Hi Lotus, 1) When something disperses, it spreads out. For example, "The pollen dispersed into the air." Another example, a teacher can say to a group of children, "disperse among yourselves," which is simply asking them to spread out. 2) To dispel something means to put an end to it, and is usually something intangible (you can not physically touch it), like a feeling (dispel your worries), a belief (i.e. dispel a myth, dispel a rumor = putting an end to the myth or rumor, saying it is not true), a thought (dispel a suspicion), and so on.

To disperse: To scatter. An act that divides a whole into many smaller portions. The police dispersed the crowd with a water cannon. The drop of dye dispersed in the water and it turned a bright blue. To dispel: To show to be untrue. To eliminate. To get rid of. We worked hard to dispel the rumor that she was pregnant. There is a meaning of "dispel" that is similar to "disperse" as in" to get rid of by scattering". He dispelled the angry mob with his reasoning. However, they are different enough that they are only mild synonyms.