Derivation of Ideal Gas Equation

What is Ideal Gas?

There are many topics that chemistry students have to learn to prepare for their final examinations. One of the most important topics that students have to prepare for their final examination is the derivation of the ideal gas equation. In this article, students will be able to learn the answer to questions like what is ideal gas, what are ideal gas laws, why is the ideal gas equation important, and what are some important ideal gas examples.

Let us first define ideal gas. According to experts, ideal gas can be described as a theoretical gas that comprises a set of randomly-moving point particles. These particles only interact with one another through elastic collisions. It is easy to define ideal gas, but the ideal gas meaning extends beyond that. This concept of the ideal gas formula is important as it obeys all ideal gas law equations, provides a simple equation of state, and is also amenable to analysis by employing statistical mechanisms. Further, students might be interested to note that the requirement of zero interaction can also be relaxed for an ideal gas. This ideal gas meaning is possible if interactions between all particles are perfectly elastic or regarded as simple point-like collisions.

It is almost important for students to note that under various conditions of pressure and temperature, many gases actually qualitatively behave like an ideal gas. In those cases, the ideal gas formula is somewhat bent as the gas molecules or atoms for monatomic gas play the role of ideal particles. If one relaxes the ideal gas definition a bit, then many gases like oxygen, nitrogen, noble gases, hydrogen, and some heavier gases like carbon dioxide, and a mixture of gases in the air can be treated as ideal gases. 

However, one must remember that all of this is done with reasonable tolerances to the ideal gas definition and ideal gas law equation. This is done over various parameter ranges around standard pressure and temperature. Usually, gases are more likely to behave like an ideal gas and follow the ideal gas constant at lower pressure and higher temperature. Can you take a guess as to why this happens?

The simple reason behind this is that the potential energy becomes less significant in comparison to the kinetic energy of the particles. This happens due to the intermolecular forces of attraction. Also, the size of the molecules becomes less significant, too, when compared to the empty spaces between the particles.

This concept of an ideal gas constant can also be illustrated by the fact that one mole of an ideal gas has a capacity of 22.710947(13) liters at standard pressure and temperature (S.T.P.). According to the ideal gas law formula, the standard temperature is often measured at 273.15 K, and absolute pressure is identified at 105 Pa. These values have also been defined by IUPAC since 1982.

Ideal Gas Examples

The ideal gas law definition and some concepts related to the ideal gas law definition are discussed in the previous section. Hence, now we will take a look at some ideal gas law examples. Some of the common ideal gas law examples are given below. 

  1. Oxygen

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  1. Nitrogen

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  1. Hydrogen

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Ideal Gas Laws

In this section, students will be able to find out the answer to the question of what is the ideal gas law.

According to experts, ideal gas laws are laws that state the behaviour of ideal gases. These laws were primarily formulated by the observational work of Boyle in the 17th century and Charles in the 18th century. Both of these ideal gas laws are stated below.

1. Boyle's Law: According to Boyle's Law, if a given mass of a gas is being kept at a constant temperature, then the pressure of that gas is inversely proportional to its volume. 

2. Charles Law: This law states that for any given fixed mass of a gas that is held at constant pressure, the volume of the gas is directly proportional to its temperature.

Ideal Gas Equation

Let us look at some ideal gas equations now. The ideal gas equation is formulated as:

PV = nRT

In this equation, P refers to the pressure of the ideal gas, V is the volume of the ideal gas, n is the total amount of ideal gas that is measured in terms of moles, R is the universal gas constant, and T is the temperature.

This means that according to the ideal gas equation, the product of pressure and volume of a gas bears a constant relation (it is proportional) with the product of the universal gas constant and the temperature.

Here, the universal gas constant is denoted by R. The universal gas constant is the product of the molecular mass of any gas multiplied with the specific gas constant. According to the S.I. system, the value of the universal gas constant is 8.314 J mol-1K-1.

Deriving the Ideal Gas Equation

Let us assume that the pressure of a gas is ‘p,’ and the volume of the gas is ‘v.’ Also, let the temperature be ‘T,’ R is the universal gas constant, and n is the number of moles of gas. Hence, according to Boyle's Law, if the values of n and T are kept constant, then the volume is inversely proportional to the pressure that is exerted by the gas. This can be represented as: 

V ∝ 1/P

According to Charle’s Law, if the values of p and n are kept constant, then the volume of the gas is directly proportional to the temperature. This can be represented as:

V ∝ T

According to Avogadro’s Law, if both P and T are kept constant, then the volume of the gas would be directly proportional to the number of moles of the gas. This can be represented by

V ∝ n

If we combine all the three equations, then

V ∝ n T or PV = nRT

Fun Facts About Ideal Gas

Did you know that there are three basic classes of ideal gases? These types of ideal gases are the classical or Maxwell-Boltzmann ideal gas, the ideal quantum Bose gas that is composed of bosons, and the ideal quantum Fermi gas that is composed of fermions.

Most of these gases have the same characteristics. However, there are some minute differences that students should have a clear idea of.

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FAQs (Frequently Asked Questions)

1. Let us assume that there is one mole of an ideal gas that is filled inside a closed container. The container has a volume of 1 cubic meter. The temperature is set at 300K. Using this information, find the pressure that is exerted by the gas on the walls of the container.

PV = nRT

P x 1 = 1 x 25 / 3 x 300

P = 2500 Pa

2. What is R?

R is called the gas constant. This gas constant was first discovered in the mid-1830s by Emil Clapeyron. That discovery is now better known as the ideal gas law. In some cases, R is also regarded as the universal constant.

This is because this constant shows up in many non-gas-related situations. This is why, depending on the units that are selected, the value of R can take many different units and forms.

3. What is the compressibility factor of an ideal gas?

The compressibility factor of an ideal gas is always 1. 

PV = nRT where 

Z is the compressibility factor at P, T for a given composition.

P is the  absolute pressure.

T is the temperature.

R is the gas constant

4.What is the compressibility factor for H₂ and He ?

The compressibility factor of H₂ and He is greater than 1. This is because both of them are gases and they show real behaviour of gases. Moreover, these gases are heavy so they occupy more volume than usual.

5. Cylinder A contains H₂ gas and cylinder B contains CH₄ gas have the same mass and volume at 30K and 60K respectively. Which of the cylinders will have a greater compressibility factor? Assuming ideal behavior for both the gases.

Both of the cylinders will have equal compressibility because it is assumed that both gases are ideal gases.

and, PV = nRT is the equation for ideal gas

So, Z = PV/ RT = 1

Therefore, both gases will have the same compressibility factor.