The hydroxyl radical (OH) is present in the diffuse molecular and partially atomic phases of the interstellar medium (ISM), but its abundance relative to hydrogen is not clear. We aim to evaluate the abundance of OH with respect to molecular hydrogen using OH absorption against cm-continuum sources over the first Galactic quadrant. This OH study is part of the HI/OH/Recombination line survey (THOR). THOR is a Karl G. Jansky Very Large Array large program of atomic, molecular and ionized gas in the range 15{deg}$leq$l$leq$67{deg} and |b|$leq$1{deg}. It is the highest-resolution unbiased OH absorption survey to date towards this region. We combine the derived optical depths with literature 13CO(1-0) and HI observations to determine the OH abundance. We detect absorption in the 1665 and 1667 MHz transitions for continuum sources stronger than $F_{rm cont}geq$0.1 Jy/beam. OH absorption is found against $sim$15% of these continuum sources with increasing fractions for stronger sources. Most of the absorption is associated with Galactic HII regions. We find OH and 13CO gas to have similar kinematic properties. The OH abundance decreases with increasing hydrogen column density. The OH abundance with respect to the total hydrogen nuclei column density (atomic and molecular phase) is in agreement with a constant abundance for $A_V$ < 10-20. Towards the lowest column densities, we find sources that exhibit OH absorption but no 13CO emission, indicating that OH is a well suited tracer of the low column density molecular gas. We present spatially resolved OH absorption towards W43. The unbiased nature of the THOR survey opens a new window onto the gas properties of the ISM. The characterization of the OH abundance over a large range of hydrogen gas column densities contributes to the understanding of OH as a molecular gas tracer and provides a starting point for future investigations.
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