ラウールの法則

\documentclass[b5j,9pt,twoside]{tetsujsarticle}
\usepackage{tetsuchem}
\pagestyle{fancy}

\begin{document}

\小見出し{ラウールの法則}

不揮発性物質が溶けた溶液の飽和蒸気圧は，純溶媒の飽和蒸気圧に比べて低下する。この現象を，\gt{蒸気圧降下}という。その原因は，不揮発性物質が溶媒分子の蒸発を妨げることにある。

$N\〔\mol〕$の純溶媒に，$n\〔\mol〕$の不揮発性物質を溶かした希薄溶液の蒸気圧降下を，ミクロな観点から定量的に導出する。ただし，温度は常に一定とし，この温度における純溶媒の飽和蒸気圧を$P_0\〔\Pa〕$，溶液の飽和蒸気圧を$P\〔\Pa〕$とする。

\begin{center}
\begin{tikzpicture}[>=latex]
\node at (1.5,-0.4) {\small\gt{純溶媒}};
\draw[fill=black!35!white] (0,0) -- (0,1.25) -- (3,1.25) -- (3,0) --cycle;
\draw (0,1.25) -- (0,2.5) -- (3,2.5) -- (3,1.25);
\node[left,draw,single arrow,single arrow head extend=0.25cm,fill=black!40!white,rotate=90,scale=0.65] at (1.5,2.5) {\color{black!40!white}bp};%ここまで左容器
\draw[fill=white] (0.45,2) circle (0.15cm);
\draw[fill=white] (1.1,1.75) circle (0.15cm);
\draw[fill=white] (2.1,1.9) circle (0.15cm);
\draw[fill=white] (2.6,2.2) circle (0.15cm);%気相中の分子
\draw[fill=white] (0.35,1.05) circle (0.15cm);
\draw[fill=white] (0.95,1.05) circle (0.15cm);
\draw[fill=white] (1.4,1) circle (0.15cm);
\draw[fill=white] (1.8,1.05) circle (0.15cm);
\draw[fill=white] (2.65,1.05) circle (0.15cm);%液相１段目
\draw[fill=white] (0.3,0.72) circle (0.15cm);
\draw[fill=white] (0.7,0.79) circle (0.15cm);
\draw[fill=white] (1.1,0.71) circle (0.15cm);
\draw[fill=white] (1.6,0.7) circle (0.15cm);
\draw[fill=white] (2.2,0.76) circle (0.15cm);
\draw[fill=white] (2.8,0.65) circle (0.15cm);%液相２段目
\draw[fill=white] (0.2,0.35) circle (0.15cm);
\draw[fill=white] (0.75,0.45) circle (0.15cm);
\draw[fill=white] (1.35,0.5) circle (0.15cm);
\draw[fill=white] (1.93,0.58) circle (0.15cm);%液相３段目
\draw[fill=white] (0.5,0.2) circle (0.15cm);
\draw[fill=white] (1.15,0.2) circle (0.15cm);
\draw[fill=white] (1.75,0.25) circle (0.15cm);
\draw[fill=white] (2.35,0.3) circle (0.15cm);
\draw[fill=white] (2.75,0.2) circle (0.15cm);%液相４段目，ここまで左粒子
\draw[->] (0.35,1.2) -- (0.35,1.6);
\draw[<-] (0.65,1.25) -- (0.65,1.65);
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\draw[->] (1.8,1.2) -- (1.8,1.6);
\draw[<-] (2.225,1.25) -- (2.225,1.65);
\draw[->] (2.65,1.2) -- (2.65,1.6);%矢印
\node at (6.5,-0.4) {\small\gt{溶液}};
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\draw[fill=white] (5.45,2) circle (0.15cm);
\draw[fill=white] (7.1,1.9) circle (0.15cm);%気相中の分子
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\draw[fill=white] (6.8,1.05) circle (0.15cm);
\draw[fill=black] (7.65,1.05) circle (0.15cm);%液相１段目
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\draw[fill=black] (5.7,0.79) circle (0.15cm);
\draw[fill=white] (6.1,0.71) circle (0.15cm);
\draw[fill=white] (6.6,0.7) circle (0.15cm);
\draw[fill=black] (7.2,0.76) circle (0.15cm);
\draw[fill=white] (7.8,0.65) circle (0.15cm);%液相２段目
\draw[fill=black] (5.2,0.35) circle (0.15cm);
\draw[fill=white] (5.75,0.45) circle (0.15cm);
\draw[fill=white] (6.35,0.5) circle (0.15cm);
\draw[fill=white] (6.93,0.58) circle (0.15cm);%液相３段目
\draw[fill=white] (5.5,0.2) circle (0.15cm);
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\draw[fill=black] (7.35,0.3) circle (0.15cm);
\draw[fill=white] (7.75,0.2) circle (0.15cm);%液相４段目，ここまで右粒子
\draw[->] (5.95,1.2) -- (5.95,1.6);
\draw[<-] (6.375,1.25) -- (6.375,1.65);
\draw[->] (6.8,1.2) -- (6.8,1.6);
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\draw (5.2,0.35) -- (5.05,-0.75) -- (4.75,-0.75) node[left] {\parboxx{\footnotesize 不揮発性の\\溶質粒子}};
\end{tikzpicture}
\end{center}

溶液表面に存在する溶媒分子のみが液体部分から気体部分へと蒸発することが出来る。今，溶液表面に存在する粒子のうち，溶媒粒子の占める割合は$\bun{N}{N+n}$であるから，溶液表面から溶媒粒子が蒸発する勢いは純溶媒の時と比べて$\bun{N}{N+n}$倍になる。飽和蒸気圧の大きさは，粒子の蒸発する勢いに比例するので，溶液の飽和蒸気圧$P\〔\Pa〕$は，純溶媒の飽和蒸気圧$P_0\〔\Pa〕$の$\bun{N}{N+n}$倍になる。これを\gt{ラウールの法則}という。


\br2
\小見出し{蒸気圧降下}

下図のように，濃度の異なる溶液（溶質は不揮発性）の入った容器を連結して十分長時間放置すると，最終的にはどうなるか。

\begin{center}
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\end{tikzpicture}
\end{center}


\br2
\小見出し{過冷却}

凝固点降下を測定するために溶液を冷却する実験を行うと，\gt{過冷却}という現象が観察される。

\begin{center}
\begin{tikzpicture}[>=stealth]
\node at (1.2,3.65) {\small 純溶媒};
\node at (2.25,3) {\small 溶液};
\draw[-latex] (0,0) -- (6.25,0) node at +(-0.5,-0.2) {\small 冷却時間};
\draw[-latex] (0,0) -- (0,4) node at +(0,0.2) {\small 温度};
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\end{center}


\br2
\小見出し{U字管}

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\draw[densely dashed] ($(r)+(0.6,2.5)$) -- +(1.3,0);
\draw ($(r)+(1.3,0)$) arc (-90:90:0.1 and 0.3);
\draw ($(r)+(1.25,0.6)$) -- +(-0.06,-0.06);
\draw ($(r)+(1.28,0.58)$) -- +(-0.11,-0.11);
\draw ($(r)+(1.3,0.55)$) -- +(-0.145,-0.145);
\draw ($(r)+(1.32,0.52)$) -- +(-0.17,-0.17);
\draw ($(r)+(1.33,0.48)$) -- +(-0.18,-0.18);
\draw ($(r)+(1.34,0.44)$) -- +(-0.19,-0.19);
\draw ($(r)+(1.345,0.395)$) -- +(-0.19,-0.19);
\draw ($(r)+(1.35,0.35)$) -- +(-0.19,-0.19);
\draw ($(r)+(1.35,0.3)$) -- +(-0.183,-0.183);
\draw ($(r)+(1.35,0.25)$) -- +(-0.17,-0.17);
\draw ($(r)+(1.34,0.19)$) -- +(-0.14,-0.14);
\draw ($(r)+(1.33,0.13)$) -- +(-0.13,-0.13);
\draw ($(r)+(1.31,0.06)$) -- +(-0.06,-0.06);
\node at ($(r)+(1.25,-0.4)$) {状態２};
\node at ($(r)+(1.25,-0.8)$) {\gt{（つりあっている）}};
\end{tikzpicture}
\end{center}


\br2
\小見出し{ヘンリーの法則（定圧容器）}

\begin{center}
\begin{tikzpicture}
\draw (0,2.5) -- (0,0) -- (1.5,0) -- (1.5,2.5);
\draw (0,0.85) -- (1.5,0.85);
\draw[fill=black] (0,1.8) -- (1.5,1.8) -- (1.5,2) -- (0.85,2) -- (0.85,3.2) -- (0.65,3.2) -- (0.65,2) -- (0,2) -- cycle;
\node at (0.75,1.3) {\footnotesize\ce{CO2}};
\node at (0.75,0.4) {\footnotesize 水 500\,\mL};
\draw[fill=black] (0.1,0.13) circle (0.02);
\draw[fill=black] (0.4,0.1) circle (0.02);
\draw[fill=black] (0.7,0.1) circle (0.02);
\draw[fill=black] (1,0.1) circle (0.02);
\draw[fill=black] (1.3,0.1) circle (0.02);
\draw[fill=black] (1.45,0.27) circle (0.02);
\draw[fill=black] (0.12,0.5) circle (0.02);
\draw[fill=black] (0.25,0.63) circle (0.02);
\draw[fill=black] (0.55,0.63) circle (0.02);
\draw[fill=black] (0.85,0.63) circle (0.02);
\draw[fill=black] (1.15,0.63) circle (0.02);
\draw[fill=black] (1.45,0.63) circle (0.02);
\draw[fill=black] (0.15,0.77) circle (0.02);
\draw[fill=black] (0.45,0.77) circle (0.02);
\draw[fill=black] (0.75,0.77) circle (0.02);
\draw[fill=black] (1.05,0.77) circle (0.02);
\draw[fill=black] (1.35,0.77) circle (0.02);
\end{tikzpicture}
\end{center}


\br2
\小見出し{ヘンリーの法則（定積容器）}
\begin{center}
\begin{tikzpicture}
\draw[fill=gray] (0,0) -- (0,1) -- (3.75,1) -- (3.75,0) -- cycle;
\draw (0,1) -- (0,2.5) -- (3.75,2.5) -- (3.75,1);
\draw (0,2.5) arc (90:270:0.4 and 0.75) node[left,fill=white,yshift=0.75cm] {$V_1\,〔\L〕$};
\draw (0,1) arc (90:270:0.2 and 0.5) node[left,fill=white,yshift=0.5cm] {$V_2\,〔\L〕$};
\node at (1.875,1.75) {$n$〔\mol〕};%ここまで左の容器
\draw[fill=black] (0.5,1.5) circle (0.07);
\draw[fill=black] (1.2,1.35) circle (0.07);
\draw[fill=black] (2.5,1.25) circle (0.07);
\draw[fill=black] (3.4,1.3) circle (0.07);
\draw[fill=black] (3,1.75) circle (0.07);
\draw[fill=black] (3.2,2.15) circle (0.07);
\draw[fill=black] (2.35,2.3) circle (0.07);
\draw[fill=black] (1.2,2.1) circle (0.07);
\draw[fill=black] (0.82,1.93) circle (0.07);
\draw[fill=black] (0.35,2.2) circle (0.07);%ここまで左の分子
\draw[-stealth] (4.25,1.25) -- (5.25,1.25);
\draw[fill=gray] (6.5,0) -- (6.5,1) -- (10.25,1) -- (10.25,0) -- cycle;
\draw (6.5,1) -- (6.5,2.5) -- (10.25,2.5) -- (10.25,1);
\draw (6.5,2.5) arc (90:270:0.4 and 0.75) node[left,fill=white,yshift=0.75cm] {$V_1\,〔\L〕$};
\draw (6.5,1) arc (90:270:0.2 and 0.5) node[left,fill=white,yshift=0.5cm] {$V_2\,〔\L〕$};%ここまで右の容器
\draw[fill=black] (7,1.5) circle (0.07);
\draw[fill=black] (8,1.65) circle (0.07);
\draw[fill=black] (8.6,1.5) circle (0.07);
\draw[fill=black] (9.75,1.3) circle (0.07);
\draw[fill=black] (9.7,2.15) circle (0.07);
\draw[fill=black] (6.85,2.2) circle (0.07);
\draw[fill=black] (7,0.5) circle (0.07);
\draw[fill=black] (7.7,0.25) circle (0.07);
\draw[fill=black] (8.6,0.45) circle (0.07);
\draw[fill=black] (9.9,0.8) circle (0.07);%ここまで右の分子
\end{tikzpicture}
\end{center}

\end{document}