This web site won't allow posts longer than 10,000 characters, so I had to split this post into 2 pieces; the first post is the answer to your questions and the second is where you can find some real good info on electric motors for future reference.
A "dual run" capacitor for HVAC equipment is nothing more than two capacitors encased in the same housing. This is common in HVAC equipment because it's so often that you have both a refrigeration compressor and a fan operating together, and both the compressor and fan being driven by capacitor start or capacitor run motors. So, a "dual run" capacitor is simply a part with two capacitors in it insead of only one.
Troubleshoot a Capacitor
So, when you say you have a 5 microfarad "dual run" capacitor, then we know that ONE of the capacitors in the housing is 5 uF, but it doesn't say anything about the other capacitor in there. If you need a 7.5 uF capacitor for the new motor, and the OTHER capacitor in the dual run capacitor you have just happens to be 7.5 microfarads, then yes, you should be able to use the old "dual run" to run the new motor.
However, what you need to do is check the capacitance of the second capacitor inside your dual run capacitor, and make sure that it's amperage rating is sufficient to handle the current through the new motor.
Now, on capacitor start and capacitor run motors, and split phase electric motors, and any electric motor that has separate start and run windings, there will be a centrifugal switch that kicks out the start winding once the motor comes up to speed. Consequently, the importance of having the capacitance correct differs depending on whether the capacitor is on the start winding or the run winding. That is, whether the motor uses a "start capacitor" or a "run capacitor". If it's on the start winding, it's less critical as long as the motor has enough torque to start quickly, and that's simply because the start capacitor is taken out of the circuit along with the start winding once the motor comes up to speed. But, if the capacitance is too far out, the motor won't have enough torque to start, or will take a long time to start and the heat produced by the high current during the start-up can wreck the motor by melting the thin lacquer insulation on the wires of the windings.
Conversely, if the capacitor is on the run winding it's a "run capacitor", and that capacitor will affect the motor's torque and power, operating temperature and how smooth and quiet the motor runs all the time it's running. In that case, I'd wait until you get the right capacitor from the manufacturer of the motor.
So, check to confirm it's capacitor run motors you have and don't use the dual run capacitor if the capacitance isn't the 7.5 ufd you need.
You don't need to know the rest...
You might be curious as to what the capacitor in a capacitor start or capacitor run motor actually does:
Basically, an electric motor is nothing more than a device that creates (or simulates) a rotating magnetic field. In an electric motor, it's the stator's job to create that rotating magnetic field, and the magnetic rotor naturally follows that magnetic field, and that's what causes the rotor to turn.
With three phase electric power, it's easy to create a REAL rotating magnetic field. Since you have three phases that are 120 degrees apart in TIME, you simply position three windings of copper wire on your stator 120 degrees apart (in SPACE) around the circumference of your stator and your stator will produce a near perfect rotating magnetic field. Ditto for two phase power, only the two phases and copper windings have to be 180 degrees apart instead of 120 degrees.
The problem arises when we get down to single phase power like the 120 VAC 15 amp power you typically have in a house.
With 120 volt AC power you only have a single phase to work with, and so putting that power through a winding of copper wire will only produce an OSCILLATING magnetic field, not a rotating one. With only single phase power, we have to use various tricks to simulate a rotating magnetic field.
ALL of the different kinds of single phase 120 VAC electric motors in your house, from the "capacitor run" motor driving your heat pump fan, to the "split phase" motor in your washing machine or dryer to the "shaded pole" motor in your electric clock all use a different "trick" to simulate a rotating magnetic field.
In the case of capacitor start electric motors, what they do is put two windings in the stator; a start winding and a run winding. If both these windings were to be connected to the same 120 VAC power supply, then both would develop their magnetic fields at exactly the same time, and you would simply be left with an oscillating magnetic field oriented somewhere between the poles of the two windings. The result would be that the rotor shaft would simply vibrate back and forth inside the stator at 60 cycles per second, just as you would with a single winding in the stator.
But, in a capacitor start electric motor, they put a capacitor in SERIES with the start winding and that makes all the difference.
Now, if you look at the applied voltage and resulting current through a resistor, both the current and voltage will always be in synch. When the voltage is at a max, then the current is at a max, and when the voltage is at a minimum the current is too.
However, a capacitor has the quality of being able to change the phase relationship between the applied voltage and resulting current. That's because the current OUT of a capacitor is at a maximum when the RATE OF CHANGE IN VOLTAGE is at a maximum, and that actually happens when the applied voltage sine wave crosses the 0 volt axis. Thus, the current out of a capacitor is at a maximum when the applied voltage is at a minimum, or 0 volts. Similarily, the current out of a capacitor is at a minimum when the RATE OF CHANGE IN VOLTAGE is at minimum, and that actually happens at the tops of the peaks and bottoms of the valleys of the applied voltage sine wave. Thus the current out of a capacitor is at a minimum when the applied voltage is at a maximum. Consequently, you can delay the CURRENT coming out of a wire by 90 degrees by putting a capacitor in that wire.
In a capacitor start electric motor, the start and run windings are positioned 90 degrees apart around the circumference of the stator. Since the capacitor in the start winding delays the current coming into the start winding by 90 degrees, the start winding develops it's magnetic field 90 degrees (or 1/4 of 1/60th of a second) later than the run winding develops it's magnetic field. And, the result of this time lag is that the rotor sees the magnetic field moving from the north pole of the run winding to the north pole of the start winding to the south pole of the run winding to the south pole of the start winding and back to the north pole of the run winding to complete a full circle, and that happens 60 times a second. This is how capacitor start and capacitor run motors simulate a rotating magnetic field.
In a "split phase" electric motor typically found in laundry equipment like washers and dryers, the start winding consists of a LOT of turns of a very thin wire, whereas the run winding consists of a much smaller number of turns of a much thicker wire. The difference in the impedance of the two windings causes one winding to develop it's magnetic field earlier or later than the other, and the result is the same as a capacitor start motor; the appearance of the magnetic field moving around the circumference of the stator, and therefore simulating a rotating magnetic field.
Ditto for even the tiny motors in your home's electric clock. A "shaded pole" electric motor uses a thick wire running between the north and south poles of a single winding to cause one side of each pole to develop it's magnetic field earlier than it otherwise would, and the other side to develop it's magnetic field later than it otherwise would. Again, the end result is the appearance of the magnetic field sweeping across each of the two poles of the motor, thereby simulating a rotating magnetic field.
The thing to remember here is that there are only capacitor start, capacitor run, split phase, resistance start, and shaded pole motors for single phase electric power like the 120 VAC power typically available from a wall outlet. Electric motors that run on two and three phase power don't need to use any of these various designs to simulate a rotating magnetic field because they are inherantly well suited to produce real rotating magnetic fields. If you remember nothing else from this post, remember that.
PS: the letter "u" denotes "micro", or "millionths". It's basically a lookalike for the Greek letter "mu" which denotes "millionths" in scientific literature and is written:
Technically, the letter "m" stands for "milli" or thousands, as in millimeter (mm) or milliliter (ml). (Still, everyone understands that "mfd" means microfarad because no one uses the term "millifarad".)