Recall the drug capsule described in Example 2. The present drug capsule consists of a 0.652-cmdiameter spherical bead (radius of 0.326 cm) containing a uniform initial concentration of 68.9 mg/cm³ dramamine. (a) what is the residual concentration of dramamine at the center of the spherical bead

Question

Recall the drug capsule described in Example 2. The present drug capsule consists of a 0.652-cmdiameter spherical bead (radius of [latex]0.326 \mathrm{~cm}[/latex] ) containing a uniform initial concentration of [latex]68.9 \mathrm{mg} /[/latex] [latex]\mathrm{cm}^{3}[/latex] dramamine. (a) what is the residual concentration of dramamine at the center of the spherical bead after [latex]48 \mathrm{~h}[/latex] ? (b) the capsule is now a cube [latex]0.652 \mathrm{~cm}[/latex] on a side. Recalculate part (a) above. (c) the capsule is now a cylindrical tablet of diameter [latex]0.652 \mathrm{~cm}[/latex] and thickness [latex]0.3 \mathrm{~cm}[/latex]. Recalculate part (a) above. The diffusion coefficient of dramamine (species [latex]A[/latex] ) in the gel matrix (species [latex]B[/latex] ) is [latex]3 imes[/latex] [latex]10^{-7} \mathrm{~cm}^{2} / \mathrm{s}[/latex] at a body temperature of [latex]37^{\circ} \mathrm{C}[/latex]. The three capsules are presented in Figure 27.7.

Accepted Answer

This problem is readily solved using the charts given in Appendix F.

(a) Spherical capsule: First, calculate the relative time [latex]\left(X_{D} ight)[/latex], relative position [latex](n)[/latex], and relative resistance [latex](m)[/latex] based on the spherical coordinate system

[latex] \begin{gathered} X_{D}=\frac{D_{A B} t}{R^{2}}=\frac{\left(3 imes 10^{-7} \frac{\mathrm{cm}^{2}}{\mathrm{~s}} ight)\left(48 \mathrm{~h} \frac{3600 \mathrm{~s}}{1 \mathrm{~h}} ight)}{(0.326 \mathrm{~cm})^{2}}=0.488 \ n=\frac{r}{R}=\frac{0 \mathrm{~cm}}{0.326 \mathrm{~cm}}=0( ext { center of sphere }) \ m=\frac{D_{A B}}{k_{c} R} \approx 0 \end{gathered} [/latex]

From Figure F.1 or Figure 18.3, the value for [latex]Y[/latex], which in this case is the unaccomplished concentration change at the center of the spherical bead, is about 0.018. We can now calculate [latex]c_{A}[/latex]

[latex]Y=0.018=\frac{c_{A s}-c_{A}}{c_{A s}-c_{A o}}=\frac{0-c_{A}}{0-68.9 \mathrm{mg} / \mathrm{cm}^{3}}[/latex]

The residual Dramamine concentration at the center of the bead after [latex]48 \mathrm{~h}\left(c_{A} ight)[/latex] is [latex]1.24 \mathrm{mg} / \mathrm{cm}^{3}[/latex].

(b) For the cube-shaped capsule the distance from the midpoint of the cube to any of the six faces is [latex]0.652 \mathrm{~cm} / 2[/latex]. The relative time [latex]X_{D}[/latex] is now defined as

[latex]X_{D}=\frac{D_{A B} t}{x_{1}^{2}}=\frac{\left(3 imes 10^{-7} \frac{\mathrm{cm}^{2}}{\mathrm{~s}} ight)\left(48 \mathrm{~h} \frac{3600 \mathrm{~s}}{1 \mathrm{~h}} ight)}{(0.326 \mathrm{~cm})^{2}}=0.488[/latex]

Values for [latex]n[/latex] and [latex]m[/latex] are unchanged, with [latex]n=0[/latex] and [latex]m=0[/latex]. As all of the faces of the cube are of equal dimension, let

[latex]Y=Y_{a} Y_{b} Y_{c}=Y_{a}^{3}[/latex]

From the appendix Figure F.4, given [latex]X_{D}=0.488, \mathrm{~m}=0[/latex] and [latex]n=0, Y_{a}[/latex] is 0.4 for a flat plate of semi-thickness [latex]x_{1}=a=0.326 \mathrm{~cm}[/latex]. Extending this value to a three-dimensional cube using the above relationship, we have

[latex]Y=Y_{a}^{3}=(0.4)^{3}=0.064[/latex]

Finally,

[latex]Y=0.064=\frac{c_{A s}-c_{A}}{c_{A s}-c_{A o}}=\frac{0-c_{A}}{0-68.9 \mathrm{mg} / \mathrm{cm}^{3}}[/latex]

with [latex]c_{A}=4.41 \mathrm{mg} / \mathrm{cm}^{3}[/latex] after [latex]48 \mathrm{~h}[/latex].

(c) For a cylindrical capsule with exposed ends, [latex]R=0.652 \mathrm{~cm} / 2[/latex] for the radial coordinate, and [latex]x_{1}=a=0.3 \mathrm{~cm} / 2[/latex] for the axial coordinate. The relative times are

[latex]X_{D}=\frac{D_{A B} t}{R^{2}}=\frac{\left(3.0 imes 10^{-7} \frac{\mathrm{cm}^{2}}{\mathrm{~s}} ight)\left(48 \mathrm{~h} \frac{3600 \mathrm{~s}}{1 \mathrm{~h}} ight)}{(0.326 \mathrm{~cm})^{2}}=0.488[/latex]

for the cylindrical dimension and

[latex]X_{D}=\frac{D_{A B} t}{x_{1}^{2}}=\frac{\left(3.0 imes 10^{-7} \frac{\mathrm{cm}^{2}}{\mathrm{~s}} ight)\left(48 \mathrm{~h} \frac{3600 \mathrm{~s}}{1 \mathrm{~h}} ight)}{(0.15 \mathrm{~cm})^{2}}=2.30[/latex]

for the axial dimension. Values for [latex]n[/latex] and [latex]m[/latex] are unchanged, with [latex]n=0[/latex] and [latex]m=0[/latex]. From Figures F.1 and F.2, respectively, [latex]Y_{ ext {cylinder }}=0.1[/latex] for the cylindrical dimension and [latex]Y_{a}=[/latex] 0.006 for the axial dimension. Therefore,

[latex]Y=Y_{ ext {cylinder }} Y_{a}=(0.1)(0.006)=0.0006[/latex]

and finally

[latex]Y=0.006=\frac{c_{A s}-c_{A}}{c_{A s}-c_{A o}}=\frac{0-c_{A}}{0-68.9 \mathrm{mg} / \mathrm{cm}^{3}}[/latex]

with [latex]c_{A}=0.413 \mathrm{mg} / \mathrm{cm}^{3}[/latex] after [latex]48 \mathrm{~h}[/latex]. As [latex]Y_{a} \ll Y_{ ext {cylinder }}[/latex], the flux directed out of the exposed ends of the cylindrical tablet along the axial dimension dominates.

The above calculations assume that convective mass-transfer resistances associated with external fluid flow over the surface of the capsule are negligible. Problems in Chapter 30 will reconsider the drug release for unsteady-state diffusion and convection in series.

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