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We present a systematic study of the electronic specific heat jump ($\DeltaC_{\rm el}$) at the superconducting transition temperature $T_c$ ofK$_{1-x}$Na$_x$Fe$_2$As$_2$. Both $T_c$ and $\Delta C_{\rm el}$ monotonouslydecrease with increasing $x$. The specific heat jump scales approximately witha power-law, $\Delta C_{\rm el} \propto T_c^{\beta}$, with $\beta \approx 2$determined by the impurity scattering rate, in contrast to most iron-pnictidesuperconductors, where the remarkable Bud'ko-Ni-Canfield (BNC) scaling $\DeltaC_{\rm el} \propto T^3$ has been found. Both the $T$ dependence of $C_{\rmel}(T)$ in the superconducting state and the nearly quadratic scaling of$\Delta C_{\rm el}$ at $T_c$ are well described by the Eliashberg-theory for atwo-band $d$-wave superconductor with weak pair-breaking due to nonmagneticimpurities. The disorder induced by the Na substitution significantlysuppresses the small gaps leading to gapless states in the slightly disorderedsuperconductor, which results in a large observed residual Sommerfeldcoefficient in the superconducting state for $x > 0$.

Superconducting specific heat jump $\Delta C_{\rm el} \propto T_c^\beta \ (\beta \approx 2)$ for K$_{1-x}$Na$_x$Fe$_2$As$_2$