Answer to Question #134797 in Mechanics | Relativity for gee

Question

Pressure is dependent on volume, temperature, amount of substance and ideal gas constant of R. Use dimensional analysis to derive the formula of the ideal gas law.

Answer 1

To derive the ideal gas law, we consider the relationship between it's volume, $V$ and pressure, $P$, temperature, $T$, and amount of substance, $n$.

From Boyle's law, $V$ $\propto \frac{1}{P}$ at constant $n$ and $T$

From Charle's law $V \propto T$ at constant $n$ and $P$

From Avogadro's law, $V \propto n$ at constant $T$ and $P$

The three expressions can be combined to give,

$V \propto \frac{nT}{P}$ which can also be written as,

$V = constant(\frac{nT}{P})$

The constant is called the ideal gas constant and, is represented by $R$.

Therefore, $V = \frac{RnT}{P} = \frac{nRT}{P}$

multiply both sides by $P$ to get,

$PV = nRT$ (the ideal gas law equation)