Talk:PlanetPhysics/Dalton's Law

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Dalton's law %%% Primary Category Code: 51.30.+i %%% Filename: DaltonsLaw.tex %%% Version: 3 %%% Owner: pahio %%% Author(s): pahio %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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The gases are mixable with each other in all proportions.\, Since the \htmladdnormallink{ideal gas law}{http://planetphysics.us/encyclopedia/IdealGasLaw.html} \begin{align} pV = nRT \end{align} is valid for any ideal gas, one may think that it's insignificant whether the mole number $n$ concerns one single gas or several gases.\, It is true, which can be shown experimentally.

Let's think that we mix the \htmladdnormallink{volumes}{http://planetphysics.us/encyclopedia/Volume.html} $V_1$, $V_2$, ..., $V_k$ of different gases having an equal pressure $p$ and an equal \htmladdnormallink{temperature}{http://planetphysics.us/encyclopedia/BoltzmannConstant.html} $T$.\, If one measures the volume $V$ of the mixture in the same pressure and temperature, one notices that $$V = V_1\!+\!V_2\!+\!...\!+\!V_k.$$ Each of the gases satisfies an equation\, $pV_i = n_iRT$,\, and thus \begin{align} pV = pV_1\!+\!pV_2\!+\!...\!+\!pV_k = (n_1\!+\!n_2\!+\!...\!+\!n_k)RT. \end{align} This is similar as the general equation (1).\, If we think that the same volume $V$ would be filled by any of the gases alone, we had an equation $$p_iV = n_iRT$$ for each gas; here the pressure $p_i$, i.e. $n_i\frac{RT}{V}$, is called the {\em partial pressure} of the gas $i$.\, By (2), we have $$p = (n_1\!+\!n_2\!+\!...\!+\!n_k)\frac{RT}{V} = n_1\frac{RT}{V}\!+\!n_2\frac{RT}{V}\!+\!...\!+\!n_k\frac{RT}{V} = p_1\!+\!p_2\!+\!...\!+\!p_k.$$ Accordingly we have obtained the

\textbf{Dalton's law.}\, The pressure of a gas mixture is equal to the sum of the partial pressures of the component gases.

This law was invented by J. Dalton in 1801.

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