixed amount of gas is directly proportional to its temperature.
• The equation for Charles's law is given by $$\frac VT=\text{constant}$$
• The equation can also be written as $$\frac{{\mathrm V}_1}{{\mathrm T}_1}=\frac{{\mathrm V}_2}{{\mathrm T}_2}$$.
• The temperature in the above equation is always measured in degrees Kelvin.
• Gasses expand when their temperature increases due to the increase in kinetic energy of their particles.
• The working of a hot air balloon is a simple example of the application of Charles' law.

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##### Frequently Asked Questions about Charles law

What does the Charles law state?

Charles law states that at constant pressure, the volume of a fixed amount of gas is directly proportional to its temperature.

What relationship does Charles law describe?

Charles law describes the effect change in temperature on the volume of a gas at constant pressure.

What is the formula for charles law?

The formula for Charles law is given by

V1/T1=V2/T2

What is Charles Law example?

The working of a hot air balloon is a simple example of the application of Charles law.

What is the equation for charles law?

The equation for Charles law is given by

V1/T1=V2/T2

## Test your knowledge with multiple choice flashcards

What happens when the temperature of a gas increases under constant pressure?

How does the Kinetic energy of gas molecules change when the temperatures increased?

Which of the following are assumptions of Charles's law?

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