DaltonsLaw 2006-06-02 12:55:14 2006-06-02 12:55:14 Derivation ideal gas law 1801 partial pressure % this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The gases are mixable with each other in all proportions.\, Since the ideal gas law \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 volumes $V_1$, $V_2$, ..., $V_k$ different gases having an equal pressure $p$ and an equal temperature $T$.\, 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 the 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. $n_iRT/V$, 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 the component gases.