Derivation of ideal gas equation — task. Science TNSB Mentoring, Class 10 Crash course.

Answer the following question in detail:

Derive the ideal gas equation.

Benoît Paul Émile Clapeyron first proposed the ideal gas equation in as a combination of Boyle's law, Charles's law and Avogadro's law.

An ideal gas Boyle’s law and Charles’ law and Avogadro’s law.

According to Boyle’s law,

i α \frac{i}{i} at constant n and i

----- (Eq. 1)

According to Charles’s law,

i α i at constant n and i

----- (Eq. 2)

According to Avogadro’s law,

i α i at constant i and i

----- (Eq. 3)

Combining equations (Eq. 1), (Eq. 2) and (Eq. 3),

\(V\propto\frac{PT}{n}\)
\(V\propto\frac{nT}{P}\)
\(V\propto\frac{n}{PT}\)
\(V\propto\frac{T}{nP}\)

Rearranging the equation,

\(\frac{VT}{nP}\ =\ Constant\)
\(\frac{PV}{nT}\ =\ Constant\)
\(\frac{PT}{nV}\ =\ Constant\)
\(\frac{PVT}{n}\ =\ Constant\)

----- (Eq. 4)

The above relation is called the combined law of gases.

Consider a gas, which contains

μ

moles of the gas, the number of atoms contained will be equal to,

\(n\ =\ \mu \times N_{A}\)
\(n\ =\frac{N_{A}}{\mu }\)
\(n\ =\frac{\mu }{N_{A}}\)
\(n\ =\frac{N_{R}}{\mu }\)

----- (Eq. 5)

Substitute the Eq. 5 to Eq. 4, then the Eq. 4 can be written as,

\(\frac{PT}{\mu \times N_{A} \times V}\ =\ Constant\)
\(\frac{PVT}{\frac{\mu }{N_{A}}}\ =\ Constant\)
\(\frac{PV}{\mu \times N_{A} \times T}\ =\ Constant\)
\(\frac{VT}{\frac{\mu \times P }{N_{A}}}\ =\ Constant\)

----- (Eq. 6)

The value of the constant in the above equation is taken to be

K_{B}

, which is called as Boltzmann constant (

1.38 \times 10^{- 23} J K^{- 1}

Hence, we have the following equation:

\(\frac{PVT}{\frac{\mu }{N_{A}}}\ =\ K_{B}\)
\(\frac{VT}{\frac{\mu \times P }{N_{A}}}\ =\ K_{B}\)
\(\frac{PT}{\mu \times N_{A} \times V}\ =\ K_{B}\)
\(\frac{PV}{\mu \times N_{A} \times T}\ =\ K_{B}\)

----- (Eq. 6)

Here,

\(\mu \times N_{A}\times K_{B}\ =\ R\)
\(\frac{K_{B} \times N_{A}}{\mu}\ =\ R\)
\(\frac{\mu \times N_{A}}{K_{B}}\ =\ R\)
\(\frac{\mu \times N_{A}}{K_{B}}\ =\ A\)

which is termed as universal gas constant whose value is

8.31 J {mol}^{- 1} K^{- 1}

Substitute the value of \(R\) in the equation,

\(PV\ =\ RT\)
\(\frac{P}{V}\ =\ \frac{R}{T}\)
\(PV\ =\ \frac{R}{T}\)
\(\frac{P}{R}\ =\ \frac{R}{T}\)

Ideal gas equation is also called as equation of state because it gives the relation between the state variables and it is used to describe the state of any gas.

7. Derivation of ideal gas equation

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