Any chemical reaction will proceed by having molecule(s) of the reactant react to form the product(s). Although it is random when an individual molecule will react, due to the large number of molecules, it will appear as if it is a continuous process. This will occur at a certain speed (or rate). It has been found that the rate of reaction (in mol dm-3 s-1) is (usually) dependant on the concentration of the reactants. This can be expressed as a rate equation. In an example reaction, A + B -> C, the rate could be proportional to the concentration of A, times the concentration of B squared. This can be written using an equation: rate = [A] (B)^2 (B should be in square brackets, but that messes up the formatting). However, this doesn't tell us whether a reaction is fast or slow; it only tells us how changing the concentrations changes the rate of reaction.
To express the rate of reaction, we need to define standard conditions for us to measure. It has been chosen that all reactants will have a (hypothetical) concentration of 1 mol dm-3. k is therefore the rate that the reaction would occur if all reactants were at this concentration.
To calculate k, it is first required to determine the exponents of the rate equation (usually referred to as finding the order of each reactant). This easiest way to do this is to find two examples of the rate, where the only difference is the concentration of one reactant. For example, if when the concentration of reactant D doubles, the rate is 8 times higher, D would be a third order reactant, so [D]^3 appears in the rate equation (2^3 = 8). Likewise, if increasing the concentration of E by 50% multiplies the rate by 2.25, E would be second order (1.5^2 = 2.25). If you are asking this question in this forum it is unlikely you will come across non-integer rate orders, but be aware that these sometimes exist.
Once you have all of the reactant orders, you can then use these to "adjust" any information you have to work out what the rate would be if everything had a concentration of 1 mol dm-3. If you are unfamiliar I recommend only changing one reactant at a time to 1 mol dm-3, and writing down your result, and repeating for every reactant.
For the unit, it is important that [unit on the left of equation] = [unit on the right of equation]. The rate (lhs) has units of mol dm-3 s-1. The units of the right hand side are [units of k] * [units of concentration of reactants]. Bear in mind that if a given reactant is second order, the unit must also be squared (etc.). With some rearranging, it can be found that the units of k are: (mol dm-3)^(1 - [sum of reactant orders]) s-1. It is useful to check at the end that the units on both sides of the rate equation are the same.
