(Solved): For thermodynamic calculations, the volumetric coefficient of expansion is a fundamental property. ...

For thermodynamic calculations, the volumetric coefficient of expansion is a fundamental property. Looking up $\mathrm{Cu}$ in the $\mathrm{CRC}$, only the linear coefficient of expansion can be found and is equal to $16.5\mathrm{ppm}/\mathrm{K}$ ( $\mathrm{ppm}$ is part-per-million which is $10^{?6}$ ). Formally, the linear coefficient ${?}_L$ and the volume expansion coefficient $?={?}_V$ are defined mathematically as: $\begin{array}{rl}{?}_L& =\frac{1}{L}{\left(\frac{?L}{?T}\right)}_P\\ {?}_V& =\frac{1}{V}{\left(\frac{?V}{?T}\right)}_P\end{array}$ Derive a general relationship between linear and volume expansion coefficients assuming isotropic behavior, and determine the volume expansion coefficient for $\mathrm{Cu}$. Since the relationship can't depend on the specific shape, it may be helpful to consider either a sphere or a cube in linking volume and length.