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

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.