Energy and Human Ambitions on a Finite Planet, 2021a

5 <strong>Energy</strong> <strong>and</strong> Power Units 74
(1 kcal), equating to 4,184 J. Most memorably, it is the amount of energy it
takes to heat one kilogram (or one liter; 1 L) of water by 1 ◦ C. Due to the
tragic convention of Calorie, we will opt for kcal whenever possible.
Food labels in the U.S. are in Calories, describing the energy content of the
food we eat. 15 We would all do ourselves a favor by calling these kcal instead
of Calories (same thing). Many other countries sensibly use either kJ or kcal
for quantifying food energy.
15: <strong>Human</strong> metabolism is not the same as
heating water, but the energy involved can
still be counted in an energy unit that is
defined in terms of heating water. It’s still
just energy.
20C + 50 calories (209 J) 70C
Figure 5.3: Following the definition of a
calorie, adding 50 cal to one gram of water
1 cm 3 = 1 mL = 1 g
raises its temperature by 50 ◦ C.
Example 5.5.1 To change 30 mL (30 g) of water by 5 ◦ C requires 150 cal,
or a little over 600 J.
Injecting 40 kcal of energy into a 2 L (2 kg) bottle of water will heat it
by 20 degrees.
Drinking 250 mL of ice-cold water <strong>and</strong> heating it up to body temperature
(thus raising its temperature by approximately 35 degrees) will
take about 8,750 cal, or 8.75 kcal, or a bit over 36 kJ of energy.
It is usually sufficient to remember that the conversion factor between
calories <strong>and</strong> Joules is about 4.2—or just 4 if performing a crude calculation.
1 cal = 4.184 J ≈ 4.2 J ∼ 4J
1 kcal = 4,184 J ≈ 4.2 kJ ∼ 4kJ
Two examples will help cement use of the kcal (a more useful scale in
this class than the much smaller calorie).
Example 5.5.2 A typical diet amounts to a daily intake of about
2,000 kcal of food energy. If you think about it, 2,000 kcal/day is a
power (energy per time). We can convert to Watts by changing kcal to J
<strong>and</strong> one day to seconds. 2,000 kcal is 8.368 MJ. One day has 86,400
seconds. The division of the two is very close to 100 W. 16
A second example hews closely to the definition of the kcal: heating
water.
Example 5.5.3 Let’s say you want to heat a half-liter (0.5 kg) of water
from room temperature (20 ◦ C) to boiling (100 ◦ C). Since each kcal can
heat 1 kg by 1 ◦ C, that same energy will raise our half-kg by 2 ◦ C. 17 So
raising the temperature by 80 ◦ C will require 40 kcal, or 167 kJ.
If the water is heated at a rate of 1,000 W (1,000 J/s), it would take 167
seconds for the water to reach boiling temperature.
Notice that we did not apply an explicit formula in Example 5.5.3. By
proceeding stepwise, we attempt to keep it intuitive. We could write a
No deep significance attaches to the fact that
1 cal happens to equate to 4.184 J, other than
to say this describes a property of water
(called specific heat capacity).
16: It would serve little purpose to perform
exact math here—producing 96.85 W in
this case—since the idea that someone’s
daily diet is exactly 2,000.00 kcal is pretty
preposterous. It will likely vary by at least
10% from day to day, <strong>and</strong> by even larger
amounts from individual to individual, so
that 100 W is a convenient <strong>and</strong> approximate
representation.
17: Make sure this is clear to you; by underst<strong>and</strong>ing,
we are installing concepts instead
of formulas, which are more powerful <strong>and</strong>
lasting.
Appendix Sec. A.8 (p. 368) addresses this
philosophy in a bit more detail.
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.