The melting process of a spherical nanoparticle is analysed using a mathematical model derived from continuum theory. The standard model for macro-scale melting is modified to include melting point depression using the Gibbs-Thomson equation. The key difference between the current and previous work in the melting of nanoparticles is that the difference in densities between the solid and liquid phases is accounted for. This modifies the energy balance at the phase change interface to include a kinetic energy term, which then changes the form of the equation, and it also requires an advection term in the heat equation for the liquid phase. Approximate analytical and numerical solutions are presented for the melting of particles in the range 10-100 nm. It is shown that when the density difference is included in the model, melting is significantly slower than when density is assumed constant throughout the process. This is attributed to the flowing liquid providing a sink term, namely kinetic energy, in the energy balance. The difference in results is greatest for small particles; however, it is concluded that the varying density model will never reduce to the constant density model resulting in a difference of around 15 % even at the macro-scale.