# Ideal Gas Equation

In thermodynamics, Ideal gas law is also called a general gas equation is a well-defined approximation of the behaviour of many gases under diverse conditions. Ideal Gas Equation is the combination of empirical laws like Charle’s law, Boyle’s law, Gay-Lussac’s law and Avogadro’s law.

## Ideal Gas Equation Definition

Ideal Gas Equation is the equation defining the states of the hypothetical gases expressed mathematically by the combinations of empirical and physical constants.

## Equation of Ideal Gas Law

State of an ideal gas is determined by the macroscopic and microscopic parameters like pressure, volume, temperature. Thus, the ideal gas equation is often written as-

PV = nRT

Where,

• P is the pressure of the ideal gas.
• V is the volume of the ideal gas.
• n is the amount of ideal gas measured in terms of moles.
• R is the gas constant.
• T is the temperature.

## Ideal Gas Equation Units

The Ideal gas equation units, terms and corresponding definitions can be expressed as follows-

 Terms Symbol Definition Units Pressure P Force per units area Pa or N/m2 Volume V Three-dimensional space enclosed by the closed surface m3 Amount of substances/number of moles n The ratio of the mass of the gas(m) to its molar mass(M). mole Ideal gas constant R Physical constant relating the average kinetic energy of an ideal gas with temperature. It is expressed in units of energy per temperature increment per mole 8.3144598(48) J.K-1.mol-1 Temperature T Measure of heat K or 0C

## What is Ideal gas?

Ideal gas does not exist in reality. It is a hypothetical gas proposed to simplify the calculations. In Ideal gas the gas molecules move freely in all direction and collision between them are considered to be perfectly elastic. Which implies no loss in the Kinetic energy due to the collision.

Although there is no ideal gas, all real gas tend to approach that property when the density gets low enough. This is possible because the molecules of the gas are so far apart from one another that they do not interact with each other. Thus, the ideal gas concept helps us in studying real gases.

## What is Ideal Gas Equation?

To study the property of gasses we need to have a standard gas to study, but which gas should it be? Hydrogen, oxygen, helium, nitrogen, carbon dioxide to name a few, and there are thousands of other gasses we could study.

The researchers have found that no matter what gas you study if you take a one-mole sample of that gas and put it in the same container and maintain a constant temperature, the pressure is almost the same, and at lower densities, even those tiny differences in the measurements also disappear. Thus, at really low densities, all the real gases tend to obey one universal law called ideal gas law.

This law is described by an equation known as the Ideal gas equation,

PV = nRT

Although the ideal gas equation has multiple limitations. This equation holds well as long as the density is kept low, this equation is applicable for single gas or even mixture of multiple gasses where n will stand for the total moles of gas particles in the given mixture.

## Ideal Gas Equation of States

The equation of states of an ideal gas explains the simple relation between the parameters or properties that are very generic and accurate. In general, an equation relating the relation between P,V,T of an ideal gas is called Equation of states.

Sometimes relation involving other parameters of a substance at equilibrium state is also referred to as the equation of states. The simplest equation of state for substances in the gas phase is the ideal-gas equation of state as-

PV =RT

## Ideal Gas Equation in Other Forms

The ideal gas equation can be rewritten in multiple ways depending upon the desciplines. Some of them are given below

### Common form

Most frequently used form of the ideal gas equation is-

PV = nRT = NkBT

Where,

• P is the pressure of the ideal gas.
• V is the volume of the ideal gas.
• n is the amount of ideal gas measured in terms of moles.
• R is the gas constant.
• T is the temperature.
• N is the number of gas molecules (N=NAn)
• kB is the Boltzmann constant.
• NA is the Avogadro’s constant.

### Molar form

This alternate form of the ideal gas equation is useful because it specifies the amount of gas accurately. Here instead of the mass of the gas molecules its chemical equivalent mass is used. Thus, the molar form is given as –

Pv = RspecificT

Where,

• P is the pressure of the ideal gas
• v is the specific volume given by $v=\frac{1}{\rho }=\frac{1}{\left ( \frac{m}{V} \right )}$
• Rspecific is the specific gas constant given by $R_{specific}=\frac{R}{M }$
• T is the temperature.

### Statistical Mechanics

In statistical mechanics, the ideal gas equations are given by-

$P=\frac{k_{B}}{\mu m_{u}}\rho T$

Where,

• P is the pressure of the ideal gas
• T is the temperature.
• μ average particle mass
• mu is the atomic mass constant
• ρ is the density given by ρ = m/V = nμmu
• kB is the Boltzmann constant.

## Ideal Gas Equation Important Questions

Q1. What is the shape of the P-T curve for an ideal gas?

Ans: The shape of the P-T curve for an ideal gas is a straight line.

Q2. What is the shape of the V-T curve for an ideal gas?

Ans: The shape of the V-T curve for an ideal gas is a straight line.

Q3. What is the shape of the P-V curve for an ideal gas?

Ans: The shape of the P-V curve for an ideal gas is a hyperbola.

Q4. What is the compressibility factor of an ideal gas?

Ans: It is always 1.

Q5. What is the compressibility factor?

Ans: Compressibility factor is defined as the ratio of a molar volume of a gas to a molar volume of an ideal gas at the same temperature and pressure. It is also known as gas deviation factor or compression factor and is denoted by Z.

Q6. What are the units of universal gas constant?

Ans: Following are the units of universal gas constant:

• R=0.0821 l.atm.mol-1.K-1
• R=8.3145 J.mol-1.K-1
• R=8.2057 m3.atm.mol-1.K-1

Q7. What is the standard pressure and temperature?

Ans: The standard pressure and temperature is 1 atm pressure and 0℃.

Q8. What happens to the pressure P atm, if the volume of a gas is doubled at constant temperature?

Ans: The pressure P atm becomes P/2 atm as P is inversely proportional to the temperature and volume is constant.

#### Practise This Question

A cubical vessel has a side with 'I' cm lenth contained a gas at a pressure of 'p'. when the side of the vessel is made 12 cm, the pressure of gas becomes?