Derivation Of Ideal Gas Equation

In thermodynamics, the ideal gas law is the well defined estimation of the behaviour of many gases under diverse conditions. The Ideal Gas Equation is a mathematical formula which expresses the states of hypothetical gases by combining empirical and physical constants. Ideal gas equation is also known as the general gas equation. The Ideal Gas Equation is made up of empirical laws including Charle’s law, Boyle’s law, Gay-law, Lussac’s and Avogadro’s law.

Equation of an Ideal Gas

It is the equation of state of a hypothetical ideal gas that is known as the Ideal gas law. The model is a good approximation to the behaviour of numerous gases under a wide range of situations, despite the fact that it has several drawbacks. The ideal gas equation can be expressed as follows:

PV = nRT is a mathematical formula.

Where,

• The ideal gas’s pressure is denoted by the letter P.
• The volume of the ideal gas is denoted by the letter V.
• The amount of ideal gas measured in moles is denoted by the letter n.
• The universal gas constant is denoted by the letter R.
• The temperature is denoted by the letter T.

In accordance with the Ideal Gas Equation –

In a gas, the relationship between the product of pressure and volume is constant, and the relationship between the product of universal gas constant and temperature is constant as well.

i.e. pv=nRT 

The Universal Gas Constant (R)

Observe that the product R for every given gas is always the same when the molecular mass of the gas is multiplied by the specific gas constant (R) of the gas. This product is referred to as the universal gas constant, and it is symbolised by the letter R.

Please keep in mind that the universal gas constant is 8.314 kJ/mole in the SI system.

The Ideal Gas Equation Derivation 

Take for example the pressure of a gas, denoted by the letter ‘p,’ and the volume of the gas, denoted by ‘v.’ Also, let the temperature be denoted by the letter ‘T.’ This equation denotes the universal gas constant R, where n denotes the number of moles of gas. Because of this, if the values of n and T are held constant, the volume of the gas is found to be inversely proportional to the pressure that is exerted by the gas according to Boyle’s Law. This can be expressed as follows:

V ∝ 1/P

With fixed p and n values, Charle’s Law states that the volume of a gas is directly proportional to its temperature when the values of p and n are maintained constant. This can be expressed as follows:

V ∝ T

The volume of a gas is directly proportional to the number of moles of the gas, according to Avogadro’s Law, if both P and T are held constant at their respective values. represented as following:

V ∝ n

If we add the results of all three equations, we get

V ∝ n T or PV = nRT

Combined gas law

Gay-law Lussac’s P/T = K is an amalgamation of three previously known gas laws: Boyle’s law PV = K, Charles law V/T = K, and Charles law P/T = K. The combined gas law is an amalgamation of three previously known laws. This results in a formula for combined gas law that reads as PV/T = K.

Where P denotes pressure, T is temperature, V denotes volume, and K denotes constant.

It is possible to modify the formula for the combined gas law in order to compare two different sets of conditions in the same substance. When the subscripts of one are used in the equation, the figures for temperature (T), pressure (P), and volume (V) indicate the initial condition. In addition, those with subscripts of two are reflective of the final condition of the experiment.

P1V1/T1 = P2V2 /T2

It is vital to note that for the purposes of calculation, the temperature should always be expressed in degrees Celsius (K). As a result, if the units are available on the Celsius scale, they must be converted to the kelvin scale. Furthermore, the conversion from degrees Celsius to degrees Kelvin is straightforward and may be accomplished by simply adding 273 to the specific unit.

Conclusion 

Known alternatively as the general gas equation, the ideal gas law describes how a hypothetical ideal gas behaves in terms of its equation of state. The model is a good approximation of the behaviour of various gases under a wide range of situations, despite the fact that it has several drawbacks. Clapeyron’s law was initially stated in 1834 as a mixture of the empirical Boyle’s law, Charles law, Avogadro’s law, and Gay-law, Lussac’s and it has been in use ever since.