The 2.5L is a really tough engine capable of unbelievable horsepower, but one real common problem is intermittent cutting out due to the Hall Effect pickup assembly in the distributor. They often become heat-sensitive and fail when they get warm, then work again an hour or two later. I never condone replacing random parts without doing a thorough diagnosis first, but in this case, since it's so common and the part is pretty inexpensive, start with that. It's a thin plastic disc with a three-wire connector, that sits under the rotor in the distributor. Any auto parts store will have it. If it doesn't solve the problem, keep the new one in the glove box for a spare.
The charging system is real easy to diagnose and repair, (with my help, of course), but since you correctly found 14.5 volts, we know that's working.
Your description of having to shift to neutral has me confused. A lot of people do not understand that "turn over" and "crank" are the same thing. They think "turn over" means to turn the ignition switch. To be clear, the engine was cranking normally but not starting. If that is correct, but it did start and run after you shifted to neutral, that was just a coincidence. If there was a no-crank condition until you shifted to neutral, that would be a starting system issue related to the neutral safety switch or the shift cable adjustment. That would be an entirely separate problem because that switch is out of the picture once you release the ignition switch. It will not cause stalling while you're driving, and it won't cause a failure to start if the starter is cranking the engine properly.
To interpret the oxygen sensor fault code, I have to know the exact code number. It's important to understand that fault codes never say to replace parts or that they're bad. They only indicate the circuit or system that needs further diagnosis. When a sensor is referenced in a fault code, it is only the cause of that code about half of the time.
Based on the readings from the O2 sensor, the Engine Computer will only modify the fuel / air mixture plus or minus about ten percent. That alone is not enough to explain the rotten fuel mileage. They used to have a lot of trouble with the MAP sensors in the early '90s. GM developed the design and had a huge problem with them, ... So they sold them to Chrysler. I'm sure yours isn't original. They all failed within the first few years, then were replaced with a better design that caused very little trouble.
The MAP sensor has the biggest say in how much fuel enters the engine, unlike all other cars. As one very high-level national trainer used to say, "Chrysler is the only manufacturer in the world that has been able to make an engine run right without a mass air flow sensor". The thing is though, there is an acceptable range of signal voltage the MAP sensor can produce, and as long as that voltage remains within that range, no fault code will be set. It can, however, report the WRONG value within that range. The computer will act on that incorrect information and command an incorrect amount of fuel. Look for a dry-rotted rubber vacuum hose going to the MAP sensor, or some other vacuum leak. Either of those will reduce the vacuum the sensor sees, which corresponds to what it would normally see during hard acceleration. You need more fuel when accelerating, and that's what the computer would command. The MAP sensor circuit can be responsible for the Engine Computer commanding more than double the correct amount of fuel needed.
MAP sensors fail very quickly when they do. The engine will go from running fine to not running at all within a day, ... Usually a few hours. They don't take many days or weeks to slowly fail completely.
Also remember that if you disconnect the battery or let it run dead, you'll erase any diagnostic fault codes stored in the Engine Computer. Chrysler makes reading those codes yourself easier than any other manufacturer. There's a link on this site to tell you how to do it, otherwise I can describe it here. If there are no related codes if the engine continues to stall, you'll need to find a scanner to view live data so you can see what is happening. Many scanners can record a few seconds of sensor data that you can replay later, if necessary. Also check for spark when the engine doesn't start. If you find that spark is missing, the next step will be to determine if the automatic shutdown (ASD) relay is turning on. I can describe that too. It's real easy, but you'll need a test light.
Finally, be aware that you caused another problem by removing the battery. The Engine Computer lost its memory and has to relearn "minimum throttle" before it will know when it must be in control of idle speed. Until that occurs, the engine may be hard to start unless you hold the accelerator pedal down 1/4", you won't get the nice idle flare-up to 1500 rpm when starting the engine, and it will tend to stall when you come to a stop, again, unless you hold the pedal down a little. It will not cause stalling while you're driving at a steady speed. Only the idle speed is too low.
To meet the conditions for the relearn to take place, drive at highway speed with the engine warmed up, then coast for at least seven seconds without touching the pedals.
Thursday, November 28th, 2013 AT 1:15 AM