Ideal Gas Equation in Terms of Density

The Ideal Gas Law is an equation in thermodynamics describing the relationship between temperature, pressure, and volume of gases. The Ideal Gas Law equation is PV = nRT, where P is the pressure, V is the volume, ‘n’ is the number of moles of gas molecules, R is the universal gas constant, and T is the absolute temperature. 

Practically, there is no such thing as an ideal gas. However, in conditions like high temperatures and low pressures where individual particles are moving so fast and are too much away from each other such that their interaction is almost zero, gases behave like an ideal gas. Ideal Gas Law is a useful approximation.

The Ideal Gas Equation

The Ideal gas equation is a mathematical formula that combines empirical laws such as Charle’s law, Boyle’s law, and Avogadro’s law.

Charles’ Law

The volume of a gas is in direct proportion to the temperature at relentless pressure, according to Charle’s Law. It implies that as the temperature increases, the volume of the gas rises as well. V ∝ T is a mathematical representation of Charles’s Law.

Boyle’s Law

It represents that the volume of a gas is inversely proportional to pressure at a constant temperature. It signifies that as the pressure rises, the volume of the gas decreases. Boyle’s law is expressed mathematically as V ∝ 1/P.

Avogadro’s Law

Avogadro’s law implies that the volume of a gas is directly proportional to the number of molecules in the gas at normal temperature and pressure conditions. As a result, we write it as V ∝ n in mathematics.

Gay-Lussac’s Law

Gay-Lussac’s Law describes that the pressure exerted by a gas of a particular mass, when kept at a constant volume, varies in a direct proportion with the absolute temperature of the gas. In simpler terms, we can say that the pressure exerted by a gas is directly proportional to the temperature of the gas when the volume is constant, and the mass is of a fixed quantity.

The mathematical expression of Gay-Lussac’s Law can be written as follows:

                          P ∝ T or P/T = k

Where,

             P is the pressure exerted by the gas

             T is the absolute temperature of the gas

             k is a constant.

The purpose of creating Ideal gas law was to show the relationship between pressure, volume, moles of gas, and temperature. It is a hypothetical or speculative equation for an ideal gas. Pressure and volume have an inverse relationship; however, temperature has a direct relationship. The equation for the Ideal Gas Law is:

P × V = n × R × T

Where,

‘P’ denotes the ideal gas pressure. ‘V’ denotes the volume of the ideal gas. ‘n’ represents the quantity of ideal gas measured in moles. ‘R’ is the proportionality constant or the gas constant and ‘T’ stands for the ideal gas temperature.

Derivation Of Ideal Gas Equation:

Let us consider,

The pressure exerted by the gas: P

The volume of the gas: V

Temperature: T

The number of moles of gas: n

Universal gas constant: R

According to Boyle’s Law, At constant n & T, the volume makes an inverse relation with the pressure exerted by a gas.

i.e. 

      v ∝ 1/P ————————-(i)

According to Charles’ Law, When P & n are constant, the volume of gas directly relates to the temperature.

i.e. 

       v ∝ T     ————————-(ii)

According to Avogadro’s Law, When P & T are constant, then the volume of gas makes a direct relation with the number of moles of gas.

i.e. 

       v ∝ n     ———————–(iii)

Combining all the three equations, we have-

V ∝ nT/P 

or 

PV = nRT

Where R is the Universal gas constant, which has a value of 8.314 J/mol-K

Ideal Gas Equation in Terms of Density

Ideal gas law: PV = nRT = NkT

Where,

             n: number of moles

             R: universal gas constant = 8.314J/molK

             N: number of molecules

             k: Boltzmann constant = 1.38066×10-23 J/K

            NA: Avogadro’s number = 6.0221 x 1023 /mol

As a result, if the number of moles (n) is 1, pV=RT is obtained. pV=(m/M) RT; (n=m/M)

Because pV=mRT/M = (mRT)/VM,

As a result, p = (dRT)/M (since d =m/V).

As a result Ideal Gas Equation in Terms of Density:

 (pM) /RT = d

M is equal to dRT/P. This equation leads to the conclusion that the density of a gas is directly proportional to the molar mass of the gas under constant temperature and pressure, which is the ideal behaviour of gases.

Conclusion:

The ideal gas is a hypothetical gas in which the molecules take up very little space and have no interactions, and thus obeys the gas laws perfectly. The study of the ideal behaviour of gases allows us to understand the matter at its most fundamental level: individual particles acting independently, almost completely free of interactions and interference.