Question

A cell in your adrenal gland has about $\mathbf{2}\mathbf{.}\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{4}}$tiny compartments called vesicles that contain the hormone epinephrine (also called adrenaline).

(a) An entire cell has about $\mathbf{150}\mathbf{}\mathbf{fmol}\mathbf{}$f epinephrine. How many attomoles (amol) of epinephrine are in each vesicle?

(b) How many molecules of epinephrine are in each vesicle?

(c) The volume of a sphere of radius r is $\frac{\mathbf{4}}{\mathbf{3}}\mathit{\pi }{\mathit{r}}^{\mathbf{3}}$. Find the volume of a spherical vesicle of radius 200nm. Express your answer in cubic meters$\left(m^3\right)$ and liters, remembering that$\mathbf{1}\mathbf{L}\mathbf{=}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}{\mathbf{m}}^{\mathbf{3}}$ .

(d) Find the molar concentration of epinephrine in the vesicle if it containsof epinephrine.

Short answer

a) There are $6amol$of epinephrine in each vesicle.

b) There are $3.6\times {10}^6$molecules of epinephrine in each vesicle.

c) The volume of a spherical vesicle of radius 200nm is role="math" localid="1663299886472" $3.35\times {10}^{-17}\mathbf{L}$.

d) The molar concentration of epinephrine in the vesicle is $0.298\mathbf{}\mathbf{M}$.

Step-by-step solution

Definition of molality.

The number of moles of solute in a solution equivalent to $\mathbf{1}\mathbf{}\mathit{k}\mathit{g}$or $\mathbf{1000}\mathbf{}\mathit{g}\mathbf{}$ of a solvent is known as its molality. The definition of molarity, on the other hand, is based on a certain volume of solution

Finding the attomoles of epinephrine in each vesicle.

(a)

It is given that the moles of epinephrine in a cell $=150\mathbf{}\mathbf{fmol}$

Note: $(1femto={10}^{-15}).$

The moles of epinephrine in cell $=150\times {10}^{-15}\mathbf{mol}$

Now, calculate the moles of epinephrine in each vesicle as shown below:

$$\begin{gathered} Molesofepinephrineineachvesicle=\frac{totalmolesincell}{totalvesicleincell} \\ =\frac{150\times {10}^{-15}}{2.5\times {10}^4} \\ =6\times {10}^{-18}mol \\ =6amol\left[1atto={10}^{-18}\right] \end{gathered}$$

Finding the molecules of epinephrine in each vesicle.

(b)

The molecules of epinephrine in each vesicle = (Moles in each vesicle) (Avogrado constant)

Molecules of epinephrine in each vesicle$$\begin{gathered} =(6\times {10}^{-18})(6.023\times {10}^{23}) \\ =3.6\times {10}^6 \end{gathered}$$

Finding the volume of spherical vesicle.

(c)

The volume of a spherical vesicle is calculated as shown below:

$$\begin{gathered} Volume=\frac{4}{3}\pi r^3 \\ =\frac{4}{3}\pi \left(200{)}^3\right(nm{)}^3 \\ =\frac{4}{3}\pi \times 8\times {10}^6({10}^{-9}m{)}^3 \\ =\frac{32\pi }{3}\times {10}^{21})m^3\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}[Note:1\mathrm{nm}={10}^{-9})m] \\ =\frac{32\pi }{3}\times {10}^{18})L\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}[Note:1L={10}^{-3})m^3] \\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}=3.35\times {10}^{-17})L. \end{gathered}$$

The molar concentration of the vehicle.

(d)

The molar concentration is calculated as:

$$\begin{gathered} Molar\hspace{0.33em}concentration=\frac{\mathrm{moles}\mathrm{of}\mathrm{epinephrine}}{\mathrm{volume}(\mathrm{in}\mathrm{L})} \\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}=\frac{10\times {10}^{-18}\mathrm{mol}}{3.35\times {10}^{-17}\mathrm{L}} \\ \hspace{0.33em}=0.298\mathrm{M}. \end{gathered}$$