Normally, you can you ESR for quantify free radicals. ESR is electron spin resonance. in this technique, you can use resonance frequency of radical that you want to measure or quantify.
It depends on the radical, but detecting hydroxyl radicals has been the most straighforward and can be considered "prototypical" example as far as practice goes. Essentially, the most convenient way has been to use salicylic acid as a trapping agent. 2,3-Dihydroxybenzoic acid will show the relative quantity of hydroxyl radicals, when you compare samples to a control (Absolute quantification is inherently difficult). You may use high-performance liquid chromatography with electrochemical detection (HPLC-ECD), if you have patience with the system, or use gas chromatography (GC) or gas chromatography-mass spectrometry (GC-MS) that is fairly robust and reliable. See attached paper.
If your intention is to study the kinetics of the reaction "radical + organic compound", I would recommend the relative rate measurement (RRM). RRMs using a reference compound with a well-established rate constant are widely used for the determination of rate constants of gas-phase reactions. The knowledge of the absolute radical concentration is not required. You are just comparing the decay rate of an organic reference compound (with known rate constant) to the decay rate of your target organic compound (with unknown rate constant). See, e.g., J. Phys. Chem. Ref. Data, Vol. 28, No. 2, 1999 and references therein
How to quantify hydroxy radicals generated by the photocatalytic oxidation process? Please suggest the method and research institute where the such type research work going on.
How do you plan to make a sulfate radical? Would you like anion or cation radical?
If you want OH- radical, you have to yank a short electron pulsue over the sample. Sulfites and nitrites may be easier to reduce but not simple. Reduced species always decompose in fragments, you have to know what species are you looking for.
Customarily, transient absorption spectra are recorded using ultra fast techniques, right after a very short laser pulse. You have to look for the wavelength to record the transient absorption.
Techniques are always more difficult than the theory. If you plan to shoot a laser on NO3-(aq), you shall also have to record the power and the width of the pulse applied to the sample.