# Counting Carbon Calories

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Additionally, some earlier writings:
"When you lose weight, where does it go?"

An interesting question, and one where most people haven't thought about it, don't know, and really don't care. But when I was asked, it set in motion a train of thought. Follow along, and see if it makes sense.

## Every breath you take ...

 All the numbers here are approximate, and it would be an interesting exercise to work out how you would verify them for yourself. Some are easy, others less so.
Assuming we are at rest, and we remain at rest, we take around ten breaths a minute. There are 1440 minutes in a day, so about 14400 breaths per day, again, assuming we don't exert ourselves.

In each breath we take in about 800 ml or so, although it varies from person to person. Again, it would be interesting to figure out how to measure that with just stuff you can find around the house.

 I repeat, this is for when you're at rest. It will all change radically if you start exercising.
So total volume of air breathed per day is about 11520 litres, or 11.5 $m^3.$

Of the air you breathe, about 20% is Oxygen, and when you breathe out it's about 16%, or so I remember being told at one point. If that's right, then roughly 4% of the air you breathe out is carbon dioxide. That means you breathe out around 460 litres of CO${}_2$ per day, and that's 0.46 cubic metres.

How much does that weigh? Good question.

## Heavy ...

 A lot of people are surprised by this ... how would you check it? How would you measure it?
Carbon dioxide weighs 0.001836 grams per cubic centimetre, which is 1.836 kilograms per cubic metre, so we have a total weight of around 845 grams.

(You should check my working so far.)

Now the atomic weight of carbon is 12, and of oxygen is 16, so the atomic weight of one molecule of CO${}_2$ is 44. That means the weight of carbon atoms we exhale in a day is 12/44 of that 845 grams, or about 230 grams.

 I should have done this calculation with glucose, not sucrose, but you might like to try that for yourself and see what you get.
The chemical formula for sucrose is $C_{12}H_{22}O_{11},$ so if we assume that all the carbon we exhale comes from metabolising sucrose, that means we need 230 grams worth of carbon. But the atomic weight of sucrose is 12x12+22x1+11x16, which is 342, and the atomic weight of the carbon in a sucrose molecule is 144. So the carbon is 144/342 of a sucrose molecule (which if you're interested simplifies to 8/19 (although it actually simplifies to that whether you're interested or not!)), and that means we need 342/144 grams of sucrose to get each gram of carbon.

So to get 230 grams of carbon we need (342/144)x230 grams of sucrose, which is about 550 grams.

(You should check my working again.)

## Energy ...

Depending on your source, you'll find that sucrose (which is just one example of a sugar) is about 4 Calories (that's 4 kCals, the sort of calories you count when dieting) per gram. So the number of (dietary) Calories in 550 grams of sucrose is 2200.

Quoting from the NHS website[0]:

• An ideal daily intake of calories varies depending on age, metabolism and levels of physical activity, among other things.
• Generally, the recommended daily calorie intake is 2,000 calories a day for women and 2,500 for men.

Interesting how our computed value for calories consumed based purely on the carbon breathed out matches that so closely

## So when you lose weight, where does it go?

 The real situation is incredibly complex - don't for a moment assume that this is everything. It's not, it's really not. but this is a start on how we can think about these things.
I haven't answered that directly, but the ideas are all here. The nutrients in the food you eat are metabolised, and either get stored as fat (or other energy stores), or broken down to provide energy, and the by-products are almost entirely water and carbon dioxide.

Broadly speaking, when you lose weight, you breathe it out.

But the amazing thing is that with some simply sums we can start to analyse how this all works.

## References

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