The Electronic Specific Heat of Ba$_{1-x}$K$_x$Fe$_2$As$_2$ from 2K to 380K*

Using a differential technique, we have measured the specific heats of polycrystalline Ba$_{1-x}$K$_{x}$Fe$_2$As$_2$ samples with $x$ = 0, 0.1 and 0.3, between 2K and 380K and in magnetic fields 0 - 13T. From this data we have determined the electronic specific coefficient $\gamma(\equiv C^{el}/T)$ over the entire range for the three samples. The sample with $x$ = 0.3 exhibits a large SC anomaly $\Delta\gamma(T_c)$ $\sim$ 48 mJ/mol K$^2$ at $T_c$ = 36K, and we determine the energy gap, condensation energy, superfluid density and coherence length. In the normal state for the $x$ = 0.3 sample, $\gamma$ $\sim$ 45 mJ/mol K$^2$ is constant from $T_c$ to 380K. In the parent compound ($x$ = 0) there is a large almost first order anomaly at the SDW transition at $T_o$ = 136K. The corresponding anomaly for the 0.1 sample at $T_o$ $\sim$ 135K is smaller and broader than for $x$ = 0. At low T, $\gamma$ is strongly reduced by the SDW gap for both $x$ = 0 and 0.1, but above $T_o$, $\gamma$ for all three samples are similar.