Question

The lethal dose of tetanus toxin (`M_text(r) = "150 000"`) is 2.5 ng/kg. How many molecules per cell does it take for the tetanus toxin to be lethal to humans?

Answer 1

It takes about one tetanus toxin molecule per 60 cells to be lethal.

Explanation:

Let's do this in steps.

Step 1. Calculate the mass of a single toxin molecule.

Mass = `"1 molecule" × "150 000 Da"/"1 molecule" × "1 g"/(6.022 × 10^23color(white)(l)"Da") = 2.49 ×10^"-19"color(white)(l)"g"`

Step 2. Convert the lethal dose to molecules of toxin per kilogram of body mass.

`"LD" = (2.5 × 10^"-9"color(red)(cancel(color(black)("g toxin"))))/("1 kg body") × "1 molecule toxin"/(2.49 ×10^"-19"color(red)(cancel(color(black)("g toxin")))) = (1.00 × 10^10color(white)(l)"molecules toxin")/"1 kg body"`

Step 3. Calculate the lethal dose per human body.

The average mass of a human adult in the world is 62.0 kg.

`"LD" = (62.0color(red)(cancel(color(black)("kg body"))))/("1 human") × (1.00 × 10^10color(white)(l)"molecules toxin")/(1 color(red)(cancel(color(black)("kg body")))) = (6.22 × 10^11color(white)(l)"molecules toxin")/"1 human"`

Step 4. Calculate the lethal dose per cell.

There are `3.72 × 10^13` cells in the human body.

`"LD" = (6.22 × 10^11"molecules toxin")/(1 color(red)(cancel(color(black)("human")))) × (1 color(red)(cancel(color(black)("human"))))/(3.72 × 10^13color(white)(l)"cells") = "0.0167 molecules toxin"/"1 cell" = "1 molecule toxin"/"60 cells"`

Now that's toxic!