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The Equation of Boyle's Law

30th Oct 2019 @ 3 min read

Physical Chemistry

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∝1⁄V

Removing the proportionality,

P=k×1⁄V

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

Boyle's law 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,

Boyle's law formula, P1V1=P2V2

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,

P1V1=P2V2

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 … Pi and V1, V2, V3 … Vi).

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

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

P1V1=P2V2=P3V3 … PiVi

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,

P1V1=P2V2=P3V3

The volume in stage 2 is

The pressure in stage 3 is

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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