Initial parameters
Initial pressure (p₁)
Pa
Initial volume (V₁)
Final parameters
Final pressure (p₂)
Pa
Final volume (V₂)

# Boyle's Law Calculator

By Wojciech Sas, PhD candidate

This Boyle's law calculator is a great tool when you need to estimate the parameters of a gas in an isothermal process. You will find the answer to "What is Boyle's law?" in the text, so read on to find out about the Boyle's law formula, see some useful examples of Boyle's law exercises and learn how to recognize when a process satisfies Boyle's law on a graph.

## Boyle's law definition

Boyle's law (also known as Boyle-Mariotte law) tells us about the relationship between the pressure of a gas and its volume at a constant temperature and mass of gas. It states that the absolute pressure is inversely proportional to the volume.

Boyle's law definition can also be phrased in the following way: the product of the pressure and the volume of a gas in a closed system is constant as long as the temperature is unchanged.

Boyle's law describes the behavior of an ideal gas during an isothermal process, which means that the temperature of gas remains constant during the transition, as does the internal energy of the gas.

## Boyle's law formula We can write the Boyle's law equation in the following way:

`p₁ * V₁ = p₂ * V₂`,

where `p₁` and `V₁` are initial pressure and volume respectively. Similarly, `p₂` and `V₂` are the final values of these gas parameters.

Depending on which parameter we want to estimate, Boyle's law formula can be written in various ways. Let's say we change the volume of a gas under isothermal conditions, and we want to find the resulting pressure. Then, the equation of Boyle's law states that:

`p₂ = p₁ * V₁ / V₂` or `p₂ / p₁ = V₁ / V₂`.

As we can see, the ratio of the final and initial pressure is the inverse of the ratio for volumes. This Boyle's law calculator works in any direction you like. Just insert any three parameters, and the fourth one will be calculated immediately!

The whole process can be visualized on a Boyle's law graph. The most commonly used type is where the pressure is a function of the volume. For this process, the curve is a hyperbola. The transition can progress in both ways, so both compression and expansion of the gas satisfy Boyle's law.

## Boyle's law examples

Boyle's law can be used in several ways, so let's take a look at some examples:

1. Imagine that we have an elastic container that holds a gas. The initial pressure is `100 kPa` (or `10⁵ Pa` if we use scientific notation) and the volume of the container equals `2 m³`. We decide to compress the box down to `1 m³`, but we don't change the overall temperature. The question is: "How does the pressure of the gas change?". We can use Boyle's law formula:

`p₂ = p₁ * V₁ / V₂ = 100 kPa * 2 m³ / 1 m³ = 200 kPa`.

After halving the volume, the internal pressure is doubled. This is a consequence of the fact that the product of the pressure and the volume must be constant during this process.

2. The next Boyle's law example concerns a gas under `2.5 atm` of pressure while occupying `6 liters` of space. It is then decompressed isothermally to the pressure of `0.2 atm`. Let's find out its final volume. We have to rewrite the Boyle's law equation:

`V₂ = p₁ * V₁ / p₂ = 2.5 atm * 6 l / 0.2 atm = 75 l`.

You can always use our Boyle's law calculator to check if your evaluations are correct!

## Where is Boyle's law applied?

Boyle's law describes all processes for which temperature remains constant. In thermodynamics, temperature is a measure of the average kinetic energy that atoms or molecules have. In other words, we can say that the average velocity of gas particles doesn't change during that transition. Boyle's law formula is valid for a wide range of temperatures.

In `advanced mode` you can choose any temperature you like, and we will calculate the amount of molecules contained in the gas. You only have to make sure that the substance remains in the gas form (e.g. neither condensates nor crystallizes) at this temperature.

There are a few areas where Boyle's law is applicable:

• Carnot Heat Engine - consists of four thermodynamic processes, two of which are isothermal ones, satisfying Boyle's law. This model can tell us what the maximal efficiency of a heat engine is.

• Breathing also can be described by Boyle's law. Whenever you take a breath, your diaphragm and intercostal muscles increase the volume of your lungs, which results in the decrease of the gas pressure. As air flows from an area of higher pressure to an area of lower pressure, air enters the lungs and allows us to take in oxygen from the environment. During exhalation, the volume of lungs decreases, so the pressure inside is higher than outside, so the air flows in the opposite direction.

• Syringe - whenever you have to make an injection, a doctor or a nurse draws a liquid from the small vial first. To do so, they use a syringe. By pulling the plunger, the accessible volume increases, which results in the decrease of pressure, and, according to Boyle's law formula, causing the suction of the fluid.

## Other thermodynamic processes

Boyle's law, together with Charles's law and Gay-Lussac's law, are among the fundamental laws which describe the vast majority of thermodynamic processes. Other than working out the values of certain parameters like pressure or volume, it is also possible to discover something about the heat transfer and the work done by the gas during these transitions, as well as the internal energy change. We have gathered them all in our thermodynamic processes calculator, where you can choose whichever process you like, and evaluate the outcomes for a real gases.

Wojciech Sas, PhD candidate