Question

There is going to be big fight between Popeye and Bluto in few minutes. Popeye gets in energetic mode if his blood contains 0.1 mg/l concentration of iron. Due to certain health issues, the absorption of iron in his bloodstream follows rate law:
$-\dfrac{d[F{{e}_{spinach}}]}{dt}=k[F{{e}_{spinach}}]$, which earlier used to be a fast absorption. What is the minimum time he should have spinach before the fight so that he defeats Bluto? Assume initial iron concentration in blood to be negligible.
Data given:
Volume of blood in popeye: 5 liter; Volume of spinach = 1 liter; Molecular Weight of Fe: 56 g/mol: Density of spinach = 1 kg/l; k = $2.31\text{ x 1}{{\text{0}}^{-2}}{{\min }^{-1}}$ ;
$\dfrac{Weight\text{ }of\text{ }Fe}{Weight\ of\text{ }spinach}={{10}^{-3}}\dfrac{g\text{ }of\text{ }Fe}{Kg\text{ }of\text{ }spinach}$
(a)- 15 min
(b)- 30 min
(c)- 45 min
(d)- 60 min

Answer 1

We can calculate the number of moles by multiplying the concentration of the blood, density of the spinach in gram per liter, volume of the blood, and divide this value with the molecular weight of the iron. We can use the formula:
$-\dfrac{d[{{N}_{A}}]}{dt}=k[{{N}_{A}}]$

Complete step-by-step answer: When we eat the spinach, the iron in the spinach is absorbed into the blood. So we can write:
$F{{e}_{spinach}}\to F{{e}_{blood}}$
Initially, the moles in the spinach are ${{N}_{{{A}_{o}}}}$ and the mole in the blood is 0. At time t, the moles in the spinach are ${{N}_{{{\text{A}}_{o}}}}-a$ and the moles in the blood are a. t is the minimum time to have the spinach before the fight. The final concentration in blood that we want is 0.1 mg/l.
Therefore, the number of moles will be equal to:
$Moles=\dfrac{0.1\text{ x 1}{{\text{0}}^{-3}}\text{ x 5}}{56}=\dfrac{0.5\text{ x 1}{{\text{0}}^{-3}}}{56}$
Given that the amount of iron in 1 kg of spinach is ${{10}^{-3}}$. We have 1 liter of spinach, which means we have ${{10}^{-3}}$ grams of Fe present.
Therefore, ${{N}_{{{\text{A}}_{o}}}}=\dfrac{{{10}^{-3}}}{56}moles$
And ${{N}_{{{\text{A}}_{o}}}}-a$= $\dfrac{0.5\text{ x 1}{{\text{0}}^{-3}}}{56}$ moles, this can be written as ${{N}_{\text{A}}}$
The rate equation is given by:
$-\dfrac{d[{{N}_{A}}]}{dt}=k[{{N}_{A}}]$
When we integrate the above formula and rearrange the terms, we can write it as:
$\ln \dfrac{{{N}_{{{\text{A}}_{o}}}}}{{{N}_{A}}}=kt$
We are given the value of k is $2.31\text{ x 1}{{\text{0}}^{-2}}{{\min }^{-1}}$
Now, putting all the values in the given formula above, we get:
t = 30 min, so popeye has to eat spinach 30 min before the fight.

Therefore, the correct answer is an option (b)- 30 min.

Note: While doing the integration, we have written the formula in the form of a natural log, i.e., ln, if you want to convert it into a log, then multiply 2.303 on both sides of the formula.