Seems to be a J-frame, but the grips seem smallish to me... the serial is 5-digit...
From 1950 until mid-1952, the Chief's Special was manufactured with a 1/4"-shorter grip frame and a smaller, more rounded trigger guard compared to later J frames. Collectors refer to these guns as "Baby Chiefs" or "short grip" J frames. If it has a low 5-digit serial number, IMHO it is quite likely this gun is a Baby Chief. Most of these guns also have a half-moon shaped front sight rather than the later serrated ramp. All have the 5-screw frame with the large top sideplate screw.
The grip and trigger guard dimensions of the original J frame were the same as the preexisting I frame, which was the frame used at the time for compact S&W revolvers chambered in .32 S&W Long, .38 S&W (NOT .38 Special), and .22LR. As originally conceived, the J frame was basically an I frame that was lengthened to allow a .38Spl cylinder to fit, although it also incorporated a coil mainspring rather than the leaf mainspring used on the I frame. (The latter change was meant to reduce costs and simplify production.)
In mid-1952, S&W lengthened the grip frame and enlarged the trigger guard of both the I and J frames, and implemented the coil mainspring on the I frame.*
Later grips will fit the "short grip" or "Baby" Chiefs but will overhang the butt by 1/4". Earlier I frame grips will (logically) fit perfectly. As you might imagine, however, it can be difficult to find modern grips that fit these guns!
These guns may demand a ~$50-$100 premium over later J frames, although this varies, as most gun store (and pawn shop) operators don't know enough about older J frames to tell the difference!
Probably a pre-model 36, Chiefs Special??
Yes, although the pre-Model 37 Chief's Special Airweight was also introduced around this time. This gun should have "AIRWEIGHT" rollmarked on the side of the barrel under the caliber marking, and will have a non-magnetic aluminum alloy frame that will usually appear slightly different in color than the blued barrel and cylinder. (AL alloy cannot be blued, so the frames were finished using a different process, hence the different appearance.)
Very early Airweights also had an aluminum cylinder that was rapidly dropped from production due to inadequate strength. These guns are valuable collector's items but
SHOULD NEVER BE FIRED due to the high risk of a gun-destroying and shooter-injuring kB!
That said, if this gun has an alloy cylinder and costs <$500, I would recommend snapping it up, because you should be able to resell it for substantially more!
Steel-cylinder Airweights make good carry guns, but most experts recommend staying away from +P ammo, and sometimes using low-pressure "cowboy" or wadcutter loads for practice. The early aluminum J frames aren't very strong and may stretch or crack with hard use.
For this reason, these guns should be checked very carefully for signs of cracking and excess endshake (i.e. fore-aft cylinder play).
*The post-1952 I frame is often called the "Improved I frame" to reflect the changes. The I frame was dropped from production circa 1962 to simplify parts inventories; all I frame models were replaced by an equivalent J frame.