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Boyle's Law: Formula, Derivation & Real-World Applications

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How Does Boyle's Law Describe the Relationship Between Pressure and Volume?

Boyle's Law also known as the Boyle–Mariotte law, as it is known in France, is an experimental gas law which explains how the pressure of a gas has a relationship to its volume. It tells us how the pressure of gas tends to increase as the volume of the container decreases. To put it in the modern scientific form it may be stated as follows: The absolute pressure exerted by a given mass of an ideal gas is inversely proportional to the volume it resides in if the amount of gas and temperature stay unchanged surrounded by a closed system.


Here we must understand what the definition of an ideal gas is, what do we mean by a closed system and the relationship of the word temperature. A theoretical gas made up of lots of arbitrarily moving point particles whose only interactions are completely elastic collisions is known as an ideal gas. The ideal gas theory is useful because it obeys the ideal gas law, a simplified equation of state, and is modifiable to analysis under statistical mechanics.

 

But, this relationship between pressure and volume was first documented by Richard Townley and Henry Power in the 17th century. Robert Boyle conducted many experiments compiled and confirmed their data and published the results. Robert Hooke, who was Boyle's assistant built the experimental apparatus as opined by Robert Gunther and other authorities.

This law is based on experiments with air, which Boyle assumed to be a fluid of fine particles which are at rest in between small invisible springs. The common belief in those times was that Air was one of the four elements, but Boyle was not in alliance with this concept. Boyle's firmly believed that air was an essential element of life; and, he published works on the growth of plants without air.

 

Using a closed J-shaped tube and Boyle poured mercury from one side, and thus he forced the air, on the other hand, to contract under the pressure of mercury. He repeated this procedure of the experiment several times and using different amounts of mercury, and after several iterations, he found that under controlled conditions, the pressure of a gas is inversely proportional to the volume occupied by it.

 

Edge Mariotte (1620–1684) the French physicist discovered the same law independently of Boyle in 1679, but Boyle had already published it in 1662. Mariotte did, however, discover one different fact that the volume of air changes with temperature. This is the reason why this law is sometimes referred to as Mariotte's law or the Boyle–Mariotte law.

 

Newton in the year 1687 published it in the Philosophiæ Naturalis Principia Mathematica, wherein he showed and proved mathematically that in an elastic fluid consisting of particles which are at rest, and between which are repulsive forces inversely proportional to their distance, the density would be directly proportional to the pressure, But this mathematical treatise is not the physical explanation for the observed relationship. Instead of static theory, a kinetic theory was needed, which was provided two centuries later when Maxwell and Boltzmann came up with proof of theories.

 

Boyle's law was the first physical law to be expressed in the form of an equation describing the dependence of two variable quantities.

 

Boyle's law can be expressed mathematically as

P α 1/V, i.e. to say the Pressure is inversely proportional to the volume

OR

P x V = k which is to say that if we multiply the pressure by the volume, we get a constant

where P is the pressure of the gas, V is the volume of the gas, and k is a constant.

 

The equation states that if the temperature is constant, the product of pressure and volume is a constant for a given mass of confined gas and this holds.

 

Now, for comparing the same substance under two different sets of conditions, the law can be usefully expressed as

P1V1 = P2V2

 

The expression shows that, when the volume increases, the pressure of the gas decreases in proportion. Similarly, as the volume decreases, the pressure of the gas increases. Robert Boyle published the original law in 1662 and hence the law was named after him.

 

This expression means that if the temperature remains constant the same amount of energy given to the system persists throughout its operation and therefore, theoretically, the value of k will stay constant. since the pressure is derived as a perpendicular applied force and the probabilistic likelihood of collisions with other particles using collision theory, the application of power to a surface might not be infinitely constant for such values of v but will have a limit when differentiating such values over a given time. Forcing the volume V of the fixed quantity of gas to increase, keeping the gas at the initially measured temperature, the pressure p should decrease proportionally. In other words, to put it conversely, reducing the volume of the gas increases the pressure. Boyle's law is used to forecast the consequence of introducing a change, only in pressure and volume, to the initial state of a fixed quantity of gas.

 

This is the principle utilized and applied in our daily lives even when we cook food using a pressure cooker. The food says rice gets cooked when the water boils. We all live on the planet Earth under atmospheric pressure which is taken as the weight of 13mm of Mercury, Water to cook the rice boils only at 100 Degrees Celsius at atmospheric pressure. Now using this principle water is a heavy thick metal pressure cooker is made to boil at a higher temperature by increasing the weight or causing the pressure inside the cooking vessel to build up quickly. This, in turn, causes the water to attain a boiling point Higher than 100Degree Celsius quickly, and the rice also gets cooked in a much shorter time than in an open vessel.

 

We are creating a closed pressure system when we lock the lid onto the cooker because we are sealing the pot shut. Going back to the Ideal Gas Law and the equation PV=nRT, where P = pressure, V= volume and T = temperature (for completeness, n= the number of moles and R = is the gas constant, but they will not change in this case so we can ignore them, and the volume of the metal pot also will not really change, we will ignore this too). Hence this equation applies to a closed system, and this is the state inside the pressure cooker.

 

So basically here, we have an equation of P=T. If you increase the pressure, the temperature will also increase, and vice versa.

In an open or uncovered pot which is not tightly closed, water boils at 100ºC (212ºF) because it is exposed to the atmospheric pressure. The steam that evaporates from this pot is also at 100ºC (212ºF). Irrespective of the heat supplied to the water, it will remain at 100ºC (212ºF). If the cover is put on the pot to seal it tight and trap the steam inside the vessel, the pressure inside the pot goes up. As the pressure rises, the steam inside the pressurized pot and the temperature of the water also increases over the normal 100ºC (212ºF) boiling point temperature.

 

So, when we begin heating the cooker, the internal pressure will increase, and as the pressure increases, the internal temperature also rises. This will continue to increase until it reaches the trigger pressure of the safety valves, about double the psi (Pressure reading) at sea level. Generally, for most chemical reactions, for every 10 deg C increase, the rate of reaction is doubled, which allows us to reduce the cook time in half. By raising the pressure by ~15 psi, you can braise your food at ~250 deg F (121 deg C) and cut your cook time to a quarter of the standard time. However, you also must factor in the time it will take to come up to pressure as well as the cooldown time, so it ends up being about one third the original time, still a huge reduction. This is the reason why pressure cookers cook the food very quickly.


FAQs on Boyle's Law: Formula, Derivation & Real-World Applications

1. What is Boyle's Law?

Boyle's Law is a fundamental gas law in Physics which states that for a fixed mass of an ideal gas kept at a constant temperature, the pressure is inversely proportional to the volume. In simpler terms, if you increase the pressure on a gas, its volume will decrease, and vice-versa, provided the temperature does not change.

2. What is the mathematical formula for Boyle's Law, and what do the variables represent?

The mathematical representation of Boyle's Law is P ∝ 1/V, which means Pressure (P) is proportional to the inverse of Volume (V). This can be written as an equation: PV = k, where 'k' is a constant. For comparing the same gas under two different conditions at the same temperature, the formula is written as P₁V₁ = P₂V₂. Here, P₁ and V₁ are the initial pressure and volume, and P₂ and V₂ are the final pressure and volume.

3. What are some common real-world applications of Boyle's Law?

Boyle's Law has numerous real-world applications that we encounter daily. Some key examples include:

  • Human Breathing: When we inhale, our diaphragm and rib cage muscles increase the volume of our lungs, which decreases the internal pressure. This causes outside air (at a higher pressure) to rush in. The opposite happens when we exhale.

  • Syringes: Pulling the plunger of a syringe increases the volume inside, which decreases the pressure, drawing liquid into the barrel. Pushing the plunger decreases the volume, increases the pressure, and expels the liquid.

  • Aerosol Cans: The contents of a spray can are under high pressure. When the nozzle is pressed, the volume expands suddenly, causing a sharp drop in pressure that propels the contents out as a fine mist.

  • Scuba Diving: As a diver descends, the increasing water pressure compresses the air in their lungs and equipment. They must exhale during ascent to release this expanding air safely, preventing lung over-expansion injuries.

4. How is the relationship in Boyle's Law represented graphically?

The relationship between pressure and volume in Boyle's Law is represented by a graph called an isotherm (a curve at constant temperature). When plotting Pressure (P) on the y-axis and Volume (V) on the x-axis, the graph is a rectangular hyperbola. This curve shows that as volume increases, pressure decreases non-linearly. However, if you plot Pressure (P) against the inverse of Volume (1/V), you get a straight line passing through the origin, which clearly demonstrates the direct proportionality between P and 1/V.

5. Why must temperature and the amount of gas remain constant for Boyle's Law to apply?

The conditions of constant temperature and mass are crucial for Boyle's Law. Here's why:

  • Constant Temperature: Temperature is a measure of the average kinetic energy of gas particles. If the temperature increases, particles move faster and collide with the container walls more frequently and forcefully, which increases pressure, even if the volume is unchanged. To isolate the relationship between only pressure and volume, the kinetic energy of the particles must be kept constant.

  • Constant Mass (Amount of Gas): The amount of gas determines the number of particles in the container. If you add more gas (increase the mass), you increase the number of particles, which will lead to more collisions and a higher pressure at the same volume. Therefore, the number of gas particles must be fixed to observe the inverse P-V relationship.

6. How does Boyle's Law differ from Charles's Law?

Boyle's Law and Charles's Law both describe the behaviour of ideal gases, but they focus on different relationships. The key difference lies in which variable is held constant.

  • Boyle's Law describes the relationship between pressure and volume (P and V) when the temperature is constant. It is an inverse relationship (P₁V₁ = P₂V₂).

  • Charles's Law describes the relationship between volume and temperature (V and T) when the pressure is constant. It is a direct relationship (V₁/T₁ = V₂/T₂).

7. What are the limitations of Boyle's Law?

Boyle's Law is an ideal gas law and works very well under certain conditions, but it has limitations. The law assumes that gas particles themselves have no volume and that there are no intermolecular forces of attraction between them. Therefore, Boyle's Law is most accurate at low pressures and high temperatures. It becomes less accurate at very high pressures (when the volume of gas particles becomes significant compared to the container volume) and very low temperatures (when intermolecular forces become strong enough to cause the gas to liquefy).

8. How does the Kinetic Theory of Gases explain Boyle's Law?

The Kinetic Theory of Gases provides a microscopic explanation for Boyle's Law. According to this theory, gas pressure is caused by the collisions of gas particles with the walls of the container. If you decrease the volume of the container while keeping the temperature constant, the gas particles become more crowded. This means they will collide with the container walls more frequently, leading to an increase in pressure. Conversely, increasing the volume gives particles more space, reducing the frequency of collisions and thus lowering the pressure.