Calculating diffusion coefficient
You want to make meatballs according to a recipe that says to marinate the meatballs in garlic and soy sauce before baking them in the oven. You prepare the marinade and immerse the meatballs, which are 6 cm in diameter, in it for 1 hour. When cutting into one of these rather large meatballs, you can clearly see how far the dark colored soy sauce has diffused in the white colored pork. You notice that only the outer 1 cm is marinated.
a) What is the diffusion coefficient of marinade in these meatballs? Give your answer in units of cm2/sec, but without entering your units.
b) Given your answer to the previous question, how long would it take to marinate the meatballs all the way to the center? Give your answer in hours, but without entering the unit.
To calculate the diffusion coefficient of the marinade in the meatballs, we need to use Fick's second law of diffusion. Fick's second law states that the rate of diffusion is proportional to the concentration gradient and the diffusion coefficient.
a) To find the diffusion coefficient, we can use the equation:
D = (X^2) / (4t)
Where:
D is the diffusion coefficient,
X is the distance the marinade has diffused (which is 1 cm in this case),
t is the time of marination (which is 1 hour).
Plugging in the values, we get:
D = (1^2) / (4 * 1)
D = 1 / 4
Therefore, the diffusion coefficient of the marinade in these meatballs is 0.25 cm^2/sec.
b) To calculate how long it would take to marinate the meatballs all the way to the center, we can rearrange the equation:
t = (X^2) / (4D)
Where:
t is the time it would take to marinate the meatballs all the way to the center,
X is the radius of the meatball (which is half the diameter, so 3 cm),
D is the diffusion coefficient we calculated (0.25 cm^2/sec).
Plugging in the values, we get:
t = (3^2) / (4 * 0.25)
t = 9 / 1
Therefore, it would take 9 hours to marinate the meatballs all the way to the center.