The equation of state of an ideal elastic substance is
$$
7=K T\left(\frac{L}{L_{0}}-\frac{L_{0}^{2}}{L^{2}}\right)
$$
where $K$ is a constant and $L_{0}$ (the value of $L$ at zero tension) is a function of temperature only.
(a) Show that the isothermal Young's modulus is given by
$$
Y=\frac{7}{A}+\frac{3 K T L_{0}^{2}}{A L^{2}}
$$
(b) Show that the isothermal Young's modulus at zero tension is given by
$$
Y_{0}=\frac{3 K T}{A}
$$
(c) Show that the linear expansivity is given by
$$
\alpha=\alpha_{0}-\frac{7}{A Y T}=\alpha_{0}-\frac{1}{T} \frac{L^{3} / L_{0}^{3}-1}{L^{3} / L_{0}^{3}-2}
$$
where $\alpha_{0}$ is the value of the linear expansivity at zero tension, or
$$
\alpha_{0}=\frac{1}{L_{0}} \frac{d L_{0}}{d T}
$$
(d) Assume the following values for a sample of rubber: $T=300 \mathrm{~K}$, $K=1.333 \times 10^{-2} \mathrm{~N} / \mathrm{K}, A=1 \times 10^{-6} \mathrm{~m}^{2}, \alpha_{0}=5 \times 10^{-4} \mathrm{~K}^{-1}$. When this sample
is stretched to length $L=2 L_{0}$, calculate $7, Y$, and $\alpha$.