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# The compressibility $\kappa$ of a substance is defined as the fractional change in volume of that substance for a given change in pressure:$\kappa=-\frac{1}{V} \frac{d V}{d P}$(a) Explain why the negative sign in this expression ensures $\kappa$ is always positive. (b) Show that if an ideal gas is compressed isothermally, its compressibility is given by $\kappa_{1}=1 / P$. (c) Show that if an ideal gas is compressed adiabatically, its compressibility is given by $\kappa_{2}=1 /(\gamma P) .$ Determine values for (d) $\kappa_{1}$ and (e) $\kappa_{2}$ for a monatomic ideal gas at a pressure of 2.00 atm.

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The compressibility k of a substance is defined as:

$k=-\dfrac{1}{V}\cdot \dfrac{dV}{dP}$

$\textbf{A)}$ If the pressure increases, the volume will decreases, and otherwise. So the - sign will ensure that compressibility of a substance always stays positive.

$\textbf{ B)}$ Ifan ideal gas is compressed isothermally, its compressibility is given by:

\begin{align*} k_1&=-\dfrac{1}{V}\cdot \dfrac{dV}{dP}=-\dfrac{1}{V}\cdot \dfrac{d}{dP}\left(\dfrac{nRT}{P}\right)=-\dfrac{nRT}{V} \dfrac{d}{dP}\left(\dfrac{1}{P}\right)\\\\ &=\dfrac{nRT}{VP^2} =\dfrac{P}{P^2}\\\\ &=\boxed{\dfrac{1}{P}}\\ \end{align*}

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