# Question #28264

Mar 29, 2017

#### Explanation:

Here's how to derive the law from Boyle's Law and Charles' Law.

Consider an ideal gas at conditions ${p}_{1} , {V}_{1} , {T}_{1}$.

Now, keep $T$ constant and vary $p$ and $V$ to bring the gas to a second state ${p}_{2} , V , {T}_{1}$.

According to Boyle's Law:

(1) ${p}_{1} {V}_{1} = {p}_{2} V$

Now, keep $p$ constant and vary $V$ and $T$ to bring the gas to a third state ${p}_{1} , {V}_{2} , {T}_{2}$.

According to Charles Law,

(2) $\frac{V}{T} _ 1 = {V}_{2} / {T}_{2}$

From (1),

(3) $V = \frac{{p}_{1} {V}_{1}}{p} _ 2$

From (2)

(4) $V = \frac{{V}_{2} {T}_{1}}{T} _ 2$

Equating the right hand sides of (3) and (4), we get

$\frac{{p}_{1} {V}_{1}}{p} _ 2 = \frac{{V}_{2} {T}_{1}}{T} _ 2$

or

$\frac{{p}_{1} {V}_{1}}{T} _ 1 = \frac{{p}_{2} {V}_{2}}{T} _ 2 = {k}^{'}$ (a constant)

In general, we can write this as

$\frac{p V}{T} = k '$ or $p = \left({k}^{'} / V\right) T$

Now, if we hold the volume $V$ constant, and let ${k}^{'} / V = k$, we get

$p = k T$,

which is Gay-Lussac's Law.