2009-01-13

How much does water weigh?

Update 2009-08-07: Now the slightly lower density of water at room temperature is taken into account, improving the answer by 0.3%.

Is this a silly question? Not at all, if you're using either American or Imperial measurements.

I just ran across two web pages that say one cup of water weighs 8 ounces (oz). To protect the guilty, I won't mention the sites except to say that one is from a physicist and the other from a cooking magazine, both of whom should know better—though from different perspectives—than to publish this incorrect information. Let's do the math:

By definition, 1 liter (L) holds 1000 cubic cm (cm3, or cc) of volume, thus 1 milliliter (mL) is the same as 1 cc in volume. The weight of water was also initially defined as being 1 kilogram (kg) per liter, but that definition has been discarded since the density of water changes with temperature and atmospheric pressure (if you're curious, this PDF outlines some factors to consider when measuring water in a lab). Taking this into account, the following calculations are based on data at room temperature (25 °C/77 °F) and 1 atmosphere (basically, air pressure at sea level). Tap water is slightly denser than pure water, but the difference is negligible at room temperature. Under these conditions, water has a density of 997.13 kg/m3. For the calculations, we'll also need a few conversion factors:

1 m = 100 cm
1 kg = 1000 g
1 L = 1.0567 US quarts (qt)
1 qt = 4 cups
1 lb = 16 oz = 453.59 g
Just to be explicitly clear, all non-metric weights in this post are expressed in avoirdupois units. This is the ordinary weight system used in the US (as opposed to troy units, which are used for weighing precious metals). Since avoirdupois is the system in common usage, I think it'll be a bit cleaner to omit this extra notation in the equations. Now we calculate,
(997.13 kg/m3) × (1000 g/kg) × (1 m/100 cm)3 × (1000 cm3/L) = 997.13 g/L
Converting to US units,
(997.13 g/L) × (1 lb/453.59 g) × (1 L/1.0567 qt) = 2.0803 lb/qt
Finally, converting to cups,
(2.0803 lb/qt) × (16 oz/lb) × (1 qt/4 cups) = 8.3214 oz/cup (US)

In the US customary measurement system, then, one cup of water, which is 8 fluid ounces (fl. oz) in volume, is actually 8.3214 oz in weight. Is it close enough? I don't think so. If you were measuring water on a digital scale that displays 1/8-oz increments, the nearest reading would be 8-3/8 oz. For scales with 1/4-oz resolution, the nearest reading would be 8-1/4 oz.

Based on the above, we can make some handy tables for various volumes. Let's start with metric figures first:

Weight of water at 25 °C (77 °F) at sea level.
Metric
Volume
US
Volume
Imperial Volume Metric
Weight
Avoirdupois Weight
1 milliliter
(= 1 cc or cm3)
0.03381 fl oz 0.03520 fl oz 0.9971 g 0.03517 oz
1 liter 1.057 qt 0.8799 qt 0.9971 kg 2.198 lb
1 cubic meter 264.2 gal 220.0 gal 997.1 kg 2198 lb

In US units:

* Based on these defined (i.e., exact) figures: 1 US liquid gallon = 231 in3 and 1 inch = 2.54 cm. See the Wikipedia links below for source data.
US Customary Volume Multiplier (exact) Metric Volume* Avoirdupois Weight Metric Weight
1 teaspoon N/A 4.929 mL 0.1734 oz 4.915 g
1 tablespoon = 3 tsp 14.79 mL 0.5201 oz 14.74 g
1 fluid ounce = 2 tbsp 29.57 mL 1.040 oz 29.49 g
1 cup = 8 fl oz 236.6 mL 8.321 oz 235.9 g
1 pint = 2 cup 473.2 mL 16.64 oz
(1.040 lb)
471.8 g
1 quart = 2 pt 0.9464 L 33.29 oz
(2.080 lb)
0.9436 kg
1 gallon = 4 qt 3.785 L 8.321 lb 3.775 kg
1 cubic inch N/A 16.39 mL 0.5764 oz 16.34 g
1 cubic foot = 1728 in3 28.32 L 62.25 lb 28.24 kg
1 cubic yard = 27 ft3 764.6 L 1681 lb 762.4 kg

Let's pause for a moment and think about terminology a bit. It seems that some people have trouble distinguishing between how weight and volume are defined. What's the difference? Simply,

Weight = how heavy
Volume = how big
It's important to remember that different substances filled to the same size (volume) can have different weights. To use a classic example, a bucket full of feathers will be lighter in weight than the same bucket filled with rocks. The same principle applies to liquids in the kitchen as well. A cup (that is, 8 fluid ounces) of oil will weigh less than a cup of water.

Speaking of fluid ounces, what do you think happens when the name of a unit normally associated with weight (oz) gets included in a unit for measuring volume (fl. oz)? Constant confusion, that's what. The nomenclature makes for a huge mess, but it's what we're stuck with.

It also doesn't help matters that the US measures are colloquially called "English" or "British" units, even though the official US and Imperial systems differ significantly when it comes to measuring volume. As it happens, 1 Imperial fl. oz equals 0.96076 US fl. oz, so 8 Imperial fl. oz of water is equivalent to 7.9948 oz by weight, which I believe is actually close enough to 8 oz for cooking-measurement purposes.

Further, not only do the measurements differ between US customary and Imperial units, so do the multipliers. An Imperial pint, for example, is 20 Imperial fl oz, whereas a US pint is 16 US fl oz. So a pint of water weighs either 16.643 oz (US) or 19.987 oz (Imperial). Not quite "a pint's a pound the world around," there. At any rate, we can also calculate a table for Imperial units:

* Based on these defined (i.e., exact) figures: 1 Imperial gallon = 4.54609 L & 1 Imp. tbsp = 5/8 Imp. fl. oz. See the Wikipedia links below for source data.
Imperial (UK) Volume Multiplier (exact) Metric Volume* Avoirdupois Weight Metric Weight
1 teaspoon N/A 5.919 mL 0.2082 oz 5.902 g
1 dessertspoon = 2 tsp 11.84 mL 0.4164 oz 11.80 g
1 tablespoon = 3 tsp 17.76 mL 0.6246 oz 17.71 g
1 fluid ounce = 1.6 tbsp 28.41 mL 0.9994 oz 28.33 g
1 cup = 10 fl oz 284.1 mL 9.994 oz 283.3 g
1 pint = 2 cup 568.3 mL 19.99 oz
(1.249 lb)
566.6 g
1 quart = 2 pt 1.137 L 39.97 oz
(2.498 lb)
1.133 kg
1 gallon = 4 qt 4.546 L 9.994 lb 4.533 kg
Cubic inch, foot, and yard figures are the same as US measures.

Notice how close an Imperial gallon of water comes to weighing 10 lbs. That's because, similar to the original kg/L equivalence, the Imperial gallon was defined as the volume occupied by 10 lbs of distilled water, but at a lower temperature and slightly higher pressure than what I used here (specifically, 62 °F and 30 inches of mercury, which is about 1.0025 atm).

Conversely, we can also figure volumes based on a starting weight, again under the condition of 25 °C (77 °F) at sea level:

Weight of water US Volume Imperial
Volume
Metric Volume
1 gram 0.03391 fl oz 0.03530 fl oz 1.003 mL
1 kilogram 33.91 fl oz
(1.060 qt)
35.30 fl oz
(0.8824 qt)
1.003 L
1 avoirdupois ounce 0.9614 fl oz 1.001 fl oz 28.43 mL
1 avoirdupois pound 15.38 fl oz
(1.923 cup)
16.01 fl oz
(1.601 cup)
454.9 mL

You know, this post started as a rant but turned into a bunch of conversion tables. How did that happen? I think I'll stop here before I get all worked up over the distinction within the US system itself of liquid quarts vs. dry quarts or how the customary measures used in cooking differ from the FDA-regulated measures used for food nutrition labeling. Makes me wish all of our recipes would be published in metric, but I know I'll never see the day.

Oh, if you've read this far (thank you, really), you may now be wondering why I haven't yet provided a conversion table between US and Imperial volumes or between cubed lengths and volumes. The reason is because this post is about the weight of water, expressed in multiple units, and not about converting a volume from one unit to another. Besides, the existing tables already contain enough information, in the form of metric volumes or weights, for you to derive these conversions yourself—or, for a more direct route, you can use any of the calculators available online. You can also check with Wikipedia for conversion tables and unit definitions:

14 comments:

  1. Wow. This was so awesome and, complete over share coming, a real turn on!

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  2. My husband and I had a discussion where this topic came up and just so I am clear, I have the following question:

    Is it just a coincidence that one fluid ounce of water weighs approximately one ounce or can it be said that the weight of water was the origin of the fluid ounce?

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  3. I don't know that there is a definitive answer, but it's unlikely that the weight of water was the origin of the fluid ounce. Instead, it was the trade of wine (which, as it happens, has a density very close to that of water) that brought about the measurement of liquid quantities. The wine gallon is the volume that holds 8 troy pounds of wine, but it's unclear to me how this is converted to the fluid ounce -- if we use the avoirdupois system of 16 ounces per pound, then the math works out neatly to one oz. of weight being equal to one oz. of volume (8 x 16 = 128 oz, and 1 gallon = 128 fl. oz.); however, if we use the troy system of 12 ounces per pound, then things don't work out so well.

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  4. answer to how much water weighs? we should be using the metric system. thank you Ronald Reagan for stopping that progress way back when.

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  5. Based on water's density at different temps, would that influence the weight of whole wheat flour at freezer temps vs. room temp? Thanks

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    Replies
    1. It's possible, but I don't really know the answer. Getting exact measurements on flour is not easy, since it settles over time and thus will have different weights for the same volume depending on whether you sifted it first or not. It also absorbs (or sheds) moisture depending on surrounding humidity.

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  6. You used the weight of water at sea level and 77F. Do you know what the weight difference would be at 5300 feet above sea level and 77F?

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    Replies
    1. For all practical purposes, the difference would be zero. At high altitudes, water's density would decrease slightly (because there's less air pressure pushing against it), but since water does not expand or compress as readily as air, the effects of altitude are a lot less pronounced. Air pressure at 5,300 ft. is roughly 84 kPa, or 8.4 decibars, compared to 101.325 kPa at sea level (I know I'm using different units from the main post, but they all convert). If we plug these temperature and pressure numbers into this calculator, we get an answer of 997.086 kg/m3, compared to 997.13 kg/m3 at sea level. That's a difference of less than 0.005%. I don't think you'd be able to measure this difference unless you used very precise instruments in a highly controlled environment.

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  7. There is some history about the origins of the relationship between fluid ounces and ounces (and pints and pounds) at this site:
    http://www.miketodd.net/encyc/measures1.htm

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  8. Badass rant/informative read!

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  9. Very informative and interesting post!! Thanks!! Just tried to search "how much a drop of water weighed" and got glued to this rant (not in an offensive way at all).. Awesome.

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  10. That is a spectacular article! Er, uh, pardon me, rant! Finally, someone writing about what real people need to know about a substance we use in the real world, not at 4 degrees C in a lab . . . . my bread baking soul thanks you from the bottom to top of its floury little heart!

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