To synthesize high molecular weight polyacrylamide (PAM), you can follow the traditional method of free radical polymerization. Here's a step-by-step procedure:

1. Materials needed: Acrylamide (AM) monomer, a crosslinking agent (e.g., N,N'-methylenebisacrylamide), a free radical initiator (e.g., ammonium persulfate), a buffer solution (e.g., Tris-HCl), a polymerization apparatus (e.g., beakers, stir bars, and a stir plate), a source of nitrogen gas, and a heating apparatus (e.g., water bath).

2. Set up your polymerization apparatus and fill it with the desired volume of buffer solution.

3. Add the necessary amounts of acrylamide monomer and crosslinking agent to the buffer solution. The concentration of acrylamide can vary based on the desired molecular weight of the polymer.

4. Dissolve the monomer and crosslinking agent completely in the buffer solution while stirring gently.

5. Add the free radical initiator (e.g., ammonium persulfate) to the solution, ensuring uniform distribution.

6. Degas the solution by bubbling nitrogen gas through it for a few minutes to remove any dissolved oxygen.

7. Place the solution in a water bath or other heating apparatus to maintain the desired temperature for polymerization. Commonly, polymerization is conducted at around 50-70°C.

8. Polymerization will begin shortly after the addition of the initiator. Allow the reaction to proceed for the desired duration to achieve the desired molecular weight. The reaction time can vary depending on factors such as temperature and monomer concentration.

9. After the desired reaction time, stop the polymerization by removing the reaction mixture from the heating apparatus or adding a quenching agent like sodium metabisulfite.

10. Purify the resulting polymer by various methods such as precipitation, filtration, and drying.

Keep in mind that acrylamide is a neurotoxin and should be handled with care in a well-ventilated area or under a fume hood. Additionally, disposal of the reaction waste and byproducts should be done following proper safety regulations.

It's important to note that the synthesis of polyacrylamide should comply with local laws, regulations, and safe handling procedures. It is always recommended to consult specific protocols and references or seek guidance from a knowledgeable expert in synthesis.