Empirical Derivation of Gas Equation

Introduction to Empirical Derivation of Gas Equation

The Gas Equation, PV=nRT, is the empirical relation between the press, volume, temperature, and several moles of a gas.  It is the empirical combination of Boyle’s law, Charles’s law, and Avogadro’s law. Benoit Paul Émile Clapeyron firstly described the Ideal Gas Law in 1834. According to the ideal gas equation, when we multiply the pressure and volume of a gas, it makes a relation directly proportional to the product of the universal gas constant and the temperature. It can be written as, 

PV = nRT

Where, 

 P = Pressure of the Ideal Gas

V = Volume of the Ideal Gas 

n = number of moles in the Ideal Gas 

R = Universal Constant, which is R = 8.314kJ/K.mol

T = Temperature 

So, The ideal gas law depicts that the product of the volume and the pressure of one gram molecule of a perfect gas is proportional to the product of the absolute temperature and the universal gas constant.

The ideal gas equation is PV=nRT.

Ideal Gas 

An ideal gas is a hypothetical gas consisting of various moving particles moving randomly and interacting through elastic collisions. However, this gas does not exist in reality. 

All the molecules present in the Ideal Gas can move freely in all directions. When they collide, this collision is perfectly elastic, where no loss in kinetic energy occurs. 

Empirical Derivation of Gas Equation

The  Gas Equation gas is derived from the experimentation or observation of Robert Boyle, Gay-Lussac and Amedeo Avogadro. When we combine the individual equation of every scientist into a single expression, we get the Ideal gas equation, which describes the relation Volume, Pressure, Temperature and number of moles in a Gas. 

Let’s take the pressure of a gas as ‘P’ and the volume of the gas as ‘V’.  Along with this, let’s assume the temperature is ‘T’. Here, We take R, the universal gas constant, and n is the number of moles of gas.

According to Boyle’s Law, when the values of n and T  are constant, the V (volume) is inversely proportional to the P (pressure) exerted by the gas. It can be written as: 

                V ∝ 1/P

According to Charles’ Law, when P and n are constant, the V (volume) is directly proportional to the T (temperature). It can be written as:

                  V ∝ T

According to Avogadro’s Law, when P and T values are constant, the V (volume) is directly proportional to the n (number of moles). It can be written as:

                  V ∝ n

If we combine the three laws, we get:

                V ∝ nT/P

This shows that the volume of any gas is directly proportional to the temperature and the number of moles present in it, while inversely proportional to the pressure exerted on it.

If we rewrite the equation, we get:

         V = RnT/P = nRT/P

                   [R = 8.314kJ/K.mol] 

After, Multiplying both the sides of the equation by P, we get:

 *it will clear the fraction. 

               PV = nRT

Importance of Gas Equation 

• The ideal gas law can be used to relate the four physical properties of any gas at a given time

• This law can solve stoichiometry problems consisting of chemical reactions related to gasses

• The ideal gas equation also facilitates the relationship between various non-constant properties of ideal gasses like n, P, V, T, in which three will remain fixed

• The ideal gas equation works as an essential tool to get the insight of gasses at high temperatures and low pressures

Universal Gas Constant (R) 

Universal Gas Constant describes the work done per mole per Kelvin. It is expressed in joules, i.e. Energy per temperature, increase in per mole. 

However, when the specific gas constant of any gas is multiplied by its constant molecular mass, then the resulting product is always the same for every gas, which is equal to the 8.314kJ/K.mol or 0.082L.atm/K.mol. So, this product is called Universal Gas Constant, denoted by R.

Whenever we use the value of the Gas constant as R = 8.31 J/K.mol, then we need to take pressure P in pascals (Pa), volume in cubic meters (m3) and the temperature T in kelvin (K). And, whenever we use the value of the Gas constant as R = 0.082 L.atm/K.mol then, we need to take pressure in atmospheres (atm), volume in liters (L) and the temperature T in kelvin (K).

Applications of Gas Equation 

If we describe the Ideal Gas equation in simple words, at the given amount of gas, the temperature will rise as the gas is compressed into a smaller volume(its particles come closer). Simultaneously, the temperature decreases as the gas expands in the larger volume (its particles move farther).

The same law applies in refrigerators; the cool gas is compressed, increasing its temperature. The hot gas, which is continuously passed through a radiator, permits extra heat to escape in the air, which allows the gas to expand into the refrigerator. As it expands, it cools off continuously, and heat is coming out from the refrigerator.

Limitations of Ideal Gas Equation 

• This equation works well when the density is low. With the increment in density, this equation does not provide relevant outcomes

• The Ideal Gas equation applies for single gasses and the mixture of various gasses where ‘n’ is the value of total moles of gas particles

Conclusion

In thermodynamics, Gas law is the best method to know the behavior of different gasses in diverse conditions. However, a Gas Equation or a General Gas Equation is used to describe the states of the hypothetical gasses, which is expressed mathematically through the combination of the empirical laws (Boyle’s law, Charles’s law and Avogadro’s law)  and physical constants.