Question #eded3

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Question #eded3

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Answer 1 · 2017-05-30T01:18:47

See Explanations Section

Explanation:

The Empirical Gas Law relationships are:

Boyles' Law: $Pressure \propto = (\frac{1}{Volume})$ $"Pressure" prop=(1/"Volume")$ ; mass & Temperture remain constant.
=> $P \propto (1/V)$ $"P"prop"(1/V)"$ => $P = k (\frac{1}{V})$ $P=k(1/V)$ => $k = (P V)$ $k = (PV)$
=> $k_{1} = k_{2}$ $k_1=k_2$ => $P_{1} V 1 = P_{2} V_{2}$ $P_1V1 = P_2V_2$

Charles' Law: $Volume \propto f(Temperature)$ $"Volume "prop" f(Temperature)"$ ; Pressure & mass remain constant.
=> $V \propto T$ $VpropT$ => $V = k T$ $V=kT$ => $k = (\frac{V}{T})$ $k = (V/T)$
=> $k_{1} = k_{2}$ $k_1=k_2$ => $(\frac{V_{1}}{T_{1}}) = (\frac{V_{2}}{T_{2}})$ $(V_1/T_1)=(V_2/T_2)$

Gay-Lussac Law: $Pressure \propto f(Temperature)$ $"Pressure "prop" f(Temperature)"$ ; mass & Volume remain constant.
=> $P \propto T$ $PpropT$ => $P = k T$ $P=kT$ => $k = (\frac{P}{T})$ $k = (P/T)$
=> $k_{1} = k_{2}$ $k_1=k_2$ => $(\frac{P_{1}}{T_{1}}) = (\frac{P_{2}}{T_{2}})$ $(P_1/T_1)=(P_2/T_2)$

Volume-Mass Law : $Volume \propto f(mass)$ $"Volume "prop" f(mass)"$ ; Pressure & Temperature remain constant.
=> $V \propto n$ $Vpropn$ => $V = k n$ $V=kn$ => $k = (\frac{V}{n})$ $k = (V/n)$ ; n = moles
=> $k_{1} = k_{2}$ $k_1=k_2$ => $(\frac{V_{1}}{n_{1}}) = (\frac{V_{2}}{n_{2}})$ $(V_1/n_1)=(V_2/n_2)$

Pressure-Mass Law: $Pressure \propto f(mass)$ $"Pressure "prop" f(mass)"$ ; Pressure & Temperature remain constant.
=> $P \propto n$ $Ppropn$ => $P = k n$ $P=kn$ => $k = (\frac{P}{n})$ $k = (P/n)$ ; n = moles
=> $k_{1} = k_{2}$ $k_1=k_2$ => $(\frac{P_{1}}{n_{1}}) = (\frac{P_{2}}{n_{2}})$ $(P_1/n_1)=(P_2/n_2)$

Combined Gas Law
=> $P V \propto n T$ $PVpropnT$ => $P V = k n T$ $PV=knT$ => $k = (\frac{P V}{n T})$ $k = ((PV)/(nT))$
=> $k_{1} = k_{2}$ $k_1=k_2$ => $(\frac{P_{1} V_{1}}{n_{1} T_{1}}) = (\frac{P_{2} V_{2}}{n_{2} T_{2}})$ $((P_1V_1)/(n_1T_1)) =((P_2V_2)/(n_2T_2))$

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.

$S T P$ $STP$ => $(P, V, n, T)$ $(P,V, n, T)$ $(1.0 A t m, 22.4 L, 1 mole, 273 K)$ $(1.0Atm, 22.4L, 1"mole", 273K)$

=> $(\frac{P_{1} V_{1}}{n_{1} T_{1}}) = \frac{(1 A t m) (22.4 L)}{(1 m o l) (273 K)}$ $((P_1V_1)/(n_1T_1)) = ((1Atm)(22.4L))/((1mol)(273K))$ = $0.08206 \frac{(L) (A t m)}{(m o l) (K)}$ $0.08206((L)(Atm))/((mol)(K))$

= $Universal Gas Constant (R)$ $"Universal" "Gas" "Constant" (R)$

Therefore, substituting 'R' into Combined Gas Law
=> $R = (\frac{P V}{n T})$ $R = ((PV)/(nT))$ => $P V = n R T$ $PV = nRT$

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

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

Graham's Law of Gas Effusion Rates => $Effusion Rate \propto (\frac{1}{\sqrt{m o l w t}})$ $"Effusion Rate" prop (1/sqrt(mol wt))$

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