How do you prepare 100 mL of a 0.1 M TRIS-HCl buffer with a pH 7.4 using solid TRIS base (MW 121.1) and 1 M hydrochloric acid?

I started with

pH=pKa + log ([Tris]/[Tris+]
[Tris]/[Tris+]= 10^(7.4-8.06]=0.219
so 0.219:1 ratio with 2.219 total

[Tris}=(0.219/1.219)*0.1 M= 0.018 M
[Tris+]= (1/1.219)*0.1 M= 0.082 M

M_[Tris]= (0.018M)(0.1L)= (0.0018 mol* 121.1 g/mol)= 0.218 g Tris

I'm not sure if this is correct but my main issues is how do I determine the amount of HCl I will need to add?

I think the 8.06 came from 14-pKb of Tris (5.924)

Since you are starting with Tris and you must generate the Tris(HCl, I would start with 0.1L x 0.1 M Tris or 0.01 moles Tris which is 1.211 grams. Now you want to add enough 1M HCl to make Tris*HCl (the acid) but keep the ratio of base/acid right for it to end up a pH of 7.4

................Tris + HCl ==> Tris*HCl
initial mols...0.010....0........0
change............-x....x........x
final.........0.010-x...x....... x

7.4 = 8.06 + log [(0.01-x)/(x)]
x = 0.0082 moles HCl (which is 8.2 mL of 1M HCl). You need to confirm that.

Check that.
1.211 g Tris = 1.211/121.1 = 0.01 moles (or 10 millimoles).
Add 8.2 mL of 1 M HCl is 8.2 millimoles HCl added.
That will form 8.2 mmoles of the Tris*HCl and will leave 10 mmoles base-8.2 mmoles HCl = 1.8 mmoles free base.
pH = 8.06 + log(1.8/8.2) = 7.401 which looks close enough to me and rounds to 7.40.

How do I calculate new pH of the buffer if I were to add 1mL of 1 M HCl to 5mL of buffer?

Where did you get a pKa of 8.06?

Find all cube roots of the complex number

64(cos(219°)+i sin(219°))
. Leave answers in polar form and show ALL work.

We can start by converting the number into polar form. We have:

r = 64
θ = 219°

Next, we can use the fact that the cube roots of a complex number can be found by taking the cube roots of its magnitude and dividing its argument by 3. In polar form, the cube roots will have magnitudes equal to the cube root of the original magnitude and arguments equal to:

θ/3
(θ/3) + 120°
(θ/3) + 240°

Therefore, the cube roots of 64(cos(219°)+i sin(219°)) are:

∛64(cos(219°/3)+i sin(219°/3)) = 4(cos(73°) + i sin(73°))

∛64(cos(73°+120°)+i sin(73°+120°)) = 4(cos(193°) + i sin(193°))

∛64(cos(73°+240°)+i sin(73°+240°)) = 4(cos(313°) + i sin(313°))

Therefore, the three cube roots of 64(cos(219°)+i sin(219°)) are:

4(cos(73°) + i sin(73°))

4(cos(193°) + i sin(193°))

4(cos(313°) + i sin(313°))

To determine the amount of hydrochloric acid (HCl) you need to add, you can follow these steps:

1. Determine the volume of the final buffer solution you want to prepare; in this case, it is 100 mL.

2. Determine the molar concentration of the hydrochloric acid. It is mentioned that you have 1 M hydrochloric acid.

3. Use the equation C1V1 = C2V2 to calculate the volume of 1 M hydrochloric acid (V1) required to prepare the desired buffer volume.

C1: Concentration of the hydrochloric acid = 1 M
V1: Volume of hydrochloric acid to be added (unknown)
C2: Concentration of the buffer solution = 0.1 M
V2: Volume of the buffer solution = 100 mL (or 0.1 L)

Rearranging the formula:
V1 = (C2V2) / C1
V1 = (0.1 M * 0.1 L) / 1 M
V1 = 0.01 L or 10 mL

Therefore, you need to add 10 mL of 1 M hydrochloric acid to obtain a 0.1 M TRIS-HCl buffer with a pH of 7.4, using solid TRIS base and the given reagents.