Boyle's Law

Boyle's Law is a very important gas law in chemistry and physics. It is also known as Mariotte's law or the Boyle-Mariotte law. The law is the oldest gas law. Boyle's law along with Charles’s law, Gay-Lussac’s law, and Avogadro’s law forms the ideal gas law. The law correlates how the pressure of a gas increases with a decrease in the volume of the gas.

Statement of Boyle's Law

Boyle's law states for a fixed amount of an enclosed ideal gas, its absolute pressure is inversely proportional to its volume at a constant temperature.

Formula

The equation of Boyle's law is given below.

Experiment by Boyle

Like many gas laws, the law was discovered empirically.

In the 17th century, Irish chemist Robert Boyle along with his assistant Robert Hooke conducted a series of experiments to study the elasticity of air which he called the “spring of air”. At the time, scientists lack the understanding of air. Air was considered as one of the four elements of nature; the others were earth, fire, and water. Boyle's interest in the study of air led him to the pressure-volume relationship, which later became the law.

The experiment setup included a J-shaped tube similar to the figure below.

The shorter end of the J-shaped tube was closed or sealed while the other end was opened to the atmosphere. On the opened end of the tube, he poured mercury. This cause the compression of the air on the closed end of the tube. The compression of the air is due to the force exerted by the weight of mercury. The same procedure was repeated for different amounts of mercury in the tube and corresponding volumes (the volume of the compressed air in the shorter end in arbitrary units) and pressures (the height difference of mercury between the two ends of the tube) were recorded.

Observation

The table below is from Boyle's original data.

All the above observations were reported in his book “A Defence of the Doctrine Touching the Spring and Weight of the Air” published in 1662. From the experiment, he concluded that the pressure of a gas is inversely proportional to its volume. The below graph is produced from Boyle's original data.

Note: In 1679 French physicist Edme Mariotte had established the same relationship independent of Boyle. Hence, the law is also known as Mariotte’s law or the Boyle-Mariotte law.

Explanation

As the law states pressure and volume are inversely proportional at a constant temperature for a given mass of an ideal gas. The statement can mathematically be expressed as:

The above expression can be rearranged as:

where k is a constant of proportionality.

The above expression is valid for a given mass of an ideal gas and at a constant temperature. Hence as the volume increases, the pressure of the gas decreases and as the volume decreases, the pressure increases.

If P₁ and P₂ are the pressures at volumes V₁ and V₂ respectively at a constant temperature for a given amount of an ideal gas, then from Boyle's law,

From the above expression, it is clear that the product of the pressure and the volume of an ideal gas is constant at a constant temperature. When the pressure of a gas at a constant temperature is double, the volume reduces to half of the initial volume. This is explained as below:

Graphical Representation of Boyle's Law

The graphical representation of Boyle's law can be demonstrated in the three graphs which are explained below.

As can be observed from the above graph, the pressure of an ideal gas decreases with an increase in the volume. The above curve is hyperbolic in nature. It can also be observed that the curve shifts up with an increase in temperature. Each curve is at a constant temperature, which is called an isotherm.

The graph of the pressure vs the inverse of the volume is a straight line passing through the origin. The slope of the line is the proportionality constant, k.

The product of the pressure and the volume is always constant at a constant temperature for an ideal gas. This is observed from the above figure. Therefore, the product of the pressure and the volume is independent of the volume (or the pressure) at a constant temperature for an ideal gas.

We can also plot the logarithmic graphs of Boyle's law equation. The equation is as followed:

Taking logarithm on both sides,

Deriving Boyle's Law from Ideal Gas Law

We can also derive the Boyle's law from the ideal gas law. The famous ideal gas equation is mentioned hereunder.

where n is the amount of gas (or moles of gas), R is the ideal gas constant, and T is the temperature.

Boyles's law has two primary conditions: constant temperature and a fixed amount of gas. Thus, T and n are constant. In the above ideal gas equation, the right-hand side of the equation, all the variables (n, R, and T) are constant. The product (nRT) of these three constant is also constant. So, we can replace nRT by the constant named k.

Finally, the equation becomes PV=k, which is nothing but Boyle's equation.

Limitation of Boyle's Law

The limiaion are as follows:

• Boyle's law is applicable only to ideal gases.
• The law holds good only at high temperatures and low pressures.
• The law fails at high pressures. The product of the pressure and volume does not remain constant at high pressures but shows a slight increase. This increase is due to an increase in the volume which is caused by repulsive forces among the molecules. At high pressures, the molecules are too slow to one another. Repulsive forces become dominant as compactness increases. This results in the expansion of the gas.

Real-World Applications of Boyle's Law

There are numerous applications of Boyle's law. Some can be observed in day to day life. A few of them are mentioned below.

Human Lungs

During the inhalation cycle of the breathing system, the lungs expand. This expansion is nothing but increase in the volume, which as per Boyle's law causes the reduction in the pressure relative to outside. Thus, air flows inside the body due to the pressure difference. By the same analogy, during the exhalation process, the volume decreases, which results in an increase in the pressure relative to the outside. And air flows from the body to outside.

Syringe

When the plunger of a syringe is pulled outside, there is the creation of vacuum i.e. a low-pressure region (due to increase in the volume) compare to outside and fluid gets suck in the syringe.

Soda Cans

As we all know when the cap of a soda can is opened, the pressurised gas inside the can expands, which results in a decrease in pressure of the gas.

Hand Bicycle Pump

When the piston of a hand pump is pushed downwards, the air inside the pump compresses i.e. the volume reduces. This reduced volume increases the pressure and the air is pumped into tires.

Scuba Divers

When a scuba diver dives beneath the water surface due to the hydrostatic pressure on divers, the air inside their lungs contracts. As a diver approaches the surface, the air inside their lungs expands since the pressure decreases on the surface of the water.

Examples

Example 1: To Calculate Expanded Volume of Methane

Statement: Consider a fixed amount of methane gas at pressure 2 × 10⁵ N m⁻² of volume 2.0 m³ goes under expansion such that the initial and final temperature remains the same. The new pressure of the gas is 1 ×10⁵ N m⁻². Calculate the new volume?

Solution: As from Boyle's law at a constant temperature and for a given mass of gas,

where P₁, V₁ and P₂, V₂ are the initial and final pressures and volumes.

Therefore, the new volume after the expansion is 4 m³.

Example 2: To Calculate New Pressure after Compression

Statement: A storage tank A of volume 750 dm³ contains a non-reactive gas at an absolute pressure of 4.0 bar. The gas is pumped into another storage tank B of volume 300 dm³. Assume the temperatures of the gas inside both the tanks are the same, and there is no loss of the gas during the transportation. Calculate the new pressure of the gas?

Solution: Since the initial and final temperature are the same, and there is complete conservation of the mass of the gas during transportation, we can apply Boyle's law as:

where P_A, V_A and P_B, V_B are the pressures and volumes of storage tank A and storage tank B respectively.

The pressure inside the storage tank B is 1.6 bar.

Example 3: To Determine Volume of Hydrogen

Statement: Hydrogen gas from four identical cylinders is charged inside a vessel. The absolute pressure inside each cylinder is 12 bar. The volume of each cylinder is 30 dm³. The reading of the pressure gauge mounted on the vessel after charging of hydrogen gas is 3.3 barg. Calculate the volume of the vessel using Boyle's law?

Solution: First, calculate the total initial volume which is the addition of all the volumes of the cylinders.

The final pressure is given in gauge, which needs to be converted into the absolute pressure. Absolute pressure is gauge pressure plus atmospheric pressure. Taking the atmospheric pressure as 1.0 bar,

Assume the temperature remains constant in both systems. From Boyle's law,

Therefore, the volume of the vessel (which the volume of gas occupied) is 33 dm³.

Example 4: To Determine Volume of Nitrogen

Statement: For a given mass of nitrogen gas, the initial volume is 15 dm³ at STP. The gas is expanded isothermally to the pressure of 5.45 × 10⁴ N m⁻². Calculate the change in the volumes of the gas?

Solution: At STP, the pressure is 1 atm. The initial pressure of the gas is

From Boyle's law,

Thus, the change in the volumes is calculated as:

The gas is isothermally expanded by 13 dm³.

Example 5: To Determine Height of Mercury

Statement: In a J-shaped tube filled with mercury, initially, mercury levels in both limbs are the same. The initial volume of the trapped gas in the closed end is 0.50 L. The volume of the gas decreases to 0.30 L after the addition of mercury from the open end of the tube. Calculate the height difference of mercury in both limbs after the change?

Solution: The opened end of the tube is subjected to the atmosphere. Initially, the mercury levels in both limbs are the same. Thus, the initial pressure is equal to atmospheric pressure. Taking atmospheric pressure as 1 atm, which is equivalent to 760 mmHg.

From Boyle's law,

The final pressure is the pressure exerted by the mercury column plus the atmospheric pressure. So, the pressure exerted by the mercury column is given as:

The height difference in mercury levels is 510 mmHg.

Associated Articles

• Charles’s law
• Gay-Lussac’s law
• Avogadro’s law
• Ideal gas law