Roughy, low amount of solvent will encrease concentration and then reaction rate.
Additional effect could be done choosing polar or non polar solvents.Polar solvents breaks dipole making loose ion pairs: this encrease the reactivity and then reaction rate.
Polar solvent may also coordinate reaction intermediate such as carbocation: usually in these cases the carbocation formation is the rate determining step so you may see an encreasing of reaction rate.
These are just examples, you should think case by case about how your solvent may stabilize the activated complex and then you'll be able to understand the possible effect on reaction rate
Polar solvents enhance the rate of reaction in which the charged intermediates are present. Non polar solvents are useful if the charge of the reactants decrease during the course of the reaction. However, in such reactions polar aprotic solvents such as DMSO are useful.
Since most organic reactions are done in solution, it is important to recognize some of the ways that solvents affect the course and rates of reactions. There are important differences between protic
solvents, those which contain hydrogens that form hydrogen bonds and can exchange rapidly (such as those bonded to oxygen, nitrogen, or sulfur), and aprotic solvents, in which all hydrogen is bound to carbon. Similarly, polar solvents, those that have high dielectric constants, have effects on reaction rates that are different from nonpolar solvent media. When discussing solvent effects, it is important to distinguish between the macro-scopic effects and those that depend upon details of structure. Macroscopic properties refer to properties of the bulk solvent. One example is the dielectric constant, which is a measure of the ability of the solvent to increase the capacitance of a condenser. Dielectric constants increase with molecular dipole moment and polarizability because of the ability of both the permanent and induced molecular dipole to align with the
external electric field. An important property of solvent molecules is the response to changes in charge distribution as reaction occurs. The dielectric constant of a solvent is a good indicator of its ability to accommodate separation of charge. It is not the only factor, however, since, being a macroscopic property, it conveys little information
about the ability of the solvent molecules to interact with the solute molecules and ions at close range. These direct solute-solvent interactions depend on the specific structures of the molecules.
Solvents that fall into the nonpolar aprotic class are not very effective at stabilizing the development of charge separation. These molecules have small dipole moments and do not have hydrogens capable of forming hydrogen bonds. Reactions that involve charge separation in the TS therefore proceed more slowly in this class of solvents than in protic or polar aprotic solvents. The reverse is true for reactions in which species having opposite charges come together in the TS. In this case the TS is less highly charged than the individual reactants and reaction is favored by weaker solvation that leaves the oppositely charge reactants in a more reactive state.
Many empirical measures of solvent polarity have been developed.
Several other treatments of solvent effects on solvolysis rates have been developed. The equations typically include several terms related to: (a) macroscopic nonspecific solvent properties, such as the dipole moment and dielectric constant; (b) empirical polarity criteria; (c) solvent electrophilicity and nucleophilicity parameters; and (d) terms related to solvent cohesivity. The last term accounts
for the difference in work required to disrupt structure within the solvent, when, for example, there is expansion in volume between reactants and the TS.
In a solution the solvent is present in high concentration.Therefore the concentration of solvent is effectively a constant.Thus the solvent concentration terms do not come in to the rate expression.However the rate constant gets modified depending on the solvent.This is because of the formation of solvent cages.The solvent molecules arrange around the reacting molecules forming solvent cages.So the molecules undergo collisions with in these solvent cages.So the overall rate is affected by the solvent.
Hashini Senanayake Finally somebody is talking about the cage effect.
There are many different ways a solvent can affect reaction and reaction rate. Some of these effects are related to the identity of the solvent. That is, physical and chemical properties. Other effects, however, are related to the volume of the solvent. This latter effect is closely related to the concept called "the cage effect" as Hashini mentioned. If you are interested, I recommend Atkins' Physical Chemistry (Topic 18B)
Polarity of solvent can also have a big influence on the products obtained in organic photochemistry. For example, in the case of photoinduced hydrogen transfer. Two mechanisms are possible : a one-step mechanism where the electron and the proton are transferred in the same time and a two-steps mechanism where electron in transferred first and proton follows. These two mechanisms cannot be distinguished in terms of energy, they are both exergonic. But, for a one-step process, the electron transfer is endergonic where as in the case of a two-steps process, the electron transfer is exergonic. As you could see in our publications (10.1039/C9NJ03061A, 10.1002/chem.200903045), different products can be obtained with different solvent (cage effect discussion is made in the former). $\Delta G^{0}_{el}$, electron transfer exergonicity, is available from the Rehm-Weller equation, as a function of polarity of solvent. For a less thermodynamic point of view and a more kinetic point of view, I suggest you to see this last publication in Science for concerted proton-electron transfer :