The Mid-InfraRed Instrument (MIRI) on board the James Webb Space Telescope (JWST) allows one to probe the molecular gas composition at mid-infrared (mid-IR) wavelengths with unprecedented resolution and sensitivity. It is important to study these features in low-mass embedded protostellar systems, since the formation of planets is thought to start in this phase. Previous studies were sensitive primarily to high-mass protostars. The aim of this paper is to derive the physical conditions of all gas-phase molecules detected toward a sample of 18 low-mass protostars as part of the JWST Observations of Young protoStars (JOYS) program and to determine the origin of the molecular emission and absorption features. This includes molecules such as CO$_2$, C$_2$H$_2$, and CH$_4$ that cannot be studied at millimeter wavelengths. We present JWST/MIRI data taken with the Medium Resolution Spectrometer (MRS) of 18 low-mass protostellar systems, focusing on gas-phase molecular lines in spectra extracted from the central protostellar positions. The column densities and excitation temperatures were derived for each molecule using local thermodynamic equilibrium (LTE) slab models. Ratios of the column densities (absorption) or total number of molecules (emission) were taken with respect to H$_2$O in order to compare these to ratios derived in interstellar ices. Continuum emission is detected across the full MIRI-MRS wavelength toward 16/18 sources; the other two sources (NGC 1333 IRAS 4B and Ser-S68N-S) are too embedded to be detected. The MIRI-MRS spectra show a remarkable richness in molecular features across the full wavelength range, in particular toward B1-c (absorption) and L1448-mm (emission). Besides H$_2$, which is not considered here, water is the most commonly detected molecule (12/16) toward the central continuum positions followed by CO$_2$ (11/16), CO (8/16), and OH (7/16). Other molecules such as 13CO$_2$, C$_2$H$_2$, 13CCH$_2$, HCN, C$_4$H$_2$, CH$_4$, and SO$_2$ are detected only toward at most three of the sources, particularly toward B1-c and L1448-mm. The JOYS data also yield the surprising detection of SiO gas toward two sources (BHR71-IRS1, L1448-mm) and for the first time CS and NH$_3$ at mid-IR wavelengths toward a low-mass protostar (B1-c). The temperatures derived for the majority of the molecules are 100--300 K, much lower than what is typically derived toward more evolved Class II sources ($ K). Toward three sources (e.g., TMC1-W), hot ($ K) H$_2$O is detected, indicative of the presence of hot molecular gas in the embedded disks, but such warm emission from other molecules is absent. The agreement in abundance ratios with respect to H$_2$O between ice and gas points toward ice sublimation in a hot core for a few sources (e.g., B1-c), whereas their disagreement and velocity offsets hint at high-temperature (shocked) conditions toward other sources (e.g., L1448-mm, BHR71-IRS1). Molecular emission and absorption features trace various warm components in young protostellar systems, from the hot core regions to shocks in the outflows and disk winds. The typical temperatures of the gas-phase molecules of $100-300$ K are consistent with both ice sublimation in hot cores as well as high-temperature gas phase chemistry. Molecular features originating from the inner embedded disks are not commonly detected, likely because they are too extincted even at mid-IR wavelengths by small, unsettled dust grains in upper layers of the disk.
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