I did a similar animation but with a guy coughing and sending up dust...
a better method rather than using an expression or keyframes is a particle collision event
use a particle collision event found under your particles menu make sure that the type is Emite and (optional)Random # Particles is checked... then set your Num Particles, the spread, and your target particle (which requires a second particle to be created) and lastly make sure that Original Particle Dies is checked....
in your (name)Shape take your conserve down to 0 and everything else should really stay at default.....also setting the lifespan mode and other lifespan properties will help in making the particles die off as they should as sspikedudley already stated but with one change and I will tell you why
lifespan mode = random range
lifespan = 1
lifespan random = no higher than lifespan and no less than 0
the reason why you set the lifespan in that manner is because what happens is the solver the particle uses will add 1 and subtract 1 to the lifespan to give the total lifespan range 0 to 2 (lifespan - lifespan random; lifespan + lifespan random).. if you set the random any higher than the lifespan then you risk taking the range into negative which will most likely (not all the time if you're lucky) make the particles go in the wrong direction instead of letting a gravity field take over.
now take note, your target particle does NOT need an emitter and with a particle collision event you do not need an expression....
see if that works out better for you