We present moderate (${sim}5^{primeprime}$) and high angular resolution (${sim}1^{primeprime}$) observations of $^{12}rm{CO,}(J=2-1)$ emission toward nearby, interacting galaxy NGC 3627 taken with the Submillimeter Array (SMA). These SMA mosaic maps of NGC 3627 reveal a prominent nuclear peak, inter-arm regions, and diffuse, extended emission in the spiral arms. A velocity gradient of ${sim}400$-$450$ km s$^{-1}$ is seen across the entire galaxy with velocity dispersions ranging from $lesssim 80$ km s$^{-1}$ toward the nuclear region to $lesssim 15$ km s$^{-1}$ in the spiral arms. We also detect unresolved $^{13}rm{CO,}(J=2-1)$ line emission toward the nuclear region, southern bar end, and in a relatively isolated clump in the southern portion of the galaxy, while no $rm{C}^{18}O(J=2-1)$ line emission is detected at a $3sigma$ rms noise level of 42 mJy beam$^{-1}$ per 20 km s$^{-1}$ channel. Using RADEX modeling with a large velocity gradient approximation, we derive kinetic temperatures ranging from ${sim}5$-$10$ K (in the spiral arms) to ${sim}25$ K (at the center) and H$_2$ number densities from ${sim}$400-1000 cm$^{-3}$ (in the spiral arms) to ${sim}$12500 cm$^{-3}$ (at the center). From this density modeling, we find a total H$_2$ mass of $9.6times10^9 M_{odot}$, which is ${sim}50%$ higher than previous estimates made using a constant H$_2$-CO conversion factor but is largely dependent on the assumed vertical distribution of the CO gas. With the exception of the nuclear region, we also identify a tentative correlation between star formation efficiency and kinetic temperature. We derive a galactic rotation curve, finding a peak velocity of ${sim}207$ km s$^{-1}$ and estimate a total dynamical mass of $4.94 pm 0.70 times 10^{10} M_{odot}$ at a galactocentric radius of ${sim}6.2$ kpc ($121^{primeprime}$).
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