Step 1: Experiments on different isotopes of the same superconducting element (for example mercury) show that heavier isotopes have a lower critical temperature. This is the isotope effect, a key clue that lattice vibrations drive conventional superconductivity.
Step 2: The empirical law is written $T_c M^{\alpha} = \text{constant}$, where $\alpha$ is the isotope exponent. For the ideal BCS case $\alpha = 1/2$.
Step 3: Substituting $\alpha = 1/2$ gives $T_c M^{1/2} = \text{constant}$, equivalently $M^{1/2}T_c = \text{constant}$. Rearranged, $T_c \propto 1/\sqrt{M}$.
Step 4: The square-root dependence comes directly from the Debye frequency $\omega_D \propto M^{-1/2}$ of the ion lattice; since the pairing energy scale follows the phonon frequency, $T_c$ inherits the same mass dependence.\[\boxed{M^{1/2}\,T_c = \text{constant}}\]