\documentclass[11pt,a4paper]{article}
\usepackage[latin1]{inputenc}
\usepackage{ngerman}
\usepackage{textcomp}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{graphics}
\usepackage{textcomp}
\usepackage{color}               % farbe

\usepackage[pdftex]{hyperref,graphicx,color} % pdflinks
\pdfoutput=1

%Maple
\usepackage{maple2e}
\def\emptyline{\vspace{12pt}}
\DefineParaStyle{maplegroup}
\DefineParaStyle{Author}
\DefineParaStyle{subtitle}
\DefineParaStyle{Bullet Item}
\DefineParaStyle{Dash Item}
\DefineParaStyle{Diagnostic}
\DefineParaStyle{Error}
\DefineParaStyle{Heading 1}
\DefineParaStyle{Heading 2}
\DefineParaStyle{Heading 3}
\DefineParaStyle{Heading 4}
\DefineParaStyle{Normal}
\DefineParaStyle{Text Output}
\DefineParaStyle{Title}
\DefineParaStyle{Warning}
\DefineCharStyle{2D Math}
\DefineCharStyle{Help Heading}
\DefineCharStyle{Maple Input}
\DefineCharStyle{FixedWidth}
\DefineCharStyle{endFixedWidth}

\newcommand{\abs}[1]{\lvert#1\rvert}


\hypersetup
{%
pdftitle = {Atmoshärenphysik, Übung 3},%
pdfsubject = {Atmosphärenphysik, Umweltnaturwissenschaften 5. Semester},%
pdfauthor = {Thomas Kuster (5. Semester, D-UWIS)},%
pdfkeywords = {Atmosphärenphysik, Übung},%
colorlinks = {true},%
linkcolor = {blue}
}

\setlength{\parindent}{0pt} %kein einzug links nach einem abschnitt

\date{\today{}}
\author{Thomas Kuster}
\title{UWIS, Atmosphärenphysik, Übung 3}

\begin{document}
\markright{UWIS, Atmosphärenphysik, Übung 3}
\pagestyle{myheadings}
\maketitle

\section{}
\subsection{Kritischer Radius und Sättigungsverhältnis}
\begin{eqnarray*}
T & = & 275\,\rm{\,K}\\
a & = & \frac{2 \sigma}{\varrho_w R_v T} = \frac{3.3\cdot 10^{-7}}{T} = 1.2 \cdot 10^{-9}\rm{\,m}\\
i & = & 2\\
m & = & 10^{-14}\rm{\,g}\\
M_s & = & \\
b & = & \frac{3 i M_w}{4 \pi \varrho_w M_s} \approx \frac{4.3 \cdot 10^{-6} i m}{M_s} =
\end{eqnarray*}

Kritischer Radius ($r^{*}$)
$$
r^{*} = \sqrt{\frac{3 b}{a}} = \left\{
\begin{array}{ll}
1.918 \cdot 10^{-6}\rm{\,m} & \textrm{Seesalz}\\
1.276 \cdot 10^{-6}\rm{\,m} & \textrm{Ammoniumsulfat}
\end{array} \right.
$$

Kritische Sättigungsverhältnis
$$
S^{*} = \sqrt{\frac{4 a^3}{27 b}} = \left\{
\begin{array}{ll}
4.171 \cdot 10^{-4}\,[] & \textrm{Seesalz}\\
6.272 \cdot 10^{-4}\,[] & \textrm{Ammoniumsulfat}
\end{array} \right.
$$

\subsection{Monoton abnehmend}
\subsubsection{Maple}
\begin{mapleinput}
\mapleinline{active}{1d}{restart;}{}
\end{mapleinput}\begin{mapleinput}
\mapleinline{active}{1d}{S_ver := (1-b_2/r_2^3)/(1-b_1/r_1^3);}{}
\end{mapleinput}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{S_ver := (1-b_2/r_2^3)/(1-b_1/r_1^3)}{%
\[
{\it S\_ver}\, := \, \left( 1-{\frac {{\it b\_2}}{{{\it r\_2}}^{3}}} \right)  \left( 1-{\frac {{\it b\_1}}{{{\it r\_1}}^{3}}} \right) ^{-1}
\]
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{b_1 := 1.47e-21;}{}
\end{mapleinput}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{b_1 := 0.147e-20}{%
\[
{\it b\_1}\, := \,{ 1.47\times 10^{-21}}
\]
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{b_2 := 6.508e-22;}{}
\end{mapleinput}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{b_2 := 0.6508e-21}{%
\[
{\it b\_2}\, := \,{ 6.508\times 10^{-22}}
\]
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{r_1_0 := 1.918e-6;}{}
\end{mapleinput}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{r_1_0 := 0.1918e-5}{%
\[
{\it r\_1\_0}\, := \, 0.000001918
\]
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{r_2_0 := 1.276e-6;}{}
\end{mapleinput}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{r_2_0 := 0.1276e-5}{%
\[
{\it r\_2\_0}\, := \, 0.000001276
\]
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{plot3d(S_ver, r_1=r_1_0..5e-6, r_2=r_2_0..5e-6);}{}
\end{mapleinput}
\mapleresult
\begin{center}
\includegraphics[width=0.8\linewidth]{maple/graph_1.jpg}

\end{center}
\begin{mapleinput}
\mapleinline{active}{1d}{r_1:=r_2; plot(S_ver, r_2=r_2_0..5e-6);}{}
\end{mapleinput}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{r_1 := r_2}{%
\[
{\it r\_1}\, := \,{\it r\_2}
\]
}

\end{maplelatex}

\mapleresult
\begin{center}
\includegraphics[width=0.8\linewidth]{maple/graph_2.jpg}

\end{center}
\begin{mapleinput}
\mapleinline{active}{1d}{}{}
\end{mapleinput}

\subsubsection{Limes bilden}
$$
S = \frac{e^{*}}{e_{sat}{(\infty{})}} = 1 - \frac{b}{r^3}
$$

$$
\frac{S_2}{S_1} = \frac{1-\frac{k_2}{r_2^3}}{1-\frac{k_1}{r_1^3}} \Rightarrow \lim_{r \rightarrow \infty{}} \frac{1-\frac{k_2}{r_2^3}}{1-\underbrace{\frac{k_1}{r_1^3}}_{\rightarrow 0}} = 1
$$

oder auch mit:

$$
S = \frac{e^{*}(r)}{e_s(\infty{})} = \left(1 - \frac{b}{r^3}\right)\exp(\frac{a}{r})
$$

$$
\frac{S_2}{S_1} = \frac{\left(1 - \frac{k_{21}}{r^3}\right)\exp{\left(\frac{k_{22}}{r}\right)}}{\left(1 - \frac{k_{11}}{r^3}\right)\exp{\left(\frac{k_{12}}{r}\right)}}
\Rightarrow
\lim_{r \rightarrow \infty{}}
{
\frac{\left(1 - \frac{k_{21}}{r^3}\right)\exp{\left(\frac{k_{22}}{r}\right)}}{\left(1 - \frac{k_{11}}{r^3}\right)\underbrace{\exp{\left(\underbrace{\frac{k_{12}}{r}}_{\rightarrow 0}\right)}}_{\rightarrow 1}}
} = 1
$$

$$
\lim_{r \rightarrow \infty}{S_x} = 1 \Rightarrow S_2 - S_1 = 1 - 1 = 0
$$

\end{document}