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We explore how well James Webb Space Telescope (JWST) spectra will likely constrain bulk atmospheric properties of transiting exoplanets. We start by modeling the atmospheres of archetypal hot Jupiter, warm Neptune, warm sub-Neptune, and cool super-Earth planets with clear, cloudy, or high mean molecular weight atmospheres. Next we simulate the $\lambda = 1 - 11$ $\mu$m transmission and emission spectra of these systems for several JWST instrument modes for single transit and eclipse events. We then perform retrievals to determine how well temperatures and molecular mixing ratios (CH$_4$, CO, CO$_2$, H$_2$O, NH$_3$) can be constrained. We find that $\lambda = 1 - 2.5$ $\mu$m transmission spectra will often constrain the major molecular constituents of clear solar composition atmospheres well. Cloudy or high mean molecular weight atmospheres will often require full $1 - 11$ $\mu$m spectra for good constraints, and emission data may be more useful in cases of sufficiently high $F_p$ and high $F_p/F_*$. Strong temperature inversions in the solar composition hot Jupiter atmosphere should be detectable with $1 - 2.5+$ $\mu$m emission spectra, and $1 - 5+$ $\mu$m emission spectra will constrain the temperature-pressure profiles of warm planets. Transmission spectra over $1 - 5+$ $\mu$m will constrain [Fe/H] values to better than 0.5 dex for the clear atmospheres of the hot and warm planets studied. Carbon-to-oxygen ratios can be constrained to better than a factor of 2 in some systems. We expect that these results will provide useful predictions of the scientific value of single event JWST spectra until its on-orbit performance is known.

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