CHARLES'S LAW: FORMULAS AND UNITS, EXPERIMENT, EXERCISES - CHEMISTRY - 2024

The law of Charles or Guy-Lussac is one that allows the statement of one of the properties of the gaseous state: the volume occupied by a gas is directly proportional to the temperature at constant pressure.

This proportionality is linear for all temperature ranges if the gas in question is ideal; real gases, on the other hand, deviate from the linear trend at temperatures near their dew point. However, this has not limited the use of this law for a myriad of applications involving gases.

Chinese lanterns or wish balloons. Source: Pxhere.

One of the quintessential applications of Charles' law is in air balloons. Other simpler balloons, such as wish balloons, also called Chinese lanterns (top image), reveal the relationship between volume and temperature of a gas at constant pressure.

Why at constant pressure? Because if the pressure were to increase, it would mean that the container where the gas is located is hermetically sealed; and with this, the collisions or impacts of the gaseous particles against the internal walls of said container would increase (Boyle-Mariotte law).

Therefore, there would be no change in the volume occupied by the gas, and Charles's law would be lacking. Unlike an airtight container, the fabric of the wish balloons represents a mobile barrier, capable of expanding or contracting depending on the pressure exerted by the gas inside.

However, as the balloon tissue expands, the internal pressure of the gas remains constant because the area over which its particles collide increases. The higher the temperature of the gas, the higher the kinetic energy of the particles, and therefore the number of collisions.

And as the balloon expands again, the collisions against its internal walls remain (ideally) constant.

So the hotter the gas, the greater the expansion of the balloon and the higher it will rise. The result: reddish (albeit dangerous) lights suspended in the sky on December nights.

What is Charles's law?

Statement

The so-called Charles's Law or Gay-Lussac's Law explains the dependence that exists between the volume occupied by a gas and the value of its absolute temperature or Kelvin temperature.

The Law can be stated in the following way: if the pressure remains constant, it is satisfied that “for a given mass of a gas, it increases its volume by approximately 1/273 times its volume at 0 ºC, for each degree centigrade (1 ºC) to increase its temperature ”.

Jobs

The research work that established the law was initiated in the 1780s by Jacques Alexander Cesar Charles (1746-1823). However, Charles did not publish the results of his investigations.

Later, John Dalton in 1801 managed to determine experimentally that all gases and vapors, studied by him, expand between two determined temperatures in the same volume quantity. These results were confirmed by Gay-Lussac in 1802.

The research works of Charles, Dalton and Gay-Lussac, allowed to establish that the volume occupied by a gas and its absolute temperature are directly proportional. Therefore, there is a linear relationship between the temperature and the volume of a gas.

Graph

Graph of T vs V for an ideal gas. Source: Gabriel Bolívar.

Graphing (top image) the volume of a gas against temperature produces a straight line. The intersection of the line with the X axis, at a temperature of 0ºC, allows obtaining the volume of the gas at 0ºC.

Likewise, the intersection of the line with the X axis would give information about the temperature for which the volume occupied by the gas would be zero "0". Dalton estimated this value at -266 ° C, close to Kelvin's suggested value for absolute zero (0).

Kelvin proposed a temperature scale whose zero should be the temperature at which a perfect gas would have a volume of zero. But at these low temperatures the gases are liquefied.

That is why it is not possible to speak of volumes of gases as such, finding that the value for absolute zero should be -273.15 ºC.

Formulas and units of measurement

Formulas

Charles's law in its modern version states that the volume and temperature of a gas are directly proportional.

So:

V / T = k

V = volume of gas. T = Kelvin temperature (K). k = constant of proportionality.

For a volume V ₁ and a temperature T ₁

k = V ₁ / T ₁

Likewise, for a volume V ₂ and a temperature T ₂

k = V ₂ / T ₂

Then, equating the two equations for k we have

V ₁ / T ₁ = V ₂ / T ₂

This formula can be written as follows:

V ₁ T ₂ = V ₂ T ₁

Solving for V ₂, the formula is obtained:

V ₂ = V ₁ T ₂ / T ₁

Units

The volume of the gas can be expressed in liters or in any of its derived units. Likewise, the volume can be expressed in cubic meters or in any derived unit. The temperature must be expressed in absolute temperature or Kelvin temperature.

So, if the temperatures of a gas are expressed in degrees centigrade or the Celsius scale, to perform a calculation with them, the amount of 273.15 ºC would have to be added to the temperatures, in order to bring them to absolute temperatures or kelvin.

If the temperatures are expressed in degrees Fahrenheit, 459.67 ºR would have to be added to those temperatures, to bring them to absolute temperatures on the Rankine scale.

Another well-known formula of Charles's Law, and directly related to its statement, is the following:

V _t = V _or (1 + t / 273)

Where V _t is the volume occupied by a gas at a certain temperature, expressed in liters, cm ³, etc.; and V _o is the volume occupied by a gas at 0 ºC. For its part, t is the temperature at which the volume is measured, expressed in degrees centigrade (ºC).

And finally, 273 represents the value of absolute zero on the Kelvin temperature scale.

Experiment to prove the law

Mounting

Setting up the experiment to demonstrate Charles's law. Source: Gabriel Bolívar.

In a container of water, which fulfilled the function of a water bath, an open cylinder was placed at the top, with a plunger that fitted to the inner wall of the cylinder (upper image).

This piston (made up of the piston and the two black bases) could move towards the top or bottom of the cylinder depending on the volume of gas it contained.

The water bath could be heated by using a burner or heating plant, which supplied the necessary heat to increase the temperature of the bath and, therefore, the temperature of the cylinder equipped with a plunger.

A determined mass was placed on the plunger in order to ensure that the experiment was carried out at constant pressure. The temperature of the bath and the cylinder was measured by using a thermometer placed in the water bath.

Although the cylinder probably did not have a graduation to display the volume of air, this could be estimated by measuring the height reached by the mass placed on the piston and the surface of the cylinder base.

Development

The volume of a cylinder is obtained by multiplying the surface area of ​​its base by its height. The surface of the cylinder base could be obtained by applying the formula: S = Pi xr ².

While the height is obtained by measuring the distance from the base of the cylinder, to the part of the piston on which the mass rests.

As the temperature of the bath was increased by the heat produced by the lighter, the plunger was observed to rise within the cylinder. Then, they read on the thermometer the temperature in the water bath, which corresponded to the temperature inside the cylinder.

They also measured the height of the mass above the plunger, being able to estimate the volume of air that corresponded to the measured temperature. In this way they made several measurements of the temperature and estimates of the volume of air corresponding to each of the temperatures.

With this, it was finally possible to establish that the volume that a gas occupies is directly proportional to its temperature. This conclusion allowed to enunciate the so-called Charles Law.

Balloon with ice in winter

In addition to the previous experiment, there is a simpler and more qualitative one: that of the balloon with ice in winter.

If a helium-filled balloon were placed in a heated room in winter, the balloon would have a certain volume; But, if it were later moved outside the house with a low temperature, it would be observed that the helium balloon shrinks, reducing its volume according to Charles's Law.

Solved exercises

Exercise 1

There is a gas that occupies a volume of 750 cm ³ at 25 ºC: what will be the volume that this gas occupies at 37 ºC if the pressure is kept constant?

It is necessary first to transform the units of temperature to kelvin:

T ₁ in degrees Kelvin = 25 ºC + 273.15 ºC = 298.15 K

T ₂ in degrees Kelvin = 37 ºC + 273.15 ºC = 310.15 K

Since V ₁ and the other variables are known, V ₂ is solved and calculated with the following equation:

V ₂ = V ₁ · (T ₂ / T ₁)

= 750 cm ³ (310.15 K / 298.15 K)

= 780.86 cm ³

Exercise 2

What would be the temperature in degrees centigrade to which 3 liters of a gas would have to be heated to 32ºC, so that its volume expands to 3.2 liters?

Again, the degrees centigrade are transformed into kelvin:

T ₁ = 32 ºC + 273.15 ºC = 305.15 K

And as in the previous exercise, we solve for T ₂ instead of V ₂, and it is calculated below:

T ₂ = V ₂ · (T ₁ / V ₁)

= 3,2 L · (305,15 K / 3 L)

= 325.49 K

But the statement asks for degrees centigrade, so the unit of T ₂ is changed:

T ₂ in degrees centigrade = 325, 49 º C (K) - 273.15 ºC (K)

= 52.34 ºC

Exercise 3

If a gas at 0ºC occupies a volume of 50 cm ³, what volume will it occupy at 45ºC?

Using the original formula of Charles's law:

V _t = V _or (1 + t / 273)

We proceed to calculate V _t directly when all the variables are available:

V _t = 50 cm ³ + 50 cm ³ · (45 ºC / 273 ºC (K))

= 58.24 cm ³

On the other hand, if the problem is solved using the strategy of examples 1 and 2, we will have:

V ₂ = V ₁ · (T ₂ / T ₁)

= 318 K · (50 cm ³ /273 K)

= 58.24 cm ³

The result, applying the two procedures, is the same because ultimately they are based on the same principle of Charles's law.

Applications

Wish balloons

The wish balloons (already mentioned in the introduction) are provided with a textile material impregnated with a combustible liquid.

When this material is set on fire, there is an increase in the temperature of the air contained in the balloon, which causes an increase in the volume of the gas according to Charles's law.

Therefore, as the volume of the air in the balloon increases, the density of the air in the balloon decreases, which becomes less than the density of the surrounding air, and that is why the balloon rises.

Pop-Up or turkey thermometers

As their name indicates, they are used during the cooking of turkeys. The thermometer has an air-filled container closed with a lid and is calibrated in such a way that upon reaching the optimum cooking temperature, the lid is lifted with a sound.

The thermometer is placed inside the turkey, and as the temperature inside the oven increases, the air inside the thermometer expands, increasing its volume. Then when the volume of the air reaches a certain value, he makes the lid of the thermometer lift.

Restoring the shape of ping-pong balls

Ping-pong balls, depending on the requirements of their use, are light in weight and their plastic walls are thin. This causes that when they are hit by the rackets they suffer deformations.

By placing the deformed balls in hot water, the air inside heats up and expands, leading to an increase in air volume. This also causes the wall of the ping-pong balls to stretch, allowing them to return to their original shape.

Bread making

Yeasts are incorporated into wheat flour that is used to make bread and have the ability to produce carbon dioxide gas.

As the temperature of the loaves increases during baking, the volume of carbon dioxide increases. It is because of this that the bread expands until it reaches the desired volume.

References

1. Clark J. (2013). Other gas laws - Boyle's Law and Charles' Law. Recovered from: chemguide.co.uk
2. Staroscik Andrew. (2018). Charles' Law. Recovered from: scienceprimer.com
3. Wikipedia. (2019). Charles Law. Recovered from: en.wikipedia.org
4. Helmenstine, Todd. (December 27, 2018). What Is the Formula for Charles' Law? Recovered from: thoughtco.com
5. Prof. N. De Leon. (sf). Elementary Gas Laws: Charles Law. C 101 Class Notes. Recovered from: iun.edu
6. Briceño Gabriela. (2018). Charles Law. Recovered from: euston96.com
7. Morris, JG (1974). Physicochemistry for biologists. (2 ^da edition). Editorial Reverté, SA