Math problem in chemistry

Question

Electomagnetic radiation equation in task A)

 

I(x) is radiation emitted through the material layer intensity (W/m²)

u is absorption coefficient (1/mm)

x is thickness

 

A)From the I(x)=I₀e^-ux , how to solve u. 

 

B) In measurements, it was found that through light passing through a 2mm thick sheet the light intensity decreased by 30%. How thick should the plate be to reduce the intensity of 30%?

Answer 1

Part A:

Starting with the formula: I(x) = I₀e^-ux, we can divide both sides by I₀ and then take the natural log (ln) of both sides and then divide by -x to solve for u as shown below:

I(x) = I₀e^-ux

I(x) / I₀ = e^-ux

ln ( I(x) / I_o ) = -ux

(-1/x) ln ( I(x) / I_o ) = u

Part B:

We know that the 2 mm thick plate decreased the light intensity by 30 %, thus reaching 0.7 I₀.

If the problem is asking for the plate thickness so that the original light intensity is reduced by 30 %, (thus reaching 0.7I₀ again), then the answer is already answered by the problem statement itself: 2 mm.

However, if the problem is asking for the thickness of the plate so that the light intensity is reduced down to 30% of the original intensity (i.e. 0.3 * I₀), then we need to figure out the absorption coefficient (u) using the experimental data to solve for the new thickness (x):

Pulling from Part A:

(-1/x) ln ( I(x) / I_o ) = u

( -1/[2 mm] ) * ln (0.7) = u

u = .178 mm^-1 rounded down.

Now that we know what the absorption coefficient (u) is, we can solve for the new thickness (x):

(-1/x) ln ( I(x) / I_o ) = u

(-1/u) ln ( I(x) / I_o ) = x

x = (-1/.178 mm^-1) ln ( .3 )

x = 6.764 mm as the new thickness.

Let's check our work:

I(x) = I₀e^-ux
I(6.764 mm) = I₀e^{-.178 * 6.764}
I(6.764 mm) = 0.3 I_0. Check

I would double-check the exact wording of the problem with your professor or instructor so that there is no confusion here.