Case equation without curly brace

You may better use aligned:

\documentclass[11pt,a4paper]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}

\begin{document}
\begin{equation}
\begin{aligned}
  \frac{I}{Z} &= 12+\frac{7}{Z}\;[eV],     && Z<13 \\
  \frac{I}{Z} &= 9.76+58.8Z^{-1.19}\;[eV], && Z\geq13
\end{aligned}
\end{equation}
\end{document}

Then use a regular array:

\documentclass{article}
\begin{document}
\begin{equation}
  \renewcommand{\arraystretch}{1.3}
  \begin{array}{ll}
    \frac{I}{Z}=12+\frac{7}{Z}[eV], & Z<13 \\
    \frac{I}{Z}=9.76+58.8Z^{-1.19}[eV], & Z\geq13
  \end{array}
\end{equation}
\end{document}

The alignment specification of {ll} inserts the contents left-aligned in both columns, while a redefinition of \arraystretch adds some vertical padding to each row. See Column padding in tables.

Since each element inside an array is set in \textstyle, fractions are smaller than usual by default. Adding \displaystyle will increase the fraction presentation, similar to amsmath's \dfrac.