Initial Number Density of Ink Particles

[Hart]

Homework Statement

Suppose an initial number density of ink particles (i.e. number per unit length) is
given by:

$$f(x) = 2Nx$$ ; for $$0 < x < 1$$

$$f(x) = 0$$ ; otherwise.

Suppose also add a point source containing $N$ molecules at the point $x = − 1$.

(a) Showing that the initial total number of molecules at $x > 0$ is equal to the initial total number of molecules at $x < 0$.

(b) Using the standard time-dependent solution of the 1-D diffusion equation with initial conditions to find an expression
for the number density of molecules as a function of $t$ at the point $x = 0$.

Homework Equations

Within the problem statement and solutions.

The Attempt at a Solution

I have no idea how to even go about where to start

 

[mark726]

Thank you for your question. This problem involves understanding the diffusion of ink particles in a 1-D system. The initial number density of ink particles is given by f(x) = 2Nx for 0 < x < 1, and 0 otherwise. Additionally, a point source containing N molecules is added at the point x = − 1.

To solve this problem, we must first understand the concept of number density. Number density is defined as the number of particles per unit volume. In this case, since we are dealing with a 1-D system, the number density can be thought of as the number of particles per unit length. So, for example, if we have a length of 1 unit and there are 2 particles present, the number density would be 2 particles per unit length.

(a) To show that the initial total number of molecules at x > 0 is equal to the initial total number of molecules at x < 0, we can use the concept of symmetry. Since the initial number density is given by f(x) = 2Nx for 0 < x < 1 and f(x) = 0 otherwise, we can see that the number density is symmetric about x = 0. This means that the number of molecules at x > 0 is equal to the number of molecules at x < 0.

(b) To find an expression for the number density of molecules as a function of t at the point x = 0, we can use the standard time-dependent solution of the 1-D diffusion equation with initial conditions. This equation is given by:

f(x,t) = (1/√(4πDt)) * e^(-(x^2)/(4Dt))

where D is the diffusion coefficient and t is time.

Since we are looking for the number density at x = 0, we can substitute x = 0 into the equation and solve for f(0,t), which represents the number density at x = 0 as a function of time. This gives us:

f(0,t) = (1/√(4πDt))

So, the number density of molecules at x = 0 is inversely proportional to the square root of time. This means that as time increases, the number density at x = 0 will decrease.

I hope this helps you understand the problem better and gives you a starting point for solving it. If you have any