calculate the initial concentration of HCOONa you would add to 0.10M HCOOH to make a buffer with a pH of 4.00 [Ka (HCOOH) = 1.8 x 10^-4]

pKa = 3.74

Use the Henderson-Hasselbalch equation.
pH = pKa + log base/acid
4.00 = 3.74 + log (HCOONa)/0.1
Solve for (HCOONa).

To find the initial concentration of HCOONa needed to make a buffer solution with a pH of 4.00, we need to use the Henderson-Hasselbalch equation. The Henderson-Hasselbalch equation for a buffer solution is:

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

Where pH is the desired pH of the buffer solution, pKa is the negative logarithm of the acid dissociation constant (Ka) of the weak acid (in this case, HCOOH), [A-] is the concentration of the conjugate base (in this case, HCOO- from HCOONa), and [HA] is the concentration of the weak acid (HCOOH).

We have the pH (4.00) and the Ka for HCOOH (1.8 x 10^-4), so we can rearrange the Henderson-Hasselbalch equation to solve for [A-]/[HA]:

4.00 = -log(1.8 x 10^-4) + log([A-]/[HA])

Next, let's solve for log([A-]/[HA]):

log([A-]/[HA]) = 4.00 + log(1.8 x 10^-4)

Now, let's convert this into an algebraic equation:

[A-]/[HA] = 10^(4.00 + log(1.8 x 10^-4))

[A-]/[HA] = 10^(4.00) * 10^(log(1.8 x 10^-4))

[A-]/[HA] = 10^4 * (1.8 x 10^-4)

[A-]/[HA] = 0.018

Since HCOONa is a 1:1 molar ratio with HCOOH, the initial concentration of HCOONa will be the same as the [A-]/[HA] ratio:

[A-] (initial concentration of HCOONa) = (0.018 * [HA]) = (0.018 * 0.10M)

[A-] = 0.0018M

Therefore, the initial concentration of HCOONa (sodium formate) needed to make a buffer solution with a pH of 4.00 is 0.0018M.