Combined Gas Law as the name suggest is combination of different gas laws. The different gas laws that combine to form Combined Gas Law include Boyle’s law, Charles’ Law and Gay Lussac’s law.

In this article, we will learn about combined gas law, its formula and derivation. We will also learn briefly the three gas laws individually.

What is Combined Gas Law?

Combined Gas Law is a gas law that combines Boyle’s Law, Charles’s Law, and Gay Lussac’s Law. The Combined Gas Law is not a new law but only a combination of these laws taken together. This law describes the relationship between the pressure (P), volume (V), and temperature (T) of a gas, assuming the amount of gas and the number of gas molecules (N) to be constant.

While maintaining the constant, these laws relate one thermodynamic variable to another. The interdependence of these variables maintains a constant ratio between a system’s temperature and its product of pressure and volume. This interdependence is reflected in this combined gas law.

Let’s learn the three gas laws, i.e. Boyle’s Law, Charles’s Law, and Gay Lussac’s Law, individually.

Boyle’s Law

According to Boyle’s Law, if a gas’s temperature stays constant, then the pressure and volume of the gas are inversely related to each other.

P ∝ 1/V

This means that as V decreases, P increases, and vice versa. When the proportionality symbol is replaced with a constant, the equation appears as follows:

P = k/V

⇒ P V = K

This indicates that the product of pressure and volume is constant represented by the value k. Since, the product of pressure and volume is constant Boyle’s Law can be used to describe the change in pressure or volume.

Boyle’s Law can be used to compare the changing condition of gas. For example, let’s say P₁ and V₁ represent the initial pressure and volume of a gas, respectively. Once an adjustment has been made, P₂ and V₂ represent the then volume and pressure, respectively. Now we know that product of pressure and volume is constant then product of pressure and volume before and after adjustment will be same. Mathematically we can represent it as follows:

P₁ V₁ = P₂ V₂

Charles’ Law

Charles’ Law describes the relationship between the volume and temperature of a gas when pressure and the amount of gas remain constant. It states that the volume of a gas is directly proportional to its temperature, expressed in Kelvin. Keep in mind that the temperature must be in Kelvin. Mathematically we can represent charles’ law as

V ∝ T

When we replace proportionality symbol with constant k we can rewrite equation as

V = kT

⇒ V/T = k

We can use Charles’ Law to compare changing conditions for a gas. The initial volume and temperature of a gas are now represented by V₁ and T_{1 ,} while V₂ and T₂ represent the temperature and final volume, respectively. Since, the ratio of volume and temperature is constant. Then to find the change Charles’s Law’s relationship can be shown as:

V₁ T₁ = V₂ T₂

Gay-Lussac’s Law

Gay-Lussac’s Law describes the relationship between the pressure and temperature of a gas when the volume and the amount of gas remain constant. It states that the pressure of a gas is directly proportional to its temperature, provided that the volume and the amount of gas are kept constant.

Mathematically we can represent Gay Lussac’s law as

P ∝ T

When we replace proportionality symbol with constant k we can rewrite equation as

P = kT

⇒ P/T = k

We can understand this from an example. For example, if we have a sample of gas in a rigid container experiences an increase in temperature, which also causes the gas’s pressure to rise. The gas molecules hit the container walls harder due to their increased kinetic energy, which raises the pressure inside the container.

According to Gay-Lussac’s Law, when the volume remains constant, the pressure of a given mass of gas varies directly with the gas’s absolute temperature. The only distinction between Charles’s Law and Gay-Lussac’s Law is the kind of container. In an experiment involving Charles’s Law, the container is flexible; in an experiment with Gay-Lussac’s Law, it is rigid.

The mathematical expressions of Gay-Lussac’s Law for change in condition of gas is similar to the Charles’s Law:

P₁ T₁ = P₂ T₂

where, P₁ and T₁ are initial temperature and pressure while P₂ and T₂ are final pressure and temperature.

Combined Gas Law Formula:

Combined gas law can be mathematically expressed as

P V / T = k

where,

• P = pressure
• T = temperature in kelvin
• V = volume
• k = constant (units of energy divided by temperature)

The combined gas law demonstrates that:

• Pressure is inversely proportional to the volume
• Pressure is directly proportional to the temperature
• Volume is directly proportional to the measure of temperature

Derivation of the Combined Gas Law equation

The three previously defined laws are given as:

• Boyle’s law: PV = k
• Charles law: V/T = k
• Gay-Lussac’s law: P/T = k

Combining three laws we get:

PV/T = k

This is the combined gas law formula,

where

• k is constant,
• V is volume,
• T is temperature,
• P is pressure.

The combined gas law formula can be modified to compare two sets of conditions in a single substance. The initial condition is represented by the values of temperature (T), pressure (P), and volume (V) in the equation with subscripts of ‘i’ . Additionally, the ultimate condition is represented by those with a subscript of ‘f’ .

When the gas is compared in two different conditions, the law can be stated as,

P_i V_i / T_i = P_f V_f / T_f

Where,

• P_i = initial pressure, V_i = initial volume, T_i = initial temperature
• P_f = final pressure, V_f = final volume, T_f = final temperature

It should be noted that the temperature should always be calculated in kelvin. It is therefore necessary to convert the units to kelvin if they are available in the Celsius scale.

Application of Combined Gas Law

The application of combined gas law is mentioned below:

• When working with gases at standard temperatures and pressures, the combined gas law is useful. Its accuracy decreases at high temperatures and pressures, just like other gas laws that are based on ideal behavior.
• In order to forecast weather, it can be used, for instance, to compute the pressure, volume, or temperature of the gas in clouds.
• Electric appliances used in everyday life like refrigerators and air conditioners are all predicted by the law.
• Helium Balloons also use this law: A balloon will have a specific pressure, temperature, and volume if it is filled with helium on Earth. The balloon will rise if it is released. The temperature and air pressure decrease as the balloon ascends higher into the atmosphere. The balloon’s volume will decrease as it becomes colder and increase as the air becomes thinner. As the balloon rises, the variations in pressure and temperature will cause it to take on a new volume.

Conclusion on Combined Gas Law

• The combined gas law is one of the ideal gas laws.
• Its combination of Boyle’s law, Charles’ law, and Gay-Lussac’s law gives it its name.
• Only temperature, pressure, and volume can vary while applying this law. There is no change in the quantity or number of moles of gas.
• The law basically says that a gas’s absolute temperature, volume, and pressure ratios all equal a certain constant. Thus, you can forecast the impact of a change in one of these variables on the others.