The diffusion coefficients of the first five n-alkanes in water at infinite dilution have been obtained from molecular dynamics simulations over a wide range of temperatures at 0.1 MPa and additionally, for methane and n-butane, at higher pressures up to 200 MPa. Comparison with available experimental data provides confidence in the accuracy of the predictions using the TIP4P/2005 model for water and the TraPPE force field for n-alkanes. Additionally, a Speedy-Angell-type phenomenological equation that captures the pressure and temperature behavior of the methane and n-butane is provided which can be used for engineering calculations. Furthermore, it is shown that the diffusion coefficients of methane and n-butane obey the Stokes-Einstein equation. Finally, the molecular structure of water - n-alkane mixtures is examined.