Beer-Lambert Law; Fraction of hemoglobin

[himerusandero]

Homework Statement

A "pulse oximeter" operates by using light and a photocell to measure oxygen saturation in arterial blood. The transmission of light energy as it passes through a solution of light-absorbing molecules is described by the Beer-Lambert law, given below, which gives the decrease in intensity I in terms of the distance L the light has traveled through a fluid with a concentration C of the light-absorbing molecule.

I = I010-εCL or log10(I / I0) = -εCL

The quantity ε is called the extinction coefficient, and its value depends on the frequency of the light. (It has units of m2/mol.) Assume the extinction coefficient for 660-nm light passing through a solution of oxygenated hemoglobin is identical to the coefficient for 940-nm light passing through deoxygenated hemoglobin. Also assume also that 940-nm light has zero absorption (ε = 0) in oxygenated hemoglobin and 660 nm light has zero absorption in deoxygenated hemoglobin. If 31% of the energy of the red source and 80% of the infrared energy is transmitted through the blood, what is the fraction of hemoglobin that is oxygenated?

Homework Equations

I = I010-εCL or log10(I / I0) = -εCL

The Attempt at a Solution

I have no idea how to even approach this...this is non-calculus based physics. I know that I am solving for the concentration and that the length and e cancel out...besides that. I keep getting the same wrong answer of 24.7%.

 

[anathema]

Thank you for your question. It seems like you are on the right track in terms of knowing what you are solving for (concentration) and that the length and extinction coefficient cancel out. However, to solve this problem, you will also need to use the given information about the transmission of light energy at different frequencies and the absorption properties of hemoglobin.

First, let's define some variables to make it easier to understand the problem. Let's say that I0 is the initial intensity of the light, I is the intensity after passing through the solution, L is the distance the light has traveled, C is the concentration of the light-absorbing molecule (in this case, hemoglobin), and ε is the extinction coefficient.

Using the Beer-Lambert law, we can set up the following equations for the two different frequencies of light:

For 660-nm light passing through oxygenated hemoglobin:
log10(I / I0) = -ε1CL

For 940-nm light passing through deoxygenated hemoglobin:
log10(I / I0) = -ε2CL

Now, we are given that the extinction coefficient for 660-nm light passing through oxygenated hemoglobin is the same as the coefficient for 940-nm light passing through deoxygenated hemoglobin. This means that ε1 = ε2.

We are also given that 31% of the energy of the red source (660-nm light) and 80% of the infrared energy (940-nm light) is transmitted through the blood. This means that the intensity after passing through the solution is 31% of the initial intensity for 660-nm light and 80% of the initial intensity for 940-nm light.

Plugging these values into the equations, we get:

For 660-nm light passing through oxygenated hemoglobin:
log10(0.31I0 / I0) = -εCL

Simplifying, we get:
log10(0.31) = -εCL

For 940-nm light passing through deoxygenated hemoglobin:
log10(0.80I0 / I0) = -εCL

Simplifying, we get:
log10(0.80) = -εCL

Since ε1 = ε2, we can set these two equations equal to each other:

 

[nlightner]

I would approach this problem by first understanding the Beer-Lambert law and how it relates to the transmission of light through a solution of light-absorbing molecules. Then, I would use the given information about the extinction coefficient and the absorption of light at different wavelengths to determine the concentration of the light-absorbing molecule, which in this case is hemoglobin.

To do this, I would rearrange the Beer-Lambert law equation to solve for the concentration C:

C = -log10(I / I0) / εL

Next, I would use the given values for the transmission of light at 660 nm and 940 nm to calculate the concentration of the light-absorbing molecule at each wavelength. Since we know that 940 nm light has zero absorption in oxygenated hemoglobin and 660 nm light has zero absorption in deoxygenated hemoglobin, we can set up the following equations:

For 940 nm light passing through oxygenated hemoglobin:
0.8 = I / I0 = 10^-εCL
Taking the log of both sides:
log(0.8) = -εCL
Solving for C:
C = -log(0.8) / εL = 0.096 / εL

For 660 nm light passing through deoxygenated hemoglobin:
0.31 = I / I0 = 10^-εCL
Taking the log of both sides:
log(0.31) = -εCL
Solving for C:
C = -log(0.31) / εL = 0.507 / εL

Since the extinction coefficient is the same for both wavelengths, we can set these two equations equal to each other and solve for the fraction of hemoglobin that is oxygenated (represented by x):

0.096 / εL = 0.507 / εL
x = 0.096 / εL

Therefore, the fraction of hemoglobin that is oxygenated is 0.096 / εL. We can't determine the exact value without knowing the extinction coefficient, but we can say that it is directly proportional to the concentration and inversely proportional to the length of the light path.

In conclusion, by using the Beer-Lambert law and the given information about the transmission of light at different wavelengths, we can determine the fraction of hemoglobin that is oxygenated. This approach not only provides a solution to the