**Charles’ Law Graph**

A person is running at some speed. Let’s count the number of times the person hits the boundaries in one minute. 1,2,3,4,5,6,7,8. Now, **what happens if he is running at a speed doubling up the speed?** Now let’s count the number of times he hits the boundaries at the same time.

That is in one minute only, 1 2 3 4 5 6 7 8 9 10 11 12, 13 14 15 16. So when he was running at a speed that was double the initial speed, he has to touch the boundaries double number of times. Now we give him a condition. We say that he can touch the boundaries eight times only when he is running at double the speed. **How is that possible?**

Now he is running to double the initial speed, so if he has to touch the boundaries eight times only, that will only be possible if he covers a greater distance and how great the distance or how big the distance should it be since now he is running to double the speed.

So now he should cover double the original distance. So now he has double the distance to cover at the same time. That is in 1 minute only. Let’s count the number of times he hits the boundaries. 1,2,3,4,5,6,7,8. So if he’s running at double the speed, in order to touch the boundaries the same number of times, that is the initial number of times. He has to cover double the original distance. So let’s compare.

So now the person is running at Double the speed. So in order to cover the distance that is in order to touch the boundaries the same number of times, if he’s running at Double the speed, he has to cover a greater distance. So here you see the touch the boundaries at the same time.

So when the person is was running at Double the speed, in order to touch the boundaries the same number of times, he had to cover double the distance. So we see that the distance covered is directly proportional to speed. So greater the speed greater the distance covered, lesser the speed lesser is the distance covered given that the number of hits per unit time remains constant.

**Charles’ Law Graph**

Something like this was used by a scientist, scientist named **Charles**. *He performed an experiment known as Charles Experiment in 1787. Let’s see what he performed. He took this apparatus.*

**Charles’ Law Graph**

This has a beaker which has water inside this. There’s a thermometer, there is a scale and a capillary, a small capillary of a very small fine tube. So a capillary is attached to this scale using rubber bands. Such an apparatus in which water is present to provide heat to the other devices present inside It is known as a water bath.

What is the purpose of a water bath? Well, when we have a system like this, the purpose of a water bath is that it provides a uniform temperature throughout, since it gains a temperature. So the temperature throughout this water, that is the temperature of the thermometer temperature of the capillary and the scale. So the temperature throughout Remains the Same. So he used this device. The pressure for this experiment was maintained constant.

How was this done? We know pressure at the same height Remains the Same so this capillary tube which he had used during the experiment is at the same height throughout the experiment, since it is Tied by rubber bands it is maintained at the same height throughout the experiment.

The height of the capillary is not changed and so throughout its experienced a constant pressure. Now if you’ll observe, there is gas enclosed in the capillary.

This area shows that there is a gas which is trapped in the capillary, the remaining volume is the water, which is filled in the capillary. So there is gas enclosed in this part of the capillary. Now he started heating this water.

As he heated the temperature started to increase, and the volume of the gas in this capillary started to increase. So you observe any close the burner, he switched it off the temperature decreases and the volume of the gas also decreases.

So, what did you observe here? As he was increasing the temperature, so he was hitting the water bath the volume of the gas trapped in the capillary also started to increase. As the volume of the gas started to increase the water in the capillary tube started to fall. That is how the volume of gas in the capillary started to increase, when he switched off the Bunsen burner, he switched off the flame.

So the temperature started to decrease as the temperature decreased the level of water increased, this was because the volume of gas enclosed started to decrease as the volume in the capillary started to decrease the water level in the capillary started to increase. So from this experiment, Charles observed as the temperature of the water bath was increased at constant pressure.

Since he was observing the gas which was trapped in the capillary tube. The capillary tube was maintained at the same height throughout the experiment, So he maintained constant pressure throughout the experiment. The observed as the temperature of the bath was increased the volume of the gas enclosed in the capillary tube increased and if the temperature was decreased the volume of the gas enclosed decreased, so he observed that when we increase the temperature the volume of gas enclosed increased when it decreased the temperature, the volume of the gas enclosed decreased.

## Charles’ Law

So based on this experiment, he gave his law which is known as **Charles law**. According to which for a particular gas if the pressure is kept constant, the volume is directly proportional to temperature. This means greater the temperature greater is the volume of the gas, Lesser the temperature lesser is the volume of the gas.

So this law is known as Charles law, so this is as we had seen before for the person running. So we had observed greater as the speed of the person greater is the distance covered by the person provided the number of hits per unit time remains constant. So for Charles’s Experiment of a Charles law, what did we observe? The distance covered is the volume of the gas, The speed is the temperature. A number of hits per unit time remains constant, which is the pressure. So for Charles law, as the pressure remains constant, the volume of the gas is directly proportional to its temperature.

So let’s revisit the **Charles’ law**. It states that for a particular gas at a constant pressure, the volume of a gas is directly proportional to temperature. This means to remove the proportionality sign. We introduce a constant. So we get that the volume of a gas is equal to constant into temperature. Now, we bring the temperature on this side.

We get that the volume of a gas divided by the temperature of the gas is constant. So according to Charles law, for a particular gas volume by temperature is constant provided the pressure is constant.

So the value on the Celsius scale, We already know can be converted into the Kelvin scale, by adding **273** to it. So in short slow, whenever we talk of temperature, we always use the Kelvin value **(K)**. This Kelvin value was given by Lord Kelvin and this scale is known as the absolute scale. The Kelvin scale is also known as the absolute scale of temperature. So whenever we talk of temperature and the Charles law, we always use the Kelvin value **(K)**.

Now, let’s perform an experiment to prove the **Charles’ law**.

**Charles’ Law Graph**

So a gas is enclosed in a container. The pressure is kept constant, which you’ll observe here. Now will increase the temperature of the gas. So observe what happens! You can see through the thermometer that it is heated. So the temperature rises, as the temperature increases the volume, which is enclosed increases, or vice versa as the volume increases the temperature increases.

So you observe these values and you see that **V** by **T (V/T)**, So according to Charles law, volume by temperature is a constant. So these values that are we know that the temperature is always taken in kelvin. So all these readings are converted to Kelvin **(K)** and volume by temperature in Kelvin, that value that is **V** by **T** temperature in Kelvin is always a constant value as you can see. So from this experiment we see that **V** by **T** is always a constant value provided the pressure is constant, so according to **Charles’ law**, as the temperature decreases, volume decreases.

*How far can this volume decrease?*

We know a temperature Zero Kelvin **(0-K)** was given by Lord Kelvin.

**0 – K ➡ Absolute Zero**

which he called absolute zero **(0)** at this temperature all molecular motions sees that is the speed of the particles become zero **(0)** and at this temperature, the volume of the gas is reduced to zero **(0)**.

So this is the temperature at which the speed of the particles becomes zero **(0)** and the volume of gas is also reduced to zero **(0)**.

**Charles’ Law Graph**

So, if a graph is plotted for the Charles law, we see that volume is directly proportional to temperature.

So we get a straight line, that is as temperature increases, volume increases. Now if this value we extend backward, we see that it needs the graph at **-273°c** degrees Celsius. This is **0** Kelvin.

That is the absolute zero temperature at which the volume of all gases becomes zero **0**. So at this temperature, we see from the graph that as the temperature is reduced to zero** 0** Kelvin, the volume of the gas becomes zero. So this is the graph for **Charles’ law**, which is also known as an **“Isobar”**.

**ISO** **means same** and **bar is used for pressure**. So in this graph, since in **Charles law**, the pressure remains constant. So this shows that for the same pressure. This is the graph that is obtained for the values of volume of a gas for a particular gas versus its temperature.

*The volume of a gas increases with the increase in temperature. Is it true or false?*

**(a) True**

**(b) False**

So from **Charles’ law**, we know that the volume is directly proportional to temperature, as temperature increases, volume increases, as temperature decreases, volume decreases provided the pressure is constant.

So we have the volume of a gas increases with the increase of temperature. **“So this is true” **we see that as the temperature increases, the volume increases.

So based on the experiment that Charles had performed and the data collected, he gave his law, his law states that the volume of a given mass of air, so notice that the law is valid for a particular gas.

The volume is directly proportional to absolute temperature. We know whenever we are talking of Charles law, the temperature which is **T**.

Today for two is always the Kelvin or the absolute temperature. So the volume is directly proportional to temperature provided its pressure remains constant. So we know that the volume is directly proportional to temperature provided the pressure remains constant. So this is the law. This is the Charles law for a particular gas.

So let’s revisit the **Charles law**. It states that **V** by **T** is constant. Let’s denote this constant by **T** at a constant pressure.

Now say this is the initial volume of a gas. This is the initial temperature of a gas. So **V one** by **T one** **(V1/T1)** is some constant **k**. If this is the final volume of the gas, Volume 2, and this is the final temperature of the gas **T2**. So **V two** by** T two** is also constant. So if we combine these two, we get that **V one** by **T one** is equal to **V** two by** T two** **V2/T2**, which is the same constant **K**. So we get that **V one** by **T one** is equal to **V two** by** T two V1/T1 = V2/T2**, this means for a particular gas at a constant pressure. The volume, the initial volume of a gas at initial temperature is equal to the final volume of a gas divided by its final temperature.

## Charles’ Law based on Kinetic theory

Let’s try to study the **Charles’ law** based on the kinetic theory. So we see that there is a direct relationship between the volume and temperature of a gas.

So as the temperature increases volume increases, as temperature decreases, volume decreases. **What is happening here?** So as you see the pressure remains constant, which you can observe from this pressure cause so the pressure remains constant.

As the temperature increases, the kinetic energy of the particles increases, as the kinetic energy of the particles increases, the speed of the particles increases. As the speed increases, they strike the walls of the container more since the pressure has to be maintained constant. So if the strike the walls of the container, the increase, the volume of the container, so, with the increase in temperature, the volume increases, similarly, we get the vice-versa case.

If we decrease the temperature, the kinetic energy decreases, the speed of the particles decreases. This means the number of strikes or the number of hits per unit time has to decrease since pressure is constant. This is Possible only for a lower volume or a lesser volume. So let’s revisit. As temperature increases, the kinetic energy of the particles increases.

As the kinetic energy increases, the speed of the particles increase as the speed increases the number of hits per unit time also increases. But in Charles law, the pressure has to be kept constant. This means that pressure is constant. So this is Possible only if the volume is increased, so the increase the number of hits increases the volume for a constant pressure. Similarly, we get the vice-versa case.

If the temperature is decreased the kinetic energy decreases, the speed decreases, as the speed decreases the number of hits per unit time. Decrease since pressure has to be kept constant. So the volume decreases only when the volume decreases, the number of hits will decrease for a constant pressure. So for **Charles law** as the temperature decreases, the volume decreases.

**Now, let’s try to solve a question.**

**To what temperature must a gas at 300K be cooled in order to reduce its volume one to third (1-3) Its original volume. The pressure remaining constant?**

Since we see that the pressure remains constant. This means this law or this condition is valid for **Charles’ law**. So let’s use **Charles’ law** here. **It states that volume is directly proportional to temperature.**

Let say **V** **one** by** T one** **(V1/T1) **is equal to** V two** by** T two** **(V2/T2)**.

Let’s write the data that we are given. So we are given that the initial temperature of the gas is **300 Kelvin**.

Keep in mind whenever we are doing **Charles’ law**, we have to use the absolute or the Kelvin values, So in this, we are already given the temperature in Kelvin. So we do not have to convert it. If the temperature was given in degree celsius. We always have to convert it to the **K** value. Now we have to find the final temperature.

Let’s take the initial volume to be **V** since we’re not given any volume, and it says that the volume is reduced to **1-3**. This means the final volume is **1-3** the original volume. Now, let’s apply Charles law to this.

So, we have **V1** by **T1**.

Is equal to **V2** by **T2**

So we can cancel this **V1** is V so we can

Substitute this by **V**, as we have taken **V1** is equal to **V**, so we can cancel this **V** on both the sides.

What do we get?

And You we get **T2**.

**T2** is equal to **300**. Divided by **3**, which is equal to hundred **100k** Kelvin.

So to what temperature should it be, cooled? It should be cool. **100K**. What do You observe here? We Know by **Charles’ law** that the volume is directly proportional to temperature. So as the final volume is reduced. This was the initial volume. The final volume is reduced. So we see that the final temperature is also reduced since the initial temperature was **300K**. The final temperature is **100K**.

Here, there is a plastic bag filled with air. So now, if this plastic bag is kept in a freezer.

It’s taken out after 20 minutes.

It’s observed that the plastic bag has deflated. So as the temperature decreases, the volume of the gas decreases and if it is heated on top of a flame, so as the temperature increases, the volume of gas increases, so you observe that the plastic bag reinflates. So this is the Charles law which states that the volume of a gas is directly proportional to its temperature at a constant pressure.

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