This work presents an equation of state at finite temperature and finite chemical potential using the PNJl model. The equation of state goes to the next to leading order O(1/N_c) including mesons participating to the pressure of the medium below T_c.Together with a new parametrisation of the effective Polyakov loop involving the presence of quarks in the determination of the critical temperature, is determined a pressure at zero chemical potential matching the result of the lattice and expand to finite chemical potential. Also derivatives as the entropy density, energy density, measured interaction and speed of sound are very similar to lattice results at zero chemical potential and can be expanded to finite chemical potential. Regarding the finite chemical potential, the Taylor expansion around zero chemical potential also matches the results of the lattice calculations and the low temperature and large chemical potential pressure is in agreement with pQCD calculations. This new parametrised pnjl model allows for the calculation of elastic cross sections entering the computation of the transport coefficients of the QGP at finite temperature and chemical potential in the relaxation time approximation.
