Determine the boundary and initial conditions appropriate for the diffusion of salt in each of the following situations. (a) A hollow tube initially containing pure water connects two reservoirs whose salt concentrations are \(C_{1}\) and 0 respectively. (b) The tube is sealed at one end. Its other end is placed in a salt solution of fixed concentration \(C_{1}\). (c) The tube is sealed at both ends and initially has its greatest salt concentrations halfway along its length. Assume the initial distribution is a trigonometric function.

Boundary conditions help define the behavior of a system at the edges. For our salt diffusion scenarios, we are interested in how salt concentration behaves at the ends of the tube:

In Scenario (a), we have a hollow tube with salt concentrations set at both ends. One end has a constant concentration, while the other end has none. This sets up a clear boundary:

• At the left end (\[x = 0\]): \[C(0, t) = C_{1} \]
• At the right end (\[x = L\]): \[C(L, t) = 0 \]

In Scenario (b), one end of the tube is sealed, meaning no salt can flow in or out. The other end is placed in a fixed concentration:

• Sealed left end: \[ \frac{\textpartial C}{\textpartial x} \bigg|_{x=0} = 0 \]
• Right end with fixed salt concentration: \[C(L, t) = C_{1} \]

Finally, in Scenario (c), both ends of the tube are sealed, ensuring no salt can escape:

• Sealed left end: \[ \frac{\textpartial C}{\textpartial x} \bigg|_{x=0} = 0 \]
• Sealed right end: \[ \frac{\textpartial C}{\textpartial x} \bigg|_{x=L} = 0 \]

These conditions are essential to defining the problem and understanding how the diffusion process will unfold.

Initial conditions help us understand the system's state at the very beginning (time \[t = 0 \]). Knowing how salt concentration starts in the tube is crucial:

In Scenario (a) and (b), we start with pure water in the tube, meaning salt concentration is zero everywhere.
\[ C(x, 0) = 0 \]

For Scenario (c), the initial salt distribution is a bit more complex. Here, the concentration is greatest at the midpoint and varies in a trigonometric manner:
\[ C(x, 0) = C_{0} \text{sin} \frac{\textpi x}{L} \]This tells us that initially, the salt is distributed sinusoidally, peaking at the midpoint and dropping off towards the ends. These initial conditions set the starting point for the diffusion equations to begin changing the salt concentration over time.

Diffusion equations describe how substances like salt spread over time. The fundamental equation governing this process is Fick's Second Law:

\[ \frac{\textpartial C}{\textpartial t} = D \frac{\textpartial^2 C}{\textpartial x^2} \]Where:

• \(C\) = concentration of salt
• \(t\) = time
• \(D\) = diffusion coefficient
• \(x\) = position in the tube

In our scenarios, we use this equation along with boundary and initial conditions to determine how concentration changes over time

• In Scenario (a), salt diffuses from the high concentration at one end toward the zero concentration at the other end
• In Scenario (b), salt diffuses from one end with a fixed concentration while no salt flows at the sealed end
• In Scenario (c), the salt concentration changes sinusoidally at the start and then diffuses with both ends sealed

Understanding these diffusional behaviors helps in predicting and analyzing the salt concentration profiles over time.

Concentration gradients represent how concentration changes in space. Diffusion occurs when there's a concentration gradient, with particles moving from high to low concentration areas.

For example, in Scenario (a), we have a clear gradient between the high salt concentration at one end and zero concentration at the other. This drives diffusion from the high concentration side to the low concentration side.

In Scenario (b), while one end has a fixed concentration and no salt movement at the sealed end, the gradient still forms, driving the diffusion process:

Sealed left end: \[ \frac{\textpartial C}{\textpartial x} \bigg|_{x=0} = 0 \]

Right end with fixed concentration:
\[C(L, t) = C_{1}\]
In Scenario (c), the initial sine wave distribution of salt implies a varying gradient, with the steepest gradient at the midpoint, facilitating the diffusion process:

\[C(x, 0) = C_{0} \text{sin} \frac{\textpi x}{L} \]

Understanding these gradients is crucial as they directly influence the rate and direction of diffusion, affecting overall concentration profiles within the tube.

Sealed and open systems dictate the boundaries through which diffusion can occur:

In an open system, salt can freely enter or exit from the boundaries. In Scenario (a), both tube ends connect to different concentration reservoirs, allowing salt to diffuse from high (one end) to low concentration (other end):

• Left end: \[C(0, t) = C_{1} \]
• Right end: \[C(L, t) = 0 \]

In sealed systems, the boundaries prevent any exchange with the environment. Scenario (b) has one end sealed, making salt diffusion unique as it can only occur through one opening:

• Sealed end: \[ \frac{\textpartial C}{\textpartial x} \bigg|_{x=0} = 0 \]
• Open end: \[C(L, t) = C_{1} \]

Scenario (c) seals both ends, requiring the salt to redistribute within the closed space, driven by the internal concentration differences:

• Left and Right sealed ends: \[ \frac{\textpartial C}{\textpartial x} \bigg|_{x=0} = 0 \], \[ \frac{\textpartial C}{\textpartial x} \bigg|_{x=L} = 0 \]

Recognizing these system types is essential for understanding how diffusion operates within different boundary conditions.