how do you prepare 100 mL of 0.200 M acetate buffer, pH5.00, starting with pure liquid acetic acid and solutions containing~3 M HCl and ~3 M NaOH

CH3COOH + NaOH ==> CH3COONa + H2O

CH3COOH is the acid; i.e., a
CH3COONa is the base; i.e., b
pKa for acetic acid is pKa = -log Ka. I think pKa for acetic acid is 4.75, at least close to that, but you should confirm that.
pH = pKa + log (b/a)]
5.00 = 4.75 + log b/a.
Solve for the ratio of b/a.That's equation 1.
Equation 2 is a + b = 0.200 M
Solve the eqn 1 and eqn 2 simultaneouly to give values for both acid and base, convert those to volume of acid and base. You probably don't have enough information about the acid such as density or molarity. Also, I note that the HCl and NaOH are "about" 3 M. In practice one calculates the volumes, mixes everything together, measures the pH with a pH meter, then adds HCl or NaOH, drop by drop with stirring until the pH meters reads 5.00.
Post your work if you get stuck with the calculation part.

To prepare a 100 mL solution of 0.200 M acetate buffer with a pH of 5.00, starting with pure liquid acetic acid, 3 M HCl, and 3 M NaOH, you can follow these steps:

1. Determine the desired final concentration and volume of the buffer solution. In this case, we want a 0.200 M concentration and a volume of 100 mL.

2. Calculate the required amounts of acetic acid and sodium acetate using the Henderson-Hasselbalch equation, which relates the pH, pKa, and ratio of the concentrations of the conjugate acid and base:

pH = pKa + log([A-]/[HA])

In this case, the pKa of acetic acid is 4.76. Rearrange the equation to solve for the ratio [A-]/[HA]:

[A-]/[HA] = 10^(pH - pKa)

Substitute the pH of 5.00 and the pKa of 4.76 to find the ratio [A-]/[HA].

3. Since the volume of the buffer solution is 100 mL, the moles of acetic acid (HA) and sodium acetate (A-) required can be calculated using the equation:

moles = concentration × volume (in liters)

Calculate the moles of acetic acid and sodium acetate required based on the determined ratio and the desired concentration:

moles of HA = (0.200 M × 100 mL) / 1000 = X moles
moles of A- = (ratio × 0.200 M × 100 mL) / 1000 = Y moles

4. Convert the moles of acetic acid and sodium acetate to grams using their molecular weights:

moles of acetic acid = X moles
moles of sodium acetate = Y moles

5. Prepare a solution of acetic acid by measuring the required quantity of pure liquid acetic acid using a balance or pipette.

6. Add the measured acetic acid solution to a container.

7. Prepare a solution of sodium acetate by dissolving the required quantity of sodium acetate (based on the calculated moles) in enough distilled water to reach approximately 80 mL.

8. Adjust the pH of the sodium acetate solution to 5.00 using a pH meter and small increments of either HCl or NaOH. Add HCl solution dropwise to decrease the pH or NaOH solution dropwise to increase the pH.

9. Once you've reached a pH of 5.00, transfer the sodium acetate solution to the container with acetic acid and mix thoroughly.

10. Finally, add distilled water to bring the total volume of the solution to 100 mL. Mix gently to ensure a homogeneous solution.

You have successfully prepared a 100 mL solution of 0.200 M acetate buffer at pH 5.00 using pure liquid acetic acid, 3 M HCl, and 3 M NaOH.