I depends on the problem. The rate constant is different depending on the problem in which it occurs.
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how does the rate law show how concentration changes after the rate of reaction
There is no difference between them they are same rate constant is another name of specific rate constant
The differentiation of a constant is zero because a constant value does not change with respect to the variable being differentiated. The derivative measures the rate of change of a function, and since a constant has no change, its derivative is zero.
constant rate means that rate which is fixed and can not be changed. varying rate is that rate which is not fixed and can be change easily.
The reaction rate is the rate at which the moles of substance change that varies with both temperature and concentration of the reactants. The specific rate constant is a proportionality constant that will vary only with temperature.