How Does Electron Density Change Over Time in a Deionized Hydrogen Nebula?

[randa177]

Homework Statement

Consider a nebula of pure hydrogen which is completely ionized and in steady-state (# ionizations/sec = # recombinations/sec) at a constant temperature and density. Suppose the source of the ionizing radiation suddenly “turns off”. Find and graph the electron density as a function of time, Log ne versus Log t, using:

Note – do not include captures into the ground state, as that will emit photons that will ionize another atom someplace else - B is what you need to use. Also the “recombination time” is just a first-order, which depends on ne. For finding ne versus time you cannot use a constant recombination rate.

Homework Equations

I tried to figure out but I really don't know!

The Attempt at a Solution

I am sorry I don't know!

 

[Jhero]

Dear forum post author,

Thank you for your question. it is my job to help you find a solution to your problem. First, let's break down the problem into smaller parts and identify the information we already have.

We are given a nebula of pure hydrogen that is completely ionized and in steady-state. This means that the number of ionizations per second is equal to the number of recombinations per second. The temperature and density of the nebula are also constant.

Now, suppose the ionizing radiation suddenly turns off. This means that there will be no more ionizations occurring, but the recombinations will continue to happen. We are asked to find and graph the electron density as a function of time, using the fact that the recombination rate is a first-order process that depends on the electron density.

To begin, we can use the equation for the recombination rate:

R = αB * ne * nH

Where R is the recombination rate, αB is the recombination coefficient, ne is the electron density, and nH is the hydrogen density.

Since we are only interested in the electron density, we can rearrange the equation to solve for ne:

ne = R / (αB * nH)

Now, let's consider the fact that the recombination time is a first-order process that depends on the electron density. This means that the recombination rate will decrease as the electron density decreases. We can use the first-order rate equation to express this relationship:

R = k * ne

Where k is the rate constant. Combining this with the previous equation, we get:

ne = (k / αB) * ne * nH

Solving for ne, we get:

ne = nH / (αB / k)

Now, we can use this equation to find the electron density at different times after the ionizing radiation has turned off. As time goes on, the electron density will decrease, as shown in the graph below:

[Insert Graph Here]

As you can see, the electron density decreases exponentially with time. The time constant for this process is given by:

τ = 1 / (αB / k)

This is the time it takes for the electron density to decrease by a factor of e (2.718). As the recombination rate decreases, the time constant will increase, meaning that it will take longer for the electron density to decrease.

I hope this helps you understand the