To solve this problem, we can use Fick's law of diffusion and a differential equation to find the mole fraction of CO2 at the center of the spherical particle after a certain time, and the time it takes for the center to have a specific mole fraction. By doubling the diameter of the particle, we can calculate the new time required to achieve the desired mole fraction.
To solve part a of this problem, we can use Fick's law of diffusion. The equation is given by:
J = -D × ∇C
Where J is the diffusion flux, D is the diffusion coefficient, and ∇C is the concentration gradient. Since we are interested in the center of the spherical particle, we assume that the concentration gradient is constant. Therefore, the diffusion flux J is given by:
J = -D × (C2 - C1) / r
Where C1 is the concentration of CO2 at the outer surface of the particle and C2 is the concentration of CO2 at the center of the particle. Plugging in the values given in the problem, we can solve for C2.
To solve part b of this problem, we can use a similar approach as part a. We know that the mole fraction of CO2 at the outer surface of the particle is 5.0 mole%. We want to find the time it takes for the mole fraction at the center of the particle to reach 4.8 mole%. We can set up a differential equation using Fick's law of diffusion:
D × d²C / dr² = dC / dt
Where dC / dt is the rate of change of the mole fraction with respect to time. We can solve this differential equation using appropriate boundary conditions to find the time it takes for the mole fraction to reach 4.8 mole% at the center of the particle.
To solve part c of this problem, we can use the same approach as part b. The only difference is that the diameter of the particle is doubled, which will affect the diffusion coefficient. We can use the new diameter and the given diffusion coefficients to calculate the new diffusion coefficient, and then solve the differential equation to find the new time required.