# Boyle's Law Calculator

Created by Wojciech Sas, PhD
Reviewed by Bogna Szyk and Jack Bowater
Last updated: Jul 14, 2023

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 practical examples of Boyle's law exercises and learn how to recognize when a process satisfies Boyle's law on a graph.

In case you need to work out the results for an isobaric process, check our Charles' law calculator.

## 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. We can characterize this gas by the ideal gas equation, which you can read more about in our ideal gas law calculator. Boyle's law tells us about an isothermal process, which means that the temperature of the 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.

We can write Boyle's law formula in various ways depending on which parameter we want to estimate. 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!

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

🔎 If a transition is an isochoric process (the constant volume), you'll find Omni's Gay-Lussac's law calculator helpful.

## Boyle's law examples

We can use Boyle's law 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 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 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 measures 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 number of molecules contained in the gas. You only have to ensure 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 decreases gas pressure. As air flows from an area of higher pressure to a place of lower pressure, air enters the lungs and allows us to take in oxygen from the environment. During exhalation, the volume of the lungs decreases, so the pressure inside is higher than outside, so the air flows in the opposite direction.

• Syringe – Whenever you have 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 decreases pressure and, according to Boyle's law formula, causes the suction of the fluid.

## Other thermodynamic processes

Boyle's law, along with Charles's and Gay-Lussac's law, are among the fundamental laws describing the vast majority of thermodynamic processes.

Other than working out the values of specific 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 combined gas law calculator, where you can choose whichever process you like and evaluate the outcomes for real gases.

## FAQ

### Why is Boyle's law also called isotherm?

Boyle's law is one of the three fundamental thermodynamic processes. In each of them, we study a variation of two out of three quantities:

• The pressure;
• The temperature; and
• The volume.

The third quantity remains constant during the process. In the case of Boyle's law, we don't change the temperature, thus we call the process isothermal.

### How much will a balloon with initial volume 1000 cm³ expand at cruising altitude?

1. Find the initial pressure. We will take atmospheric pressure at sea level: Pi = 1 atm = 101,325 Pa.
2. Find the final pressure. In a cruising plane, the cabin is usually pressurized at about Pf = 0.8 atm = 81,060 Pa.
3. Calculate the final volume with Boyle's law: Vf = Pi · Vi/Pf = (101,325 Pa · 0.001 m3)/81,060 Pa = 0.00125 m3.
4. Find the expansion by subtracting the final and initial volumes : ΔV = Vf - Vi = (0.00125 - 0.001) m3 = 0.00025 m3 = 250 cm3.

### How do I calculate Boyle's law?

To calculate Boyle's law, we need to perform a few simple steps, which depend on the initial data we know. To calculate the final pressure given the initial volume and pressure and the final volume:

1. Compute the product of the initial volume and pressure: Vi · Pi.
2. Divide the result by the final volume. The final pressure is Pf = (Vi · Pi)/Vf.
• You can invert final and initial values freely (reversible transformation).
• To find the final volume, switch it with the final pressure on the right-hand side of the formula.

### What is the final pressure if the volume reduces by half with intial pressure 1 atm?

The final pressure in a process where the volume reduces by half, starting from Pi = 1 atm, is 2. To find this result:

1. Write Boyle's law for the final pressure: Pf = (Vi · Pi)/Vf.
2. In this formula, identify the ratio of the volumes. Since we know that the volume reduces by half, we can write Vi = 2 · Vf, hence Vi/Vf = 2.
3. The final pressure is, then: Pf = 2 · Pi = 2 atm.
Wojciech Sas, PhD
Initial parameters
Initial pressure (p₁)
Pa
Initial volume (V₁)
Final parameters
Final pressure (p₂)
Pa
Final volume (V₂)
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