Tuesday, January 4, 2011

How Much Energy is in a Jelly Bean?

Consider the question, “If all of the chemical energy stored in a jelly bean (JB) were converted to mechanical energy of a projectile, at what vertical height could the projectile reach if it were shot off at ground level?”

We shall make a few reasonable assumption and approximations, being content to generally be within “factors of 2”:
Food energy in an average JB, E ~ 10 “food calories”*.
Mass of the projectile, M = 10 kg (just a little more than the mass of a typical bowling ball)
We make the assumption also that the chemical energy is 100% converted to mechanical energy (this will never be strictly true, but for purposes of illustrating how much energy is in the little JB, we shall go with this).

We also note that a “food calorie” is really a kilocalorie, as conventionally defined in physics (for example, see this site).

So, E = 10 “food calories” x 1000 cal/(food calorie) = 10,000 cal

There are 4.2 Joules per cal, so, converting to Joules, E = 42,000 Joules.

Now the projectile, if launched off at ground level with kinetic energy E, will reach its maximum height at a value h, where E = Mgh, with g = the acceleration of gravity near the earth’s surface = 9.8 m/s^2. Solving for h, we find, to the nearest 100 meters,
h = E/Mg = 4.2E4/(10*9.8) ~ 400 m
Since 1 yard ~ 1 m, we see that the height h is equal about 4 football fields. Pretty impressive, and I think surprising.

To fully appreciate this, consider the speed of the ball when it returns back to ground level (Man, get out of it’s way!). By conservation of energy, v = sqrt(2gh) = sqrt(2*9.8*400) ~ 90 meters/s, or about 300 ft/sec. We probably all recall the old conversion factor of 88 ft/sec = 60 miles/hr, so this is v ~ 200 miles/hr.

How can we understand this 42 kJ being stored in a tiny little ol JB? Well, if it were all Carbon (Z = 6, A =12), the number of C atoms would be N = Avogadro’s number divided by A, or
N = (6E23 atoms/mole)/12 g/mole = 5E22 atoms per g.
Sugar is of course a hydrocarbon molecule of C, O, and H, but for simplicity just consider it to be C, as representing an average atom. Typical energy differences in such a complex molecule are on the order of dE = 1 eV, so if the mass of a JB is m = 3 g (*see below), and if we assume that each atom can release about 1 eV as the sugar is “burned”, this is a total chemical potential energy of
E = 3 g x (5E22 atoms/g) x 1eV = 15E22 eV.
But 1.6E-19 J = 1 eV, so E = 27E3 Joules, or 27 kJ, in the ballpark of the 42 kJ used in the projectile height calculation. Of course, only a slight readjustment of dE is required to make the 42 kJ, so this shows how 10 food calories is about what we would expect for a typical JB.

It is amusing to consider two other energy levels associated with our little JB, although these might make it seem a bit more sinister. But that would not really be fair, as these energy levels are not extractable from the JB in a practical sense.

The energy difference levels in the nucleus of any atom such as carbon are on the order of 1 MeV, six orders of magnitude larger than the chemical energies of eV’s. This can be traced to the relatively large strength of the nuclear, or “strong”, force, relative to the weaker force of electro-magnetism. The latter is what is involved in chemical transitions, the former in nuclear transitions. Hence if the nucleus of each atom in the JB were to undergo a typical transition, this would correspond to 42 kJ x 1 million, or 42 Giga-Joules. To relate this to mechanical energy: this is for example the kinetic energy of a 20 ton ( ~ 20,000 kg) object moving at 4000 miles per hour (~ 2000 km/s).

Suppose our JB encountered an “anti matter JB”, and annihilated into energy. Each of the two colliding objects would release an energy of E = “M x c-squared” = (3E-3 kg) x (3E8 m/s)^2 = 3E14 J, or 300,000 GJ. A 1 megaton bomb is ~ 4 E15 J , so if just one JB completely annihilated, it would release an amount of energy within about 10% of that.

So these last two examples show that the chemical release is really just a modest amount of energy compared to what is potentially represented by that little lump of matter.

* I just purchased some jelly beans (“Palmer’s Select”) at a local 7/11. The package claims that each JB weighs about 3 g, and contains 12 “food calories”. These JB’s are the traditional JB, similar to “Brach’s”, and are much larger than the newer “Jelly Bellies”.