What is rate constant and what is specific rate constant?

Consider a simple reaction involving substance ##A##:
##Ararr”products”##
Reactions tend to be fast at the beginning when concentrations are high and then slow down to zero when the reaction is complete.
If we measure the concentration of ##A## as the reaction proceeds and plot the results against time we get a graph like this:
(chemguideUK)
We can measure the rate of the reaction by measuring the gradient, or slope, of the line.
You can do this by drawing a tangent to the curve at a particular value of ##[A]## and measuring the gradient.
You can see the line is steepest at the beginning showing that this is where the reaction is fastest.
If we plot the rate against the concentration of ##A## we get a graph like this:
(ibchem.com)
A straight line like this means that rate is proportional to concentration:
##Rprop[A]##
We can replace the ##prop## sign with an equal sign and a constant##rArr##
##R=k[A]##
##k## is referred to as “the rate constant” and can be measured from the gradient of the rate v concentration graph.
The equation is referred to as the “rate equation” or “rate expression”
We can also write it like this:
##R=k[A]^(1)##
The index ##1## is referred to as the “order of the reaction” so we can describe this as a “first order reaction”.
The specific rate constant is defined as the rate of reaction when all the concentrations are equal to 1.
If ##[A]=1## you can see that ##R=k##.