Biomod/2013/NanoUANL/Reactor

What is a reactor?

Introduction

The CCMV capsid was considered as a continuous stirred-tank reactor with accumulation of the product.

For an enzymatic reaction of the type:

[math]\displaystyle{ E + S \leftrightarrow ES \rightarrow E^0 + P }[/math]

with a reaction rate of:

[math]\displaystyle{ \frac{d[ES]}{dt}=k_1[E][S]-k_{-1}[ES]-k_2[ES] }[/math]

We established the following in our system:

• Uniform distribution throughout the reactor
• K_-1 >> K₁ and K₂
• One enzyme per reactor/VLP
• Tortuosity approaches zero during diffusion

Mass balance was presented as such:

INFLOW= OUTFLOW- DISAPPEARANCE BY REACTION + ACCUMULATION

where

Inflow= F₀

Outflow= F₀(1-X_S)

Disappearance = V(-r_S

Accumulation = [math]\displaystyle{ \tfrac{d[P]}{dt} }[/math]

F₀ = F₀(1-X_S) - V(-r_S) + [math]\displaystyle{ \tfrac{d[P]}{dt} }[/math]

The intake and outflow flux were determined by diffusion , considering a spherical container.

File:CCMV2.jpg

For the simplification of the diffusion phenomenon we considered:

• Constant temperature
• Constant pressure
• Species B stays in a stationary state (it does not diffuse in A)
• The container (VLP) has a spherical shape

A mass balance, taking into account a spherical envelope leads to:

[math]\displaystyle{ \frac{d}{dr}(r^2N_{Ar})=0 }[/math]

where N_Ar represents molar flux. For N_Br we obtain:

[math]\displaystyle{ \frac{d}{dr}(r^2\frac{cD_{AB}}{1-x_A}\frac{dx_A}{dr})=0 }[/math]

At a constant temperature the product (cD_AB) is equally constant and x_A=1-x_B, the equation can be integrated into the following expression:

[math]\displaystyle{ F_A=4\pi r_1^2N_{Ar}|_{r=r1}= \frac{4\pi cD_{AB}}{1/r_1=1/r_2} \ln\frac{x_{B2}}{x_{B1}} }[/math]

where x are the fractions, c is the concentration and r are the respective radii.

This equation defines the nanoreactor inflow; a similar analysis yields the reactor outflow.

The ionic silver diffusion coefficient in function to the solution is described by Nerst's equation (1888)¹:

[math]\displaystyle{ D_{AB}°= \frac{RT}{F^2} \frac{\lambda^o_+\lambda^o_-}{\lambda^o_++\lambda^o_-} \frac{|Z_-|+|Z_+|}{|Z_+Z_-|} }[/math]

where

• F = Faraday's constant
• D_AB°=Diffusion coefficient at infinite dilution
• λ₊°=Cationic conductivity at infinite dilution
• λ_-°=Anionic conductivity at infinite dilution
• Z⁺=Cation valence
• Z^-=Anionic valence
• T=Absolute temperature

Via Joback's method, we obtain the normal boiling temperature:

[math]\displaystyle{ T_b=\mathbf{198} + \sum_{k} N_k(tbk) }[/math]

in which N_k is the number of times that the contribution occurs in the compound.

Using a similar approach, also by Joback, we estimated the critical temperature:

[math]\displaystyle{ T_c=T_b\Bigg[ \mathbf{0.584}+\mathbf{0.965} \bigg\{\sum_{k} N_k(tck) \bigg\} - \bigg\{\sum_{k} N_k(tck) \bigg\}^2 \Bigg]^{-1} }[/math]