Boyle’S Law
Boyle’s Law
Boyle’s Law states that the pressure of a gas is inversely proportional to its volume when temperature and amount of gas remain constant. In simpler terms, as the volume of a gas increases, its pressure decreases, and vice versa. This relationship can be mathematically expressed as P₁V₁ = P₂V₂, where P₁ and V₁ represent the initial pressure and volume, and P₂ and V₂ represent the final pressure and volume. This law demonstrates that gases are compressible and that their pressure can be manipulated by changing their volume.
What is Boyle’s Law?
Boyle’s Law
Boyle’s Law states that the pressure of a gas is inversely proportional to its volume, when temperature and amount of gas remain constant. In other words, as the volume of a gas increases, its pressure decreases, and as the volume of a gas decreases, its pressure increases.
Mathematical Representation:
Boyle’s Law can be mathematically expressed as:
P₁V₁ = P₂V₂
Where:
 P₁ represents the initial pressure of the gas
 V₁ represents the initial volume of the gas
 P₂ represents the final pressure of the gas
 V₂ represents the final volume of the gas
Examples:

Balloon Inflation: When you blow air into a balloon, the volume of the balloon increases. According to Boyle’s Law, as the volume increases, the pressure inside the balloon decreases. This is why the balloon expands and becomes larger.

Scuba Diving: Scuba divers use compressed air tanks to breathe underwater. As they descend deeper into the water, the pressure around them increases. According to Boyle’s Law, the increased pressure causes the air in their tanks to compress, reducing its volume. This is why scuba divers need to ascend slowly to avoid decompression sickness, which can occur if the pressure change is too rapid and the air in their lungs expands too quickly.

Soda Can: When you open a can of soda, the pressure inside the can is released, causing the carbon dioxide gas to rapidly expand. This expansion creates bubbles and causes the soda to fizz.
Applications:
Boyle’s Law has numerous applications in various fields, including:
 Engineering: Boyle’s Law is used in the design of engines, compressors, and other devices that involve the compression or expansion of gases.
 Scuba Diving: As mentioned earlier, scuba divers rely on Boyle’s Law to understand the effects of pressure changes on their air supply.
 Food Packaging: Boyle’s Law is used in the packaging of certain foods, such as potato chips, to maintain their freshness and prevent spoilage.
 Aerosol Cans: Aerosol cans, such as those used for hairspray or deodorant, utilize Boyle’s Law to dispense their contents.
Boyle’s Law is a fundamental principle in understanding the behavior of gases and has practical applications in various aspects of our daily lives.
Formula and Derivation
Formula and Derivation
A formula is a mathematical equation that expresses a relationship between two or more variables. Formulas are used in all areas of mathematics and science, and they play a vital role in engineering, economics, and other fields.
Derivation of a Formula
The process of finding a formula is called derivation. Derivation involves using known mathematical principles and techniques to transform one equation into another. The goal of derivation is to find a formula that is simple, accurate, and easy to use.
Examples of Formulas and Derivations
Here are some examples of formulas and their derivations:
 The formula for the area of a circle:
$$A = \pi r^2$$
This formula can be derived by using the definition of the area of a circle and the properties of similar triangles.
 The formula for the volume of a sphere:
$$V = \frac{4}{3} \pi r^3$$
This formula can be derived by using the definition of the volume of a sphere and the properties of similar cones.
 The formula for the Pythagorean theorem:
$$a^2 + b^2 = c^2$$
This formula can be derived by using the properties of right triangles and the law of cosines.
Applications of Formulas
Formulas are used in a wide variety of applications, including:
 Engineering: Formulas are used to design and analyze structures, machines, and other systems.
 Economics: Formulas are used to model economic behavior and to make predictions about the economy.
 Physics: Formulas are used to describe the laws of motion, gravity, and other physical phenomena.
 Chemistry: Formulas are used to represent chemical compounds and to calculate their properties.
 Biology: Formulas are used to model biological processes and to analyze data.
Formulas are an essential tool for scientists, engineers, and other professionals. They provide a concise and accurate way to represent mathematical relationships and to make predictions about the world around us.
Conclusion
Formulas are a powerful tool that can be used to solve a wide variety of problems. By understanding the process of derivation, we can learn how to find formulas that are simple, accurate, and easy to use.
Examples of Boyle’s Law
Boyle’s Law states that the pressure of a gas is inversely proportional to its volume, when temperature and amount of gas remain constant. In simpler terms, as the volume of a gas increases, its pressure decreases, and vice versa. Here are some examples that illustrate Boyle’s Law:
1. Party Balloons: When you blow air into a balloon, you are increasing its volume. As the balloon expands, the pressure inside it decreases. This is why the balloon feels softer and less tense as you blow more air into it.
2. Scuba Diving: Scuba divers use compressed air tanks to breathe underwater. As they descend deeper into the water, the pressure around them increases. This causes the air in their tanks to compress, reducing its volume. As they ascend, the pressure decreases, and the air in their tanks expands, increasing its volume.
3. Soda Cans: When you open a can of soda, the pressure inside the can is suddenly released. This causes the dissolved carbon dioxide gas in the soda to rapidly expand, creating bubbles and fizzing.
4. Air Pumps: Air pumps work by compressing air into a smaller volume, increasing its pressure. This compressed air is then released through a nozzle, creating a powerful stream of air.
5. Syringes: Syringes are medical devices used to inject or withdraw fluids. When the plunger of a syringe is pulled back, it increases the volume of the syringe, decreasing the pressure inside. This allows fluid to be drawn into the syringe. When the plunger is pushed back in, the volume of the syringe decreases, increasing the pressure inside and forcing the fluid out.
6. Car Tires: As you drive your car, the tires flex and compress as they roll over bumps and uneven surfaces. This compression increases the pressure inside the tires, helping to maintain their shape and support the weight of the vehicle.
7. Gas Laws: Boyle’s Law is one of the fundamental gas laws, along with Charles’s Law, GayLussac’s Law, and the Ideal Gas Law. These laws describe the behavior of gases under different conditions and are essential for understanding various phenomena in chemistry and physics.
By understanding Boyle’s Law and its applications, we can better comprehend and predict the behavior of gases in various situations, from everyday life to scientific experiments and industrial processes.
Solved Exercises on Boyle’s Law
Boyle’s Law states that the pressure of a gas is inversely proportional to its volume, when temperature and amount of gas remain constant. Mathematically, it can be expressed as:
P₁V₁ = P₂V₂
where:
 P₁ is the initial pressure of the gas
 V₁ is the initial volume of the gas
 P₂ is the final pressure of the gas
 V₂ is the final volume of the gas
Solved Exercises on Boyle’s Law
Example 1: A gas occupies 500 mL at a pressure of 2 atm. What will be its volume if the pressure is increased to 4 atm?
Solution:
Using Boyle’s Law, we can calculate the final volume (V₂) as follows:
P₁V₁ = P₂V₂
2 atm × 500 mL = 4 atm × V₂
V₂ = (2 atm × 500 mL) / 4 atm
V₂ = 250 mL
Therefore, the final volume of the gas will be 250 mL.
Example 2: A balloon is filled with 10 L of air at a pressure of 1 atm. What will be the pressure inside the balloon if it is compressed to a volume of 5 L?
Solution:
Using Boyle’s Law, we can calculate the final pressure (P₂) as follows:
P₁V₁ = P₂V₂
1 atm × 10 L = P₂ × 5 L
P₂ = (1 atm × 10 L) / 5 L
P₂ = 2 atm
Therefore, the pressure inside the balloon will be 2 atm.
Example 3: A scuba diver descends to a depth of 30 m in the ocean. If the atmospheric pressure at sea level is 1 atm, what will be the pressure on the diver’s lungs? (Assume the density of water is 1000 kg/m³ and the acceleration due to gravity is 9.8 m/s².)
Solution:
The pressure on the diver’s lungs will be the sum of the atmospheric pressure and the pressure due to the water column. The pressure due to the water column can be calculated using the formula:
P = ρgh
where:
 P is the pressure
 ρ is the density of the fluid
 g is the acceleration due to gravity
 h is the height of the fluid column
In this case, the density of the fluid is 1000 kg/m³, the acceleration due to gravity is 9.8 m/s², and the height of the water column is 30 m. Therefore, the pressure due to the water column is:
P = ρgh = 1000 kg/m³ × 9.8 m/s² × 30 m
P = 294,000 Pa
Converting this pressure to atmospheres, we get:
P = 294,000 Pa / (101,325 Pa/atm)
P ≈ 2.9 atm
Therefore, the pressure on the diver’s lungs will be approximately 2.9 atm.
Frequently Asked Questions – FAQs
How does Boyle’s law work?
Boyle’s Law states that the pressure of a gas is inversely proportional to its volume, when temperature and amount of gas remain constant. In simpler terms, as the volume of a gas decreases, its pressure increases, and as the volume increases, its pressure decreases.
Mathematical Representation:
Boyle’s Law can be expressed mathematically as:
P₁V₁ = P₂V₂
Where:
 P₁ represents the initial pressure of the gas
 V₁ represents the initial volume of the gas
 P₂ represents the final pressure of the gas
 V₂ represents the final volume of the gas
Examples:

Balloon Inflation: When you blow air into a balloon, the volume of the balloon increases. According to Boyle’s Law, as the volume increases, the pressure inside the balloon decreases. This is why the balloon expands and becomes larger.

Scuba Diving: Scuba divers use compressed air tanks to breathe underwater. As they descend deeper into the water, the pressure around them increases. According to Boyle’s Law, the increased pressure causes the air in their tanks to compress, reducing its volume. This allows them to breathe comfortably at greater depths.

Soda Can: When you open a can of soda, the pressure inside the can is suddenly released. This causes the dissolved carbon dioxide gas to rapidly expand, creating bubbles and fizzing. The decrease in pressure allows the gas to escape from the liquid, resulting in the formation of foam.

Syringe: When you pull the plunger of a syringe, the volume of the syringe increases. According to Boyle’s Law, the increased volume causes the pressure inside the syringe to decrease. This creates a suction effect, drawing liquid into the syringe.
These examples illustrate how Boyle’s Law plays a crucial role in various everyday phenomena and practical applications. Understanding this law helps us comprehend the behavior of gases and predict their properties under different conditions.
Why is Boyle law important?
Boyle’s Law: Understanding the Relationship Between Pressure and Volume
Boyle’s Law, formulated by the renowned scientist Robert Boyle in the 17th century, is a fundamental principle in the field of physics and gas behavior. It describes the inverse relationship between the pressure and volume of a gas when temperature remains constant. In simpler terms, as the pressure of a gas increases, its volume decreases, and vice versa.
Importance of Boyle’s Law:

Understanding Gas Behavior: Boyle’s Law provides a crucial foundation for comprehending the behavior of gases under varying pressure and volume conditions. It allows scientists, engineers, and researchers to predict and manipulate gas properties in various applications.

Industrial Applications: Boyle’s Law finds extensive use in numerous industrial processes and technologies. For instance:

Gas Compression: In industries such as natural gas processing, refrigeration, and scuba diving, Boyle’s Law guides the compression and storage of gases in tanks and cylinders.

Pneumatic Systems: Pneumatic systems, commonly used in automation and manufacturing, rely on Boyle’s Law to regulate the pressure and flow of compressed air in actuators, valves, and other pneumatic devices.


Medical Applications: Boyle’s Law plays a vital role in medical equipment and procedures:

Anesthesia: Anesthesia machines utilize Boyle’s Law to precisely control the pressure and flow of anesthetic gases during surgical procedures.

Pulmonary Function Testing: Boyle’s Law principles are applied in pulmonary function testing to measure lung volumes and assess respiratory conditions.


Environmental Monitoring: Boyle’s Law is crucial in environmental monitoring and pollution control:

Air Pollution Studies: Scientists use Boyle’s Law to analyze the relationship between air pressure and pollutant concentrations in the atmosphere.

Scuba Diving Safety: Boyle’s Law helps divers understand the changes in gas volume and pressure during ascent and descent, ensuring their safety underwater.


Everyday Phenomena: Boyle’s Law explains various everyday observations:

Balloon Inflation: As you blow air into a balloon, its volume increases while the pressure inside decreases.

Soda Can Opening: When you open a soda can, the sudden release of pressure causes the dissolved carbon dioxide gas to expand rapidly, creating bubbles and fizzing.


Theoretical Basis for Other Gas Laws: Boyle’s Law forms the foundation for understanding other gas laws, such as Charles’s Law (relationship between volume and temperature) and GayLussac’s Law (relationship between pressure and temperature).
In summary, Boyle’s Law is of paramount importance in comprehending gas behavior, enabling various industrial, medical, environmental, and everyday applications. It provides a fundamental understanding of how pressure and volume are inversely related, allowing scientists, engineers, and individuals to manipulate and predict gas properties accurately.
What is the formula for Boyle’s gas law?
Boyle’s Gas Law Formula
Boyle’s gas law states that the pressure of a gas is inversely proportional to its volume when temperature and amount of gas remain constant. Mathematically, it can be expressed as:
P₁V₁ = P₂V₂
Where:
 P₁ is the initial pressure of the gas
 V₁ is the initial volume of the gas
 P₂ is the final pressure of the gas
 V₂ is the final volume of the gas
Examples of Boyle’s Gas Law
Here are a few examples of how Boyle’s gas law works:
 If you have a balloon filled with air and you squeeze it, the volume of the balloon will decrease and the pressure of the air inside will increase.
 If you have a scuba tank filled with compressed air and you open the valve, the air will rush out of the tank and the pressure will decrease.
 If you have a car tire that is underinflated, the pressure of the air inside the tire will be lower than the pressure of the air outside the tire. This will cause the tire to collapse.
Applications of Boyle’s Gas Law
Boyle’s gas law has a number of applications in real life, including:
 Designing scuba diving equipment
 Designing car tires
 Packaging food in cans and bottles
 Storing gases in tanks
By understanding Boyle’s gas law, we can better understand how gases behave and how to use them safely and effectively.
What is a good example of Boyle’s Law?
Boyle’s Law states that the pressure of a gas is inversely proportional to its volume, when temperature and amount of gas remain constant. In simpler terms, as the volume of a gas increases, its pressure decreases, and vice versa.
Example:
Consider a balloon filled with air. When you blow air into the balloon, its volume increases. As the volume increases, the pressure inside the balloon decreases. This is why the balloon expands and becomes larger.
Conversely, when you let air out of the balloon, its volume decreases. As the volume decreases, the pressure inside the balloon increases. This is why the balloon shrinks and becomes smaller.
Another example of Boyle’s Law can be seen in scuba diving. When a scuba diver descends deeper into the water, the pressure around them increases. This increased pressure compresses the air in their lungs, reducing its volume. As the volume of air decreases, its pressure increases, allowing the diver to breathe normally.
Boyle’s Law is an important concept in understanding the behavior of gases and has practical applications in various fields, including scuba diving, gas compression, and the design of gas containers.
Can Boyle’s law be experimentally proven?
Boyle’s Law states that the pressure of a gas is inversely proportional to its volume, when temperature and amount of gas remain constant. This means that as the volume of a gas increases, its pressure decreases, and vice versa.
Experimental Proof of Boyle’s Law:
Boyle’s law can be experimentally proven using a simple apparatus consisting of a sealed container with a movable piston, a pressure gauge, and a graduated cylinder. The following steps outline a typical experiment to demonstrate Boyle’s law:

Initial Setup:
 Fill the sealed container with a known amount of gas, such as air.
 Use the graduated cylinder to measure the initial volume (V1) of the gas in the container.
 Record the initial pressure (P1) using the pressure gauge.

Compression:
 Slowly compress the gas by pushing the piston inward, reducing the volume of the gas (V2).
 As the volume decreases, observe the pressure gauge. You will notice that the pressure increases (P2).

Expansion:
 Now, slowly pull the piston outward, allowing the gas to expand and increase its volume (V3).
 Observe the pressure gauge again. You will notice that the pressure decreases (P3).

Data Collection:
 Repeat steps 2 and 3 several times, recording the corresponding volumes and pressures for different compression and expansion cycles.

Plotting the Data:
 Plot a graph with pressure (P) on the yaxis and volume (V) on the xaxis.
 Connect the data points with a smooth curve.
Observations and Analysis:
 The graph should show an inverse relationship between pressure and volume. As the volume decreases, the pressure increases, and vice versa.
 The curve should be a rectangular hyperbola, which is the mathematical representation of Boyle’s law (P1V1 = P2V2).
Conclusion:
The experimental results confirm Boyle’s law, demonstrating that the pressure of a gas is inversely proportional to its volume, when temperature and amount of gas remain constant.
What is Boyle’s law?
Boyle’s Law
Boyle’s law, also known as the BoyleMariotte law, describes the inversely proportional relationship between the pressure and volume of a gas, when temperature remains constant. In simpler terms, as the pressure of a gas increases, its volume decreases, and vice versa. This relationship can be mathematically expressed as:
P₁V₁ = P₂V₂
Where:
 P₁ represents the initial pressure of the gas
 V₁ represents the initial volume of the gas
 P₂ represents the final pressure of the gas
 V₂ represents the final volume of the gas
Examples of Boyle’s Law:

Balloon Inflation: When you blow air into a balloon, the pressure inside the balloon increases. This causes the balloon to expand and increase in volume. As you release the air, the pressure inside the balloon decreases, causing it to shrink and decrease in volume.

Scuba Diving: Scuba divers use compressed air tanks to breathe underwater. As they descend deeper into the water, the pressure increases. According to Boyle’s law, the volume of the air in the tank decreases as the pressure increases. This means that the diver has less air to breathe at greater depths.

Soda Can: When you open a can of soda, the pressure inside the can is released. This causes the dissolved carbon dioxide gas to rapidly expand, creating bubbles and causing the soda to fizz.

Car Tire: When you inflate a car tire, you are increasing the pressure inside the tire. This causes the tire to expand and become firmer. If you overinflate the tire, the pressure can become too high and cause the tire to burst.
Boyle’s law is a fundamental principle in understanding the behavior of gases and has practical applications in various fields, including scuba diving, meteorology, and engineering.
What is the relationship between pressure and volume?
Boyle’s Law: Inverse Relationship between Pressure and Volume
The relationship between pressure and volume is described by Boyle’s Law, which states that the pressure of a gas is inversely proportional to its volume when temperature remains constant. In simpler terms, as the volume of a gas decreases, its pressure increases, and as the volume increases, its pressure decreases.
Examples:

Balloon Inflation: When you blow air into a balloon, the volume of the balloon increases, causing the pressure inside to decrease. This is why the balloon expands and becomes larger.

Scuba Diving: Scuba divers experience increased pressure as they descend deeper into the water. This is because the weight of the water above them exerts pressure on the air in their scuba tanks, compressing it and increasing its pressure.

Pressure Cookers: Pressure cookers work by trapping steam inside the pot, increasing the pressure and raising the boiling point of water. This allows food to cook faster at a higher temperature.

Soda Cans: When you open a soda can, the pressure inside the can is released, causing the carbon dioxide gas to rapidly expand and create bubbles. This is why soda fizzes when you open it.
Mathematical Representation:
Boyle’s Law can be expressed mathematically as:
P₁V₁ = P₂V₂
Where:
 P₁ represents the initial pressure of the gas
 V₁ represents the initial volume of the gas
 P₂ represents the final pressure of the gas
 V₂ represents the final volume of the gas
This equation shows that the product of initial pressure and volume is equal to the product of final pressure and volume, assuming temperature remains constant.
In conclusion, Boyle’s Law demonstrates the inverse relationship between pressure and volume in gases. As pressure increases, volume decreases, and vice versa. This principle finds applications in various fields, from scuba diving to food preparation and industrial processes.
Why does volume decrease when pressure is increased?
Why does volume decrease when pressure is increased?
When pressure is applied to an object, the particles that make up the object are forced closer together. This causes the volume of the object to decrease. The relationship between pressure and volume is inversely proportional, meaning that as pressure increases, volume decreases, and vice versa.
Examples:
 When you blow up a balloon, you are increasing the pressure inside the balloon. This causes the balloon to expand and increase in volume.
 When you squeeze a ball, you are increasing the pressure on the ball. This causes the ball to shrink and decrease in volume.
 When you put a lid on a pot of water, you are increasing the pressure inside the pot. This causes the water to boil at a higher temperature.
The ideal gas law:
The relationship between pressure, volume, and temperature of a gas is described by the ideal gas law:
PV = nRT
where:
 P is the pressure of the gas
 V is the volume of the gas
 n is the number of moles of gas
 R is the ideal gas constant
 T is the temperature of the gas
The ideal gas law shows that if the pressure of a gas is increased, the volume of the gas will decrease, assuming that the temperature and number of moles of gas remain constant.
Applications of the ideal gas law:
The ideal gas law is used in a variety of applications, including:
 Designing and operating gas compressors
 Predicting the behavior of gases in chemical reactions
 Determining the pressure of a gas in a container
 Calculating the volume of a gas at a given pressure and temperature
The ideal gas law is a fundamental principle of thermodynamics and is used in a wide range of scientific and engineering applications.
What happens to pressure if the volume is doubled?
Boyle’s Law: Pressure and Volume Relationship
Boyle’s Law states that the pressure of a gas is inversely proportional to its volume when temperature and the amount of gas remain constant. In simpler terms, if the volume of a gas increases, its pressure decreases, and if the volume decreases, its pressure increases. This relationship can be mathematically expressed as:
P₁V₁ = P₂V₂
Where:
 P₁ represents the initial pressure of the gas
 V₁ represents the initial volume of the gas
 P₂ represents the final pressure of the gas
 V₂ represents the final volume of the gas
Examples:

Balloon Inflation: When you blow air into a balloon, its volume increases. As a result, the pressure inside the balloon decreases, allowing it to expand further.

Syringe and Piston: If you pull the piston of a syringe outward, the volume of the enclosed air increases. This causes the pressure inside the syringe to decrease, drawing air in. Conversely, pushing the piston inward decreases the volume and increases the pressure, expelling air.

Scuba Diving: Scuba divers experience increased pressure as they descend deeper into the water. This is because the weight of the water above them compresses the air in their scuba tanks, reducing its volume and increasing its pressure.

HighAltitude Cooking: At higher altitudes, the atmospheric pressure is lower. This means that water boils at a lower temperature. As a result, cooking times need to be adjusted to account for the reduced pressure.

Aerosol Cans: Aerosol cans contain pressurized gas, such as butane or propane. When you press the nozzle, the gas is released and expands rapidly. This sudden expansion creates a decrease in pressure, drawing the liquid product out of the can.
Understanding Boyle’s Law is essential in various fields, including physics, chemistry, engineering, and even everyday life. It helps us comprehend and predict the behavior of gases under different conditions, enabling us to design and operate systems that involve gas compression, expansion, and flow.
Why Boyle’s law is not applicable at high pressure?
Boyle’s Law states that the pressure of a gas is inversely proportional to its volume, when temperature and amount of gas remain constant. Mathematically, it can be expressed as:
P₁V₁ = P₂V₂
Where:
 P₁ and P₂ are the initial and final pressures of the gas
 V₁ and V₂ are the initial and final volumes of the gas
This law holds true for ideal gases at low to moderate pressures. However, at high pressures, Boyle’s law begins to deviate from experimental observations. This deviation is more pronounced for gases that are highly compressible, such as carbon dioxide (CO₂) and ammonia (NH₃).
There are several reasons why Boyle’s law is not applicable at high pressures:

Intermolecular Forces: At high pressures, the molecules of a gas are forced closer together, increasing the intermolecular forces between them. These forces can cause the gas to behave more like a liquid, resulting in a decrease in its compressibility.

Molecular Volume: At high pressures, the molecules of a gas occupy a significant portion of the total volume of the gas. This means that there is less free space for the molecules to move, which reduces the gas’s ability to expand or compress.

NonIdeal Gas Behavior: Real gases exhibit nonideal behavior at high pressures, deviating from the assumptions of the ideal gas law. This deviation is due to the interactions between gas molecules and the finite size of the molecules themselves.

Phase Transitions: At extremely high pressures, some gases may undergo phase transitions, such as liquefaction or solidification. In these cases, Boyle’s law is no longer applicable as the gas is no longer in a gaseous state.
Examples:

Carbon Dioxide (CO₂): At room temperature and low pressures, CO₂ behaves as an ideal gas and follows Boyle’s law. However, at high pressures, CO₂ deviates from ideal behavior and its compressibility decreases. This is because the intermolecular forces between CO₂ molecules become significant at high pressures, causing the gas to behave more like a liquid.

Hydrogen (H₂): Hydrogen is a highly compressible gas that deviates from Boyle’s law at relatively low pressures compared to other gases. This is because hydrogen molecules are very small and have a high kinetic energy, which allows them to overcome intermolecular forces even at moderate pressures.
In summary, Boyle’s law is not applicable at high pressures due to the increased intermolecular forces, molecular volume, nonideal gas behavior, and potential phase transitions. These factors cause the compressibility of gases to decrease at high pressures, deviating from the inverse relationship predicted by Boyle’s law.