# Question eded3

May 30, 2017

See Explanations Section

#### Explanation:

The Empirical Gas Law relationships are:

Boyles' Law: "Pressure" prop=(1/"Volume")#; mass & Temperture remain constant.
=> $\text{P"prop"(1/V)}$ => $P = k \left(\frac{1}{V}\right)$ => $k = \left(P V\right)$
=> ${k}_{1} = {k}_{2}$ => ${P}_{1} V 1 = {P}_{2} {V}_{2}$

Charles' Law: $\text{Volume "prop" f(Temperature)}$; Pressure & mass remain constant.
=> $V \propto T$ => $V = k T$ => $k = \left(\frac{V}{T}\right)$
=> ${k}_{1} = {k}_{2}$ => $\left({V}_{1} / {T}_{1}\right) = \left({V}_{2} / {T}_{2}\right)$

Gay-Lussac Law: $\text{Pressure "prop" f(Temperature)}$; mass & Volume remain constant.
=> $P \propto T$ => $P = k T$ => $k = \left(\frac{P}{T}\right)$
=> ${k}_{1} = {k}_{2}$ => $\left({P}_{1} / {T}_{1}\right) = \left({P}_{2} / {T}_{2}\right)$

Volume-Mass Law : $\text{Volume "prop" f(mass)}$; Pressure & Temperature remain constant.
=> $V \propto n$ => $V = k n$ => $k = \left(\frac{V}{n}\right)$; n = moles
=> ${k}_{1} = {k}_{2}$ => $\left({V}_{1} / {n}_{1}\right) = \left({V}_{2} / {n}_{2}\right)$

Pressure-Mass Law: $\text{Pressure "prop" f(mass)}$; Pressure & Temperature remain constant.
=> $P \propto n$ => $P = k n$ => $k = \left(\frac{P}{n}\right)$; n = moles
=> ${k}_{1} = {k}_{2}$ => $\left({P}_{1} / {n}_{1}\right) = \left({P}_{2} / {n}_{2}\right)$

Combined Gas Law
=>$P V \propto n T$ => $P V = k n T$ => $k = \left(\frac{P V}{n T}\right)$
=> ${k}_{1} = {k}_{2}$ => $\left(\frac{{P}_{1} {V}_{1}}{{n}_{1} {T}_{1}}\right) = \left(\frac{{P}_{2} {V}_{2}}{{n}_{2} {T}_{2}}\right)$

Ideal Gas Law => Assumes one of the P,V,n,T condition sets of the Combined Gas Law is at Standard Temperature - Pressure conditions (STP). The other set of P,V,n,T conditions are Non-Standard Conditions.

$S T P$ => $\left(P , V , n , T\right)$ $\left(1.0 A t m , 22.4 L , 1 \text{mole} , 273 K\right)$

=> $\left(\frac{{P}_{1} {V}_{1}}{{n}_{1} {T}_{1}}\right) = \frac{\left(1 A t m\right) \left(22.4 L\right)}{\left(1 m o l\right) \left(273 K\right)}$ = $0.08206 \frac{\left(L\right) \left(A t m\right)}{\left(m o l\right) \left(K\right)}$

= $\text{Universal" "Gas" "Constant} \left(R\right)$

Therefore, substituting 'R' into Combined Gas Law
=> $R = \left(\frac{P V}{n T}\right)$ => $P V = n R T$

Note:
One can also use the same logic for deriving relationships for ...

Henry's Law of Gas Solubility => $\text{ Solubility of a gas" prop } A p p l i e d P r e s s u r e$

Graham's Law of Gas Effusion Rates => $\text{Effusion Rate} \propto \left(\frac{1}{\sqrt{m o l w t}}\right)$