A Severe Shortage of Blood!

Blood donation bags
at a blood drive in
Rochester Michigan
on March 24, 2020.

Yesterday Michigan’s governor, Gretchen Whitmer, announced a shelter-in-place order. Now I need a legitimate reason to leave home. I have one: To donate blood! The Red Cross has a severe blood shortage. I encourage all readers of Intermediate Physics for Medicine and Biology to make an appointment.

What’s the purpose of blood, anyway? To carry oxygen. Let’s estimate the concentration of oxygen in blood (Fermi problem time). As a first step, Homework Problem 1 in Chapter 1 of IPMB asks the reader to estimate how much hemoglobin is in a red blood cell.

Problem 1. Estimate the number of hemoglobin molecules in a red blood cell. Red blood cells are little more than bags of hemoglobin, so it is reasonable to assume that the hemoglobin takes up all the volume of the cell.

My name tag at the blood drive.

Russ Hobbie and I don’t send IPMB’s solution manual to anyone except instructors, but because you all are going to make blood-donation appointments as soon as you’re done reading this post, I’ll share the solution to this problem.

1.1 An important skill for students to learn is order-of-magnitude estimation. The first four problems in this chapter require the students to estimate some quantity of biological interest.

Approximate the dimensions of a red blood cell as 8 μm × 8 μm × 2 μm. Approximate the dimensions of a hemoglobin molecule as 6 nm × 6 nm × 6 nm. The number N of hemoglobin molecules is equal to the volume of a red blood cell divided by the volume of a hemoglobin molecule: 

We do not expect a “back-of-the-envelope” estimate such as this one to be accurate to, say, a factor of 2 or π. But it should give a quick order of magnitude approximation.

I had a difficult time taking
this selfie: one hand holding my
phone, the book balanced on my
chest, and a needle in the other arm.

Each hemoglobin molecule can bind with four oxygen molecules, so a red blood cell can contain 2400 million oxygen molecules. I’ll assume the hemoglobin isn’t packed too tightly, so let’s round that down to 2000. The volume of a red blood cell is 128 cubic microns. Inside a red blood cell the oxygen concentration is therefore 2000 million molecules per 128 cubic microns, or about 16,000,000/μm³. A typical hematocrit (fraction of blood volume occupied by red blood cells) is 40%. Therefore blood has an oxygen concentration of around 6 million per cubic micron.

I admit, those are strange units. A cubic micron is 10^-15 liters, and 6 million molecules is 10^-17 moles. So, the concentration of oxygen in blood is about 0.01 molar, or 10 mM.

You can estimate the concentration of oxygen in air using the ideal gas law, pV = nRT. Air is about 20% oxygen, so using p = 0.2 atm, T = 310 K, and R = 0.082 liter atm/(mole K), you get n/V = 0.008, or 8 mM. Within the uncertainty of our rough estimate, this result implies that the concentration of oxygen in blood is nearly the same as the concentration of oxygen in air. As it should be! The whole point of blood is to get oxygen from the air into the tissues.

The best part of blood donation.

Thanks to all the phlebotomists and volunteers for collecting blood, despite the risk; they’re heroes. I won’t be able to give blood again for another eight weeks. By that time I hope the @#$%&! coronavirus is gone and life has returned back to normal.

After giving blood. My daughter Stephanie,
who also donated, took the photo.