The Equation of Boyle's Law

30th Oct 2019 @ 3 min read

The equation of Boyle's law is PV = k.

Boyle's law is a gas law discovered by Robert Boyle. It states the pressure of a gas is inversely proportional to the volume of the gas at a constant temperature and for a fixed amount of the gas.

Let the pressure of a gas be P and the volume of the gas be V. Thus, the above statement can be mathematically written as follows:

$P \propto \frac{1}{V}$

Removing the proportionality,

$P = k \times\frac{1}{V}$

Here k is a proportionality constant. Rearranging the above equation,

$PV=k$

Therefore, we can conclude the product of pressure and volume remains constant.

Boyle's law under different conditions

Let a gas moves from condition 1 (P₁, V₁) to condition 2 (P₂, V₂) and the temperature and amount of the gas remains the same.

Boyle's law from P1,V1 to P2,V2 — Condition 1 to condition 2

With the help of Boyle's law, we can establish the relationship between both conditions.

As per the law, P₁V₁ = k and P₂V₂ = k.

Combining both expressions,

$P_1V_1=P_2V_2$

The formula P₁V₁ = P₂V₂ is very useful to determine unknown pressure or volume if other variables are known.

An example will give a better understanding of it.

Example 1

A hydrogen balloon of 0.50 L at a pressure of 101.325 kPa expands isothermally to a final pressure of 80.121 kPa. Determine the final volume?

To determine the unknown volume using Boyle's law — V₂ is unknown.

Initially, P₁ = 101.325 kPa and V₁ = 0.50 L and the final volume (P₂) is 80.121 kPa.

According to the law,

$P_1V_1=P_2V_2$ $V_2 =\frac{P_1V_1}{P_2} &=\frac{101.325\,\text{kPa}\times 0.50\,\text{L}}{80.121\,\text{kPa}} \\&=0.63\,\text{L}$

Therefore, the final volume of the balloon is 0.63 L.

The unknown volume is 0.63 L. — V₂ = 0.63 L

Boyle's law equation is not limited up to two conditions but can generalised to any number of conditions. Let consider a gas under i different conditions (P₁, P₂, P₃ … P_i and V₁, V₂, V₃ … V_i).

Boyle's law for different conditions — A series of i conditions

According to the law, the product of pressure and volume remains constant.

$P_1V_1=P_2V_2=P_3V_3\,\ldots\,P_nV_n$

Example 2

In a closed system, helium gas undergoes isothermal compression from stage 1 to an intermediate stage 2 and then to the final stage 3. The temperature of the gas is maintained at 30 °C. The pressure in stage 1 and stage 2 is 1.0 atm and 2.0 atm, and the volume in stage 1 and stage 3 is 20 cm³ and 5 cm³. Determine the unknown volume in stage 2 and the unknown pressure in stage 3?

Boyle's law to determine volume and pressure — V₂ and P₃ are unknown.

Using the law,

$P_1V_1=P_2V_2=P_3V_3$

The volume in stage 2 is

$V_2=\frac{P_1V_1}{P_2}&=\frac{1.0\,\text{atm}\times 20\,\text{cm}^3}{2.0\,\text{atm}}\\&=10\,\text{cm}^3$

The pressure in stage 3 is

$P_3=\frac{P_1V_1}{V_3}&=\frac{1.0\,\text{atm}\times 20\,\text{cm}^3}{5\,\text{cm}^3}\\&=4.0\,\text{atm}$

Therefore, the unknown volume is 10 cm³ and the unknown pressure is 4.0 atm.

The unknown volume is 10 cm3 and the pressure is 4 atm. — V₂ = 10 cm³ and P₃ = 4.0 atm

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Thanks for your response!

GG Supreme
01st Jul 2022

Good wanderfull

GEROME BOTON
03rd Nov 2020

I'm so thankful for finding your explanation on Boyles law which is quite easy to understand on the concepts,real life applications etc.Kudos to the writers !God bless!