Ideal Gas Equation 

Ideal gas equation gives the relations between pressure, volume and temperature of a given gas. It is known as the fundamental gas equation or general gas equation. Although it is valid only for ideal gas, which is a hypothetical gas with some approximations, it still helps to describe and understand the fundamental behaviour of gases. It was first stated by Benot Paul Emile Clapeyron in 1834. It is the combination of the basic gas laws, which are described below.

As we discussed above, the ideal gas equation is the combination of four laws. These are Boyle’s Law, Charles’s Law, Gay-Lussac’s Law and Avogadro’s Law. So let us take a brief introduction of these laws first. 

Boyle’s law

For a given mole of gas at a constant temperature, the pressure of the given gas is inversely proportional to its volume.

V ∝ 1/P

PV = k1

Where, 

V is the volume 

P is the pressure of the gas 

K1 is the constant

In another form, we can write it as, P1V1 = P2V2

Here, P1 is the initial pressure, V1 is the initial volume, P2 is the final pressure, and V2 is the final volume.

Charles’ Law

It relates the volume and temperature of the given mass of gas at constant pressure.

Statement: “The volume of a given amount of gas at a constant pressure varies as per its absolute temperature.”

Mathematically, we can state that-

V ∝ T

or V= KT 

or V/T = K = constant 

or it can be written as V1 / T1 = V2 / T2

The temperature is measured in kelvins here. Lord Kelvin, a British physicist and mathematician, devised this temperature scale (called the absolute scale of temperature.). Absolute zero, which equates to -273°C, is the scale’s lowest point. The molecular mobility of the gas stops at absolute zero, or -273°C, and the volume of the gas becomes zero. The gas transforms into a liquid or a solid. As a result, absolute zero is the temperature at which no gaseous substance exists. The temperature is always calculated by adding the temperature in degrees Celsius by 273 degrees.

Mathematically, K=(°C+273)

This absolute scale of the temperature is used in gas laws. 

Gay-Lussac’s Law

It relates the pressure and the absolute temperature of the given mass of gas at constant volume. 

Statement: Volume remains constant. The pressure of a given mass of a gas increases or decreases by 273 of its pressure at 0°C per degree change in temperature. 

P=P0(1+t273)

Or P=P0(273+t273)

Or P=P0(TT0)

Here T = (273 + t)K and T0 = 273 K

Or PT=P0T0

Or PT=constant 

“At constant volume, the pressure of the given amount of gas is directly proportional to the absolute temperature.” 

Avogadro’s Law

Statement: Equal volume of all the gases contain equal no of molecules under identical temperature and pressure conditions. 

One mole of gas (6.0231023) molecules under STP occupy 22.4L of volume. 

The molar volume of a gas at a given temperature and pressure is a specific constant. It is independent of the gas taken.

Thus, it can be written as V∝ n ( At constant temperature and pressure)

 Or V1n1=V2n2

At SATP (standard ambient temperature and pressure) means 25°C and 1 bar pressure, molal volume =24800ml. 

Ideal Gas Equation 

Now we can write the ideal gas equation by the combination of all four laws that we discussed above, 

We know , PV=constant …………….(1)

 PT=constant ……………..(2)

 V/T=constant…………….(3)

 V1n1= constant 

Hence we can conclude that PV ∝ nT

If we remove the proportionality symbol then, PV=RnT

Where R is called the ideal gas constant. This is the same for gases but can vary based on which units are being used; the most common expressions are R = 0.0821 (L*atm/mol*K) or R=8.31 (J/mol*K).

So the equation PV=nRT is called an ideal gas equation. 

As we discussed earlier, this equation is valid for ideal gas only. So what is an ideal gas? In general, we can say that the gases which obey the ideal gas equation are ideal gases.

So now the question is, can this equation be verified by some other method? Yes, ideal gas equations can also be verified by the kinetic theory of gases. 

Before going to the kinetic theory of the gases, let us discuss the properties of the ideal gas. 

1. Gas contains a very large number of molecules. 
2. Molecules have very neglected volume in comparison to the total volume of the gas.
3. Molecules always perform random motion.
4. Molecules are distributed uniformly inside the gas. This means gas molecules are isotropic in nature. 
5. The gas molecules experience force only during the collision.
6. The collision between gas molecules and walls are always elastic in nature. 
7. All molecules have unequal speed, meaning the speed of molecules is distributed from zero to infinity.
8. The duration of the collision is negligible in comparison to the time interval between two successive collisions. 
9. Gravitational potential energy does not affect the motion of the molecules.

So from the above point, we can derive another gas equation: 

PV=13mNc2

Here m is mass of gas molecule, N number of molecules and c is average speed of molecules.

c2 is proportional to T

So if T is constant, the PV is also constant, which proves Boyle’s law.

In a similar manner, we can prove the other laws from this kinetic theory equation.

Conclusion

Gas laws are the most fundamental laws to understand thermal physics. Although it’s not applicable for real gas, it helps to understand the behaviour of substances (basically gases) properly. Under some specific conditions, many gases also obey these laws. There are also other gas laws like Henry’s Law, Dalton’s law of partial pressure, Graham’s law etc.