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.
my answer 0.000069
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.
my answer 6 hours - (36 hours calculation from the coefficient above)
both are incorrect, could you help me please?
1) I think you just have to convert it in terms of 6.9 e-5.
2) I also got ~36 hours, but it is incorrect
To calculate the diffusion coefficient and the time required for marinating the meatballs all the way to the center, you can use Fick's second law of diffusion:
dC/dt = D * (d^2C/dx^2)
Where:
- dC/dt is the change in concentration with time
- D is the diffusion coefficient
- d^2C/dx^2 is the second derivative of the concentration with respect to distance
Given that only the outer 1 cm is marinated after 1 hour, we can assume that the concentration gradient is approximately linear in this case. So we have:
dC/dt = C1 - C0 / t,
where C1 is the concentration at the outer boundary and C0 is the initial concentration at the center (assumed to be 0).
a) To find the diffusion coefficient (D), we need the values of C1 and t. From the problem, we know that C1 = 1 cm and t = 1 hour. Plugging these values into the equation, we have:
dC/dt = 1 cm / 1 hour = D * (d^2C/dx^2)
Since the concentration gradient is approximately linear, the second derivative is zero. Thus, we have:
1 cm / 1 hour = D * 0,
which means the diffusion coefficient (D) is zero. Therefore, your initial answer of 0.000069 cm^2/sec is incorrect.
b) Given that the diffusion coefficient is zero, it means that there is no diffusion occurring. Therefore, it is not possible to marinate the meatballs all the way to the center, regardless of the time. Hence, your initial answer of 6 hours is also incorrect.
In conclusion, based on the given information, the diffusion coefficient cannot be determined, and the meatballs cannot be marinated all the way to the center.
To calculate the diffusion coefficient and the time required for marination, we can use Fick's second law of diffusion. Fick's law states that the diffusion flux (J) is proportional to the concentration gradient across a medium.
The equation for Fick's second law is:
J = -D * (∂C/∂x)
Where:
- J is the diffusion flux (amount of material diffusing per unit area per unit time),
- D is the diffusion coefficient,
- C is the concentration of marinade, and
- ∂C/∂x is the concentration gradient (change in concentration per unit length).
In this case, we are given that the outer 1 cm of the meatball is marinated (i.e., diffused) and the meatball is 6 cm in diameter. So, the radius of the meatball is 3 cm.
a) To find the diffusion coefficient (D), we need to calculate the diffusion flux (J) and the concentration gradient (∂C/∂x).
The concentration gradient (∂C/∂x) is equal to the difference in concentration between the surface and the center of the meatball divided by the thickness of the diffused region. Here, the thickness of the diffused region is 1 cm.
Since we are given that only the outer 1 cm is marinated, the initial concentration at the surface is 1 and at the center is 0.
∴ ∂C/∂x = (0 - 1) / 1 = -1
Now, we can solve for the diffusion flux:
J = -D * (∂C/∂x)
Given that J is equal to 1 (since the outer 1 cm is marinated within 1 hour), we can substitute the values into the equation:
1 = -D * (-1)
Simplifying the equation gives:
D = 1 cm^2/hr
So, the diffusion coefficient is 1 cm^2/hr.
b) To calculate the time required to marinate the meatballs to the center, we can use the calculated diffusion coefficient (D) and the thickness of the meatball.
The thickness of the meatball is half of its diameter, so it is 6/2 = 3 cm.
Using the formula for the diffusion coefficient:
D = ∆x^2 / (2 * t)
where ∆x is the thickness of the meatball and t is the time taken for diffusion, we can rearrange the equation to solve for t:
t = (∆x^2) / (2 * D)
Substituting the values into the equation:
t = (3^2) / (2 * 1) = 4.5 hours
Therefore, it would take approximately 4.5 hours to marinate the meatballs all the way to the center.