We study the hydrodynamic limit for a one dimensional isothermal anharmonic finite chain in Lagrangian coordinates with hyperbolic space-time scaling. The temperature is kept constant by putting the chain in contact with a heat bath, realised via a stochastic momentum-preserving noise added to the dynamics. The noise is designed to be large at the microscopic level, but vanishing in the macroscopic scale. Boundary conditions are also considered: one end of the chain is kept fixed, while a time-variable tension is applied to the other end. We show that the microscopic deformation andmomentum converge (in an appropriate sense) to a weak solution of a system of hyperbolic conservation laws (isothermal Euler equations in Lagrangian coordinates) with boundary conditions. This result include the shock regime of the system. This is done by adapting the theory of compensated compactness to a stochastic setting, as developed by J. Fritz in \cite{Fritz1} for the same model without boundary conditions. Finally, changing the external tension allows us to define thermodynamic transformations between equilibrium states. We use this to deduce the first and the second principle of Thermodynamics for our model.