Question

How many molecules are in the human body?

Answer 1

Here's what I get.

Explanation:

Scientists recently made the most precise estimate so far of the number of cells in the human body — a whopping `3.72 × 10^13` cells!

They then used tabulated values and some other tricks to arrive at a value of roughly 27 pg for the average mass of a cell.

About 65 % of a cell's mass is water.

Thus, an average cell contains about 18 pg of water and 9 pg of other substances.

With some judicious guesses as to the average molar masses, we can calculate the number of molecules in the cell ("we leave that as an exercise for the student …").

`"Molecule"color(white)(m)"% Dry Wt"color(white)(m)"Mass/pg"color(white)(mm) "MM"color(white)(mm)"Molecules"`
`stackrel(————————————————————)("Protein"color(white)(mmmm) 55color(white)(mmmm) 5.2color(white)(mmll) "30 000"color(white)(ml) "100 000 000")`
`"RNA"color(white)(mmmmm) 20color(white)(mmmml) 1.9color(white)(mm) "138 000" color(white)(mml)"8 000 000"`
`"DNA"color(white)(mmmmmll) 3color(white)(mmmml) 0.3 color(white)(mm)3×10^9 color(white)(mmmmmml)2`
`"Lipid"color(white)(mmmmm) 12 color(white)(mmmml)1.1color(white)(mmml) 1000 color(white)(ml)"700 000 000"`
`"Carbohydrate" color(white)(ml)6color(white)(mmmml) 0.6color(white)(m) "1 000 000" color(white)(mmml)"300 000"`
`"Other"color(white)(mmmmml) 4color(white)(mmmml) 0.4`
`"Water"color(white)(mmmmmmmmmll) 18color(white)(mmmmm) 18 color(white)(mmm) 6×10^11`
`stackrel(————————————————————)(color(white)(m)"CELL"color(white)(mmmmmmmmll)27 color(white)(mmmmmmmmml)6×10^11)`

Note: The "other" substances in the cell are ionic substances, not molecules.

This is the number of molecules in a cell.

The number of molecules in the body is

`3.72 × 10^13 color(red)(cancel(color(black)("cells"))) × (6 × 10^11color(white)(l) "molecules")/(1 color(red)(cancel(color(black)("cell")))) = 2× 10^25color(white)(l) "molecules"`

Thus, the body consists of about `2 × 10^25color(white)(l) "molecules"`, and more than 99 % of them are water!

Answer 2

Warning! Long Answer. It depends on mass of the body. However, the body of a 57 kg teenager would consist of about `1.2 × 10^27` molecules.

Explanation:

Here's a second answer. It shows how the numbers you get depend on the assumptions you make.

The body of a newborn infant will have fewer molecules than that of an overweight adult.

Let's assume that you are a teenager with a mass of 57 kg.

Scientists estimate that your body is

• 61 % water
• 16 % fat
• 16 % protein
• 6% minerals
• 1% carbohydrate

Minerals include inorganic ions such as sodium, potassium, chloride, hydrogen carbonate, and phosphate.

Thus, your body consists of

• `"water"color(white)(mmmm) "34.8 kg"`
• `"fat"color(white)(mmmmmm) "9.12 kg"`
• `"protein"color(white)(mmmll) "9.12 kg"`
• `"minerals"color(white)(mmml) 3.4color(white)(ll) "kg"`
• `"carbohydrate"color(white)(ll) "0.57 kg"`

With some judicious guesses as to the average molar masses, we can calculate the number of molecules.

Minerals are not molecules, so I will omit them from the calculation

Let's assume that the average molecular masses are

• `"water:"color(white)(mmmmmmm) "18.02 u"`
• `"fat":color(white)(mmmmmmmml)1000 color(white)(l)"u"`
• `"protein:"color(white)(mmmmmll)"30 000 u"`
• `"carbohydrate:"color(white)(m)"1 000 000 u"`

`"Water" = "34 800" color(red)(cancel(color(black)("g"))) × (1 color(red)(cancel(color(black)("mol"))))/(18.01 color(red)(cancel(color(black)("g")))) × (6.022 × 10^23 "molecules")/(1 color(red)(cancel(color(black)("mol"))))`

`= 1.16 × 10^27color(white)(l) "molecules"`

`"Fat" = "9120" color(red)(cancel(color(black)("g"))) × (1 color(red)(cancel(color(black)("mol"))))/(1000 color(red)(cancel(color(black)("g")))) × (6.022 × 10^23 "molecules")/(1 color(red)(cancel(color(black)("mol")))) = 5.49 × 10^24color(white)(l) "molecules"`

`"Protein" = "9120" color(red)(cancel(color(black)("g"))) × (1 color(red)(cancel(color(black)("mol"))))/("30 000" color(red)(cancel(color(black)("g")))) × (6.022 × 10^23 "molecules")/(1 color(red)(cancel(color(black)("mol"))))`

`= 1.83 × 10^23color(white)(l) "molecules"`

`"Carbohydrate" = "580" color(red)(cancel(color(black)("g"))) × (1 color(red)(cancel(color(black)("mol"))))/("1 000 000" color(red)(cancel(color(black)("g")))) × (6.022 × 10^23 "molecules")/(1 color(red)(cancel(color(black)("mol"))))`

`= 3.49 × 10^20color(white)(l) "molecules"`

Thus, the total number of molecules in your body is

`color(white)(ll)"1.16 × 10^27`
+ `"5.49 × 10^24`
+ `"1.83 × 10^23`
+ `ul("2.49 × 10^20`
`color(white)(m)1.17 ×10^27"`

So, your body consists of `1.2 × 10^27` molecules, and more than 99 % of them are water molecules.