# Gas Laws

Gas Laws are an important part of thermodynamics, which is the study of how energy moves between different systems. Even though we can't usually see gases around us (because they are transparent), we can see them in clouds of smoke. Gases are made up of tiny particles that move around freely. Scientists use something called the ideal gas model to study how gases behave under different conditions. This model helps us understand how gases interact with each other and with their environment. The laws that describe these interactions are called ideal gas laws. By using the ideal gas model, scientists can accurately predict how different gases will behave under different conditions.

## The thermodynamic properties of gases

When scientists study different systems from a thermodynamic point of view, they look at many different properties. However, they only need to study a few of these properties to understand everything about the system. For ideal gases, these properties are temperature, pressure, and volume. Thermodynamics is the study of how energy moves between different systems that are made up of many tiny particles. Because of this, all the properties that scientists study are statistical features that come from the microscopic structure of the system. By looking at these three properties, scientists can understand everything they need to know about ideal gases.

### Temperature

Temperature is way of measuring how fast particles are moving in a system. We use the letter T to represent temperature in thermodynamics, and the unit we use is called Kelvin (K). The Kelvin scale has a zero point called absolute zero, which is the lowest possible temperature. At absolute zero, particles have no kinetic energy and the temperature is 0K, which is equal to -273.15°C.

Even though all particles in a system have different kinetic energies, the average temperature gives us an important measure of how they are moving. This is because the distribution of kinetic energy in a system follows a typical pattern. The Maxwell-Boltzmann distribution is a law that describes how ideal gases behave and how their particles move. Thanks to this law, scientists can predict how ideal gases will behave under different conditions.

### Volume

Volume is represented by the letter V in thermodynamics. It is the sum of the volumes of all the particles that make up a system, or the total spatial volume occupied by the particles as they move randomly. Unlike temperature, volume is what we call an extensive property. This means that if the amount of matter in the system changes, the volume will change too. In contrast, temperature remains the same regardless of how much matter we add or remove from the system.

When we talk about volume, it's important to remember that it's not just about the physical space the particles occupy. It also includes any empty space between the particles. For example, if you have a balloon filled with air, the volume of the balloon includes both the physical space taken up by the air particles, as well as the space between them. As we add more air to the balloon, the volume increases, even though the temperature remains the same.

### Pressure

Pressure is usually represented by the letter P in thermodynamics. It's a measure of force per unit of area that the particles in a system exert on the boundaries of the volume they occupy. Pressure is an intensive property, which means that it doesn't depend on the amount of matter in the system.

We can also interpret pressure as a measure of the density of energy in a system. This is because the particles in a system exert forces on each other due to their motion, and this motion represents energy. When the particles are closely packed together, they exert more force on the boundaries of the system, leading to higher pressure. In contrast, when the particles are more spread out, they exert less force and the pressure is lower.

You might also see the letter p used for pressure, but it's important to stick to what your teacher or textbook uses to avoid confusion. Both p and P represent the same concept of pressure, which is a fundamental property of thermodynamics.

## The ideal gas laws and formulae

In the case of ideal gases, three laws capture the relationships between temperature, pressure, and volume, namely Boyle’s law, Charles’ law, and Gay-Lussac’s law. Each law shows the relationship between two properties with a third that is kept constant.

### Boyle’s law

's law is a fundamental law of thermodynamics that describes the relationship between pressure and volume for an isothermal, which means that the temperature of the system remains constant. The mathematical expression for Boyle's law is:

P1V1 = P2V2

where P1 and V1 are the initial pressure and volume of the system, P2 and V2 are the final pressure and volume of the system, and k is a constant that represents the temperature of the system.

Boyle's law tells us that for a fixed amount of gas at a constant temperature, the pressure and volume are inversely proportional to each other. In other words, if the pressure is increased, the volume of the gas will decrease, and if the pressure is decreased, the volume will increase. This relationship is often expressed in the form of the equation above, which shows that the product of pressure and volume remains constant as long as the temperature is held constant. This means that if we double the pressure of a gas, its volume will be halved, and vice versa.

### Charles’ law

Charles’ law captures the relationship between temperature and volume for an isobaric process (constant pressure). The mathematical expression for this law is

or where k is a constant, and  1 and 2 indicate two different configurations of the system. Charles’ law indicates that whenever the pressure of an ideal gas is kept constant, the volume is directly proportional to the temperature (and vice-versa).

### Gay-Lussac’s law

Gay-Lussac’s law captures the relationship between pressure and temperature for an isochoric process (constant volume). The mathematical expression for this law is

or where k is a constant, and 1 and 2 indicate two different configurations of the system. Gay-Lussac’s law indicates that whenever the volume of an ideal gas is kept constant, the pressure is directly proportional to the temperature (and vice-versa). Check out our explanation on PV Diagrams, which are diagrams used to represent the thermodynamic stages of a process.

## Combined gas law

The three gas laws - Boyle's law, Charles's law, and Avogadro's law - were discovered through experiments in laboratories, where scientists observed the behavior of gases under different conditions. It was later realized that these laws were part of a larger, more general law for ideal gases.

The mathematical expression for this general combined law is:

PV = nRT

where P is the pressure of the gas, V is its volume, n is the amount of substance that forms the system, R is the ideal gas constant (with an approximate value of 8.314 J/K·mol), and T is the temperature of the gas.

Since R is a constant and the number of particles in a system is fixed, we can re-write the equation as:

PV/T = constant

This expression shows that if we fix any two of the variables - pressure, volume, or temperature - the third variable will be determined by the constant. This means that we can derive Boyle's law, Charles's law, and Avogadro's law from this expression, depending on which variables we hold constant.

For example, if we hold the temperature constant, we can derive Boyle's law, which tells us that the pressure and volume of a gas are inversely proportional. If we hold the volume constant, we can derive Charles's law, which tells us that the volume and temperature of a gas are directly proportional. And if we hold the pressure constant, we can derive Avogadro's law, which tells us that the volume and amount of gas are directly proportional.

## Gas law equations

Here are some examples of using each law in calculations. Note that temperature is measured in K, pressure is measured in N/m2, and volume is measured in m3. Helpful tip: label each property in the example as V1, P2, T1, etc. This will help you plug in the values easily into the correct equation.

Example 1

Consider an ideal gas with a temperature of 100K. We start with the gas at 50N/m2 and 10m3. If we increase the volume to 50m3, what is the final pressure of the gas?

Solution

If we use Boyle’s law, the final volume will be

Example 2

Consider an ideal gas occupying a volume of 10m3. We start with the gas at 50N/m2 and 100K. If we increase the pressure to 100N/m2, what is the final temperature of the gas?

Solution

If we use Gay-Lussac’s law, the final temperature will be

Example 3

Consider an ideal gas at 50N/m2 of pressure. We start with the gas at 100K and 10m3. If we decrease the temperature to 10K, what is the final volume occupied by the gas?

If we use Charles’ law, the final volume will be

## Gas Laws - Key Takeaways

Thermodynamics is a branch of physics that studies the behavior of many-particle systems, including gases. Gases are particularly interesting to study because their particles have a high degree of freedom, making their behavior complex yet fascinating. To simplify the study of gases, scientists use an approximation called the ideal gas approximation.

The ideal gas approximation assumes that gas particles are point masses with no volume and that they do not interact with one another except through perfectly elastic collisions. This approximation allows scientists to model the properties of gases simply, using only three thermodynamic properties: temperature, pressure, and volume.

There are three laws that capture the relationship between these three properties for ideal gases: Boyle's law, Charles's law, and Gay-Lussac's law. Boyle's law states that, at a constant temperature, the pressure and volume of an ideal gas are inversely proportional. Charles's law states that, at a constant pressure, the volume of an ideal gas is directly proportional to its temperature. Gay-Lussac's law states that, at a constant volume, the pressure of an ideal gas is directly proportional to its temperature.

There is also a general law for ideal gases that expresses the relationship between the three quantities - pressure, volume, system known law is expressed mathematically as PV =RT, where P is the pressure of the gas, V is its volume, n is the amount of substance that forms the system, R is the ideal gas constant, and T is the temperature of the gas.

## Gas Laws

What is R in the ideal gas law equation?

R in the ideal gas law equation is the ideal gas constant, with an approximate value of 8.31J/K·mol.

What is the ideal gas law?

The ideal gas laws are the laws that capture the relationship between thermodynamic properties. In the case of ideal gases, these properties are temperature, pressure, and volume. There are three laws that capture the relationships between temperature, pressure, and volume, namely Boyle’s law, Charles’ law, and Gay-Lussac’s law.

How do you use the ideal gas law?

There are three laws that capture the relationships between temperature, pressure, and volume, namely Boyle’s law, Charles’ law, and Gay-Lussac’s law. Each law has its own equation. For Boyle’s law, the equation is P1·V1=P2·V2. For Charles’ law, the equation is V1/T1=V2/T2. For Gay-Lussac’s law, the equation is P1/T1=P2/T2.

What is the general gas law?

The general gas law is the equation that relates the temperature, the pressure, and the volume with the particle content of an ideal gas. Its expression is P·V=n·R·T

What are the three ideal gas laws?

The three ideal gas laws are Boyle’s law, Charles’ law and Gay-Lussac’s law.

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