On the back of an envelope

For some reason my mother was always concerned whether I drank enough, and of course by the time I was studying chemistry I was pretty certain I did, and that I probably could do with less.😁🍺

Water from metabolizing food

My mother wasn't talking about beer of course, and while i was writing about saving tap water the other day, I was wondering how much of our recommend daily intake of water comes from eating rather than drinking.

Because when we metabolize ("burn") our food, the reaction with oxygen not only produces carbon dioxide ($CO_2$), but also water ($H_2O$) in a roughly equal amount. Part of that water leaves our body as water vapour when we breath out, the rest follows the same path as the water we drink, i.e. we pee it out our lose it in the form of sweat.

Calculations

So how much water does an average person produce in a day when eating the recommended amount of food?

The recommended daily intake of energy varies with age, weight, sex and perhaps a couple of things more, but let's settle for 10.000 kJ/day (2200 kCal).

That energy comes in the form of protein, that is typically not used as an energy source but as a source of building blocks to build and repair our body, carbohydrates, like sugars and starches, and fat. A healthy diet should get no more that 30% of its energy from fat, and 40% from carbs (the rest is from protein), although what individuals actually need may vary from person to person.

Because metabolizing 1 gram of fat produces about 38 kJ (9 kCal) and 1 gram of carbohydrates about 17 kJ (4 kCal) we can calculate that the daily intake of fat should be

$\frac{30\% \times 10000 kJ}{38 kJ/g} = 79 g$

For carbs we calculate

$\frac{40\% \times 10000 kJ}{17 kJ/g} = 235 g$

From (bio)chemistry we know that one molecule of a typical sugar (like glucose) produces 6 molecules of water, so all we need to know is the mass of a molecule of sugar and the mass of a molecule of water. And the beauty of chemistry is that we can determine this directly from the formulas for glucose ($C_6H_{12}O_6$) and water $H_2O$ because we know the mass of carbon, oxygen and hydrogen atoms.

A single atom has a very small mass so we normally work with moles, where 1 mol is approximately $6 \times 10^{23}$ (see Avogadro's number), and people have tabulated the molar mass for all the elements. We are interested in just three of them:

1 mol H = 1 g¹

1 mol C = 12 g

1 mol O = 16 g

¹ Approximate values but close enough, see Wikipedia on isotopes

With these molar masses for individual atoms we can now add them up to calculate the molar mass of our food molecules:

1 mol glucose $C_6H_{12}O_6$ = 72 + 12 + 64 = 148g, so our 235g of carbs equate to 235/148 = 1.6 mol

This will produce 1.6 x 6 = 9.6 mol of $H_2O$ , which has a molar mass of 18g, so this equates to about 173 g

Fats, or fatty acids, come in various forms and sizes (saturated, unsaturated, and chains of anywhere between 6 and 12 carbon atoms), but they can all be thought of as repeated units of one carbon atom and two hydrogen atoms, perhaps with a few oxygen atoms at the end. For our purposes we treat them all as pure $CH_2$ , so 14 g / mol, where each mol $CH_2$ produces 1 mol of water (in practice a little less because many fats are unsaturated and contain less hydrogen per carbon on average, but for a back of an envelope calculation this will do), so our 79 grams of fat equate to 79 / 14 = 5.6 mol $CH_2$ which will produce 5.6 mol of $H_2O$, or 5.6 x 18 = 102 g

Summary

So from the daily recommended intake of food, metabolizing the fats and carbs will produce 275 g of water, or just a big mug of coffee. The solid food we eat will contain some water as well, but with these amounts you all better listen to your mothers and stay hydrated.😀

PS

And what about alcohol? Well, most of it indeed converts to carbon dioxide and water, so a quick calculation for a 330ml bottle of 5% ABV lager, gives you $330 \times 5\% = 16.5 ml$ ethanol. At a density of about 0.8 g/ml this gives 13.4g.

The molar mass of ethanol, $C_2H_5OH$ is 46g, so that 13.4g equates to 0.3 moles, which will be metabolized to three times that number of moles of water at 18g / mol, so 0.9 x 18 = 16g. Which is almost nothing compared to the water in the bottle. There are better reasons to drink beer (responsibly of course 😜).

Acknowledgements

The image at the start of the article is by Anna Tarazevich on Pexels.