Tentukan kompresibilitas dari gas ideal dan gas nyata!

Gas Ideal

Kita tahu [link]

$$ k = -\dfrac{1}{V} \left( \dfrac{\partial V}{\partial P} \right)_{\!T} \label{k}\tag{k}$$

$$ PV = nRT $$

Entah kenapa persamaan sakti ini penting banget dalam termodinamika, jangan dilupain ok ūüėČ apalagi \(-1\) nya

$$ f(P, V, T) = \left( \dfrac{\partial P}{\partial V} \right)_{\!T}
\left( \dfrac{\partial V}{\partial T} \right)_{\!P}
\left( \dfrac{\partial T}{\partial P} \right)_{\!V} = -1 \label{z}\tag{0}$$

dari situ (sekreatif mungkin)

$$ \left( \dfrac{\partial P}{\partial V} \right)_{\!T} = \dfrac{
\left( \dfrac{\partial T}{\partial V} \right)_{\!P}}{
\left( \dfrac{\partial T}{\partial P} \right)_{\!V}}$$

dibalik biar sama seperti 1, karena butuhnya \(\left(\dfrac{V}{P}\right)_{\!T}\) jadi

$$ \left( \dfrac{\partial V}{\partial P} \right)_{\!T} = -\dfrac{
\left( \dfrac{\partial V}{\partial T} \right)_{\!P}}{
\left( \dfrac{\partial P}{\partial T} \right)_{\!V}}\label{a}\tag{1}$$

untuk \( V = \dfrac{nRT}{P} \)

$$ \left( \dfrac{\partial V}{\partial T} \right)_{\!P} = \dfrac{nR}{P} $$

untuk \( P = \dfrac{nRT}{V} \)

$$ \left( \dfrac{\partial P}{\partial T} \right)_{\!V} = \dfrac{nR}{V} $$

substitusi ke persamaan \ref{a}

$$
\begin{align}
\require{cancel} \left( \dfrac{\partial V}{\partial P} \right)_{\!T} & = -\dfrac{
\dfrac{\cancel{nR}}{P}}{
\dfrac{\cancel{nR}}{V}} \\
& = -\dfrac{V}{P}
\end{align}
$$

subtitusi ke persamaan \ref{k}, sabar ya…

$$
\begin{align}
k & = -\dfrac{1}{\cancel{V}} \left( -\dfrac{\cancel{V}}{P} \right) \\
& = \dfrac{1}{P}
\end{align}
$$

DONE! Cara ini juga berlaku untuk \(\alpha\) atau \(\beta\) didapatkan

$$
\alpha = \dfrac{1}{T} \qquad \beta= \dfrac{1}{T}
$$

Gas Nyata

kita tahu pada gas nyata digunakan persamaan Van Der Waals (n=1)

$$ \left( P + \dfrac{a}{V^2}  \right)\left(V-b\right) = RT $$

langkah persamaan sakti (\ref{z})sama seperti diatas, skip aja ok :p

untuk \( V = \dfrac{RT}{\left( P + \dfrac{a}{V^2}  \right)} + b\)

$$ \left( \dfrac{\partial V}{\partial T} \right)_{\!P} =
\dfrac{R}{\left( P + \dfrac{a}{V^2}  \right)} $$

untuk \( P = \dfrac{RT}{\left(V-b\right)} – \dfrac{a}{V^2}\)

$$ \left( \dfrac{\partial P}{\partial T} \right)_{\!V} =
\dfrac{R}{\left(V-b\right)} $$

substitusi ke persamaan \ref{a}

$$
\left( \dfrac{\partial V}{\partial P} \right)_{\!T} = -\dfrac{
\dfrac{\cancel{R}}{\left( P + \dfrac{a}{V^2}  \right)}}{
\dfrac{\cancel{R}}{\left(V-b\right)}} \\
\quad\quad\quad = -\dfrac{\left(V-b\right)}{\left( P + \dfrac{a}{V^2}  \right)}
$$

subtitusi ke persamaan \ref{k}

$$
\begin{align}
k & = -\dfrac{1}{\cancel{V}} \left(-\dfrac{\left(V-b\right)}{\left( P + \dfrac{a}{V^2}  \right)} \right) \\
& = -\dfrac{\left(V-b\right)}{V\left( P + \dfrac{a}{V^2}  \right)} \\
& = \dfrac{V\left(V-b\right)}{PV^2 + a}
\end{align}
$$