Found this piece of awesomeness the other day. If the decay rate were not constant, we would all be dead!

**Were Adam and Eve Toast?**

**by Joe Meert**

**(created 1996, Updated January 2001)**

One of the issues in the creation/evolution debate is the claim that radioactive decay is not constant. In order for the Earth to appear old, creationists must assume that radioactive decay rates in the past were faster. This solves the problems of old ages in rocks, but unfortunately opens up a bigger problem. Radioactive decay gives off heat. The amount of heat generated is proportional to the rate of decay and the amount of radioactive material present at the time. In the following exercise, I show how much heat can be generated by radioactive decay IF decay rates were faster in the past. Please note that the following analysis uses present-day heat production values which are observable quantities and I have also assumed (in the creationist analysis) that each of the elements has the same rate of decay. Changing the decay rates to those observed today for each of the major heat producing elements makes the mathematics a bit more tedious, but does not significantly alter the conclusions given here.

Present day of heat production from radioactive decay in the Earth is produced mainly by the isotopes 238U, 235U, 232Th and 40K and has a value of 6.18 x 10-12 W/kg (Turcotte and Schubert, 1982). If the heat flow out of the Earth were due solely to radioactive decay in the mantle and crust, it would equal:

Heat Production * Mass of Mantle + Crust

6.18 x 10-12 W/kg * 4.0 x 1024 Kg = 2.47 x 1013 W (Eqn 1)

the heat flow out of the Earth can be calculated from Fourier’s Law of heat conduction:

Q = k dt/dz, where:

k = thermal conductivity of the rocks W/m oC

dt/dz = geothermal gradient (change in temp with depth)

Q= W/m2

To calculate the surface heat flow, we divide the result from Eqn 1 by the total surface area of the earth (5.1 x 1014 m2)

2.47 x 1013 W / 5.1 x 1014 m2 = .048 W/m2 (Eqn 2)

If we now use this to calculate the temperature-depth profile dt/dz, we need to use an average value of thermal conductivity. I use 3.00 W/m oC as an average thermal conductivity of the rocks. This yields:

dt/dz = .048 W/m2 / 3 W/m oC = 16 oC/km (Eqn 3)

Thus, for every 1 kilometer depth, the temperature will increase by about 16 oC. Special Note: As you get deeper into the crust and mantle, temperature increases are not linear because they are a function of pressure and other factors. Nevertheless, this relationship is useful to depths of 30-70 kilometers. Therefore, at a depth of 10 kilometers, the temperature would be about 160 oC. Basaltic rock melts at about 1200 oC.

Let’s look at these same relationships with a much younger earth and a faster rate of decay. In my example, I will use a ‘standard’ creationist age for the earth of 6000 years. In order to carry out the analysis, we must still assume some sort of decay ‘constant’. There are a number of ways this could be done, but the simplest is to assign an average half-life. I will use 500 years as the average ½ life for the radioactive elements listed above. In reality, I am being OVERLY generous to the young Earth crowd. If rates were variable, they would probably have been much faster than this and the resultant earth conditions would be more extreme. The interested reader can easily play with the variables on a spreadsheet and see how they affect the conclusions reached below.

In order to figure out the rate of heat production in the past, we must first calculate a decay constant lambda (l). This is calculated from

t ½ = ln 2 / l

or l = ln 2 / t ½

using 500 years for t ½ , we get l = 1.38 x 10-3.

The rate of heat production at any time in the past is given by the following formula (Turcotte and Schubert, 1982):

Hpast = Hpresent * e l (t-t0)

we use Hpresent = 6.18 x 10-12 W/kg.

and t-t0= 6000 years and l as calculated above and we get:

Hpast = 2.43 x 10-8 W/kg

We can now use this value to calculate the amount of heat flow out of the Earth due solely to this ancient (6000 year old) heat production and calculate a thermal profile as we did above in Eqn 1 and 2.

Hpast * Mass of Mantle + crust = total heat production

Hpast = 2.43 x 10-8 W/kg * 4.0 x 1024 kg = 9.75 x 1016 W

We now divide this value by the surface area of the Earth to get the heat flow Q at 6000 years ago.

Q = 9.75 x 1016 W / 5.1 x 1014 m2 = 191 W/m2

Using the same value for thermal conductivity 3.00 W/m oC, we can calculate the dt/dz profile as above in Eqn 3:

191 W/m2 / 3.00 W/ m oC = 70,000 oC/ km!!

At 6000 years ago, it is pretty obvious that the entire Earth would be molten and Adam and Eve’s goose was cooked. I must also add that because of these extreme temperatures, a purely conductive heat regime is implausible. I am not implying that this is a purely conductive regime. What I am showing is that the thermal regime on a 6000 year old earth with rampant radioactive decay is extreme. If you want to turn off the engine and let the Earth cool from this extreme you also run into time spans much greater than 6000 years because it will take a while to cool down from those extreme temperatures. It’s difficult to estimate how long it would take for the Earth to reach present-day temperatures because I frankly don’t know where the creationists would take their argument. It is clear that rapid decay results in a tremendous release of heat. I suppose you could always argue that the Earth has been cooling since this initial heating and that there is no further significant heat released by radioactive decay. In that case, the cooling of the Earth reduces to the Kelvin problem and gives an age of 20-60 million years. The graph below gives some of the major events in Creationist history along with the geothermal gradient.

Note the Scale is semi-logarithmic.

Additional Notes/Updates

(a) I did a rough calculation for the Earth’s conductive profile using the old Earth model and the average geothermal gradient 3.5 billion years ago is about 27 oC/km.

(b) Just for those who are interested, if Noah’s flood took place about 4000 years ago, this calculation yields roughly 40,000 oC/km. Guess what? Maybe there was a vapor canopy anyway since liquid water would not be present!!

(c) At the time of Jesus’ birth, the geothermal gradient would have been ~400 oC/km.

Note: Several creationists have commented that the calculations would not be correct if the speed of light has decayed over the past 6000 years. This is due to the fact that the speed of light factors into radioactive decay equations/energy. At present, there is no evidence to suggest that the speed of light (in vacuo) is a variable. Until such a time when this assertion can be substantiated with real data, the argument stands testament to one of the major problems associated with rapid decay proposals. In fact, ‘creation scientists’ at the Institute for Creation Research have noted:

“One major obstacle to accelerated decay is an explanation for the disposal of the great quantities of heat which would be generated by radioactive decay over short periods of time. For example, if most of the radioactive decay implied by fission tracks or quantities of daughter products occurred over the year of the Flood, the amount of heat generated would have been excessive, given present conditions.”

Refs: 1. Turcotte & Schubert, 1982, Geodynamics, John Wiley and Sons