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If you assume that every collision leads to a reaction, then the two main variables affecting the rate of the reaction are the concentration of the reactants and the temperature. If the reactants are gases, then the pressure at which the reaction takes place is also a key variable. Another variable is the use of a catalyst which can reduce the activation energy needed for the reaction to take place.
If you increase the concentration of the reactants, then you increase the probability that the reactants will collide - leading to the desired product.
Likewise, by increasing the temperature, you increase the energy going into the system which provides the activation energy needed for the two reactants to combine to form the desired product. And if the reactants are gases, the increase in temperature increases their velocity which again increases the probability that the reactants will collide and form the desired product.
Keep in mind that for many reactions there is an equilibrium established between the concentration of reactants and concentration of products so that every collision in fact does not result in the desired product.
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