[asdfomg12]

My attempted work will be in italics

If  ²³⁴Pa is the end product of the decay sequence of ²³⁸U    (actually it’s  ²⁰⁶Pb), that is:

²³⁸U  ²³⁴Th  ²³⁴Pa
where the half-life of ²³⁸U to ²³⁴Th is 4.5x10⁹yr and ²³⁴Th to ²³⁴Pa is 24.1day
 

Therefore: k₁ = (ln2)/4.5x10⁹yr = 1.54x10^-10yr^-1 & k₂ = (ln2)/24.1day = 2.876x10^-2day^-1

   (a)   If we start with 1 mg of  ²³⁸U , how much ²³⁸U  will be present after 60 days? 

[²³⁸U] = [²³⁸U]₀e^-k₁t = .99999999997468mg

   (b)   How much ²³⁴Th will be present after 60 days?   

[²³⁴Th] = [²³⁸U](k₁/(k₂ - k₁))(e^-k₁t - e^-k₂t) = 4.40118x10^-9mg

   (c)   How much ²³⁴Pa will be formed after 60 days?   

This is the step i don't know how to derive the proper equation for calculation

[Enthalpy]

(a) Agree

(b) Impossible. 2.5*10^-11 mg less ²³⁸U can't create 4.4*10^-9 mg ²³⁴Th.
60 days nearly reach the equilibrium, where the amount of ²³⁴Th would be 24.1 days divided by 4.5 billion years - corrected by 234/238 - equating 1.4*10^-11 mg. After only 60 days, you may get 4/5 of this asymptotic value approximately.

(c) If you know how much ²³⁸U is gone and how much ²³⁴Th is left, ²³⁴Pa is the difference. At least for the formal computation, because numbers like 1-10^-11 need careful handling.

(d) For cases more complicated, with more nuclides or a wide initial mix, electrical engineering would have all the necessary math tools, without any limit. Some redundent methods: state variables, matrix exponentials, Laplace transformation, fluence graphs... Probably a too heavy investment for your needs, I guess some software does this for you.