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We investigate the melting properties of the bcc refractory metals V and W, and the disordered equiatomic VW alloy from first principles. We show that thermal vibrations have a large impact on the electronic density of states (DOS) and thus considerably affect the electronic contribution to the free energy. For W, the impact of vibrations on the electronic free energy of solid and liquid is different. This difference substantially impacts the computed melting point and also triggers a large electronic heat capacity difference between solid and liquid. For V, although vibrations likewise affect the electronic free energy, the effect on the melting properties cancels out to a large degree. For the binary VW alloy we observe a similar impact as for W, but slightly weaker. The underlying physics is explained in terms of the electronic DOS of the solid and liquid phases. Based on our accurate first-principles results, we reveal critical limitations of the Sommerfeld approximation in predicting the electronic heat capacity difference between solid and liquid. Our results thus prompt us to scrutinize this approximation which is often used in phase diagram parametrizations in the CALPHAD approach, as well as for materials, such as W, that have a large electronic DOS difference between solid and liquid at the melting temperature.