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The LHCb experiment will be installed in the proton-proton Large Hadron Collider (LHC) at CERN, Geneva. The detector is a single arm spectrometer currently under construction: LHC running and LHCb data taking will start in 2007. LHCb will then benefit from the prolific source of B-mesons provided by the LHC. The main goals of the LHCb experiment are to measure the CP asymmetries in the B-meson sector and to study rare decays of b-hadrons. These will extend the measurements presently made with B_d mesons by the Belle (Japan) and BABAR (USA) experiments. The expected accuracy on the comprehensive measurements with both B_d and B s mesons will allow to open new windows on physics beyond the Standard Model. The Standard Model of particle physics (SM) provides the framework for the description of a possible violation of the CP symmetry in the neutral B-meson sector. In particular, it predicts an asymmetry due to CP violation in the time dependent rates for B_{d,s} and B_{d,s}- bar to a common CP eigenstate when this transition is dominated by the subprocess b --> ccs: this is due to the interference between the decay and the mixing of these neutral mesons. In this case, the CP violation is directly related to sin(phi_{d,s}) where phi_{d,s} is the B_{d,s} weak mixing phase. The B_s meson system can be used for the measurement of such an asymmetry. However it requires hadron machines and high luminosities to compensate the low branching ratio to CP eigenstates. In contrary to the ``golden channel'' B_s --> J/psi phi which demands an angular analysis to distinguish the different CP eigenstates contributions, the J/psi eta final state is a pure CP-even eigenstate. This channel can therefore provide an excellent probe to the phi_s phase. The reconstruction and performances of this decay mode will be detailed in this dissertation. They are performed with a full Monte Carlo simulation. An annual yield of about 9'000 events for the B_s --> J/psi (mu+mu-) eta (gamma gamma) channel and 3000 for the B_s --> J/psi (mu+mu-) eta (pi+pi-pi0) decay will be collected with a background-to-signal ratio smaller than 3. The mass and proper time resolutions are quite limited due to the electromagnetic calorimeter photon reconstruction low quality. Hopefully they can be improved by the application of a Kalman Filter fit. The sensitivity of the LHCb experiment to the weak mixing phase phi_s has been determined with a toy Monte Carlo. This simulation uses the B_s --> J/psi eta decay channels enriched by the B_s --> eta_c phi pure CP eigenstate channel. It takes into account the event-by-event proper time error, the time-dependent selection efficiency, the B/S ratio, the tagging efficiencies and the reconstruction performances of each decay channel. The combined sensitivity to phi_s is found to be 0.068 rad when the SM prediction for this phase is phi_s ~ -0.04 rad. These results have been finally compared to the B_s --> J/psi phi channel, whose sensitivity to phi_s reaches 0.031 rad with a annual yield of 125'000 events. The combined sensitivity of all these channels then increases to 0.028 rad. The contribution of the pure CP eigenstates is estimated to ~ 17%. Even with much smaller statistics, the pure CP eigenstate decay channels provide a non-negligible contribution to the determination of the Bs weak mixing phase phi_s.