The existing atmosphere surrounding the earth contains a little less than one percent of the inert gas Argon, an approximate total of 6.59 x 1019 grams. The isotope Argon-40 (40AR) is 99.6 percent of the total Argon. The only know source of 40Ar is the radioactive decay of Potassium-40 (40K), of which 11.2 percent decays into 40AR and the remainder into Calcium-40 (40CA). The lithosphere (outer layers) of the earth has been estimated to be 2.59 percent Potassium, of which about 0.01167 percent is 40K. An approximately total of 5 x 1021 grams of 40K in the earth of which 11.2 percent could eventually decay into 40Ar. 40K has a half life of 1.26 billion years (1.26 x 109). To generate the amount of 40Ar in the atmosphere would require in round numbers 250 million years. However the model of an 4.5 billion year old earth would estimate that around 90 percent of the 40K had decayed, so originally there could have been around 50 x 1021 grams which would decrease the generation time to about 25 million years.
But 40Ar within the earth is only a small part of the story! Just because you have generated the required amount of 40AR due to decay within the earth does not mean that it will all appear in the atmosphere in a short time. Instead the majority will be trapped within the rocks of the earth until it can some how be released to the atmosphere by some type of volcanic event. When the rocks containing the 40Ar are heated to the molten state the gas will escape to the atmosphere except when it as at considerable depth and then the pressure will allow less of the gas to escape. Since there is at present relatively little volcanic action on the earth, in main that at the mid-ocean ridges and a few other isolated volcanic areas, 40Ar gas is released to the atmosphere at a very slow rate of approximately 2.2 x 109 grams per year. The figure below represents a plot of the rate of Argon release to the atmosphere, called outgassing, that has been proposed by one scientific study.
from ref. http://www.geo.cornell.edu/geology/classes/ Geo656/00notes/656_00Lecture25.pdf
These theories provide for a much faster outgassing rate and of course allows the present atmospheric Argon levels to be developed in 4.5 billion years since that is the age of the earth that many scientists propose. The plot of the 129Xe is of interest to us only because the study used the known decay rate of 129Iodine to 129Xe and the present atmospheric content of 129Xe to derive the proposed degassing rate. This is possible because 129Iodine has a short half life of 17 million years and today for all practical purposes no longer exists in the earth. Then using this information along with the known decay rate of 40Ar they were able to arrive at a proposed degassing rate with an initial "burp" of degassing with the early earth in a molten state. As the earth solidified the degassing rate then entered a second slower phase.
The question is are their any other possibilities that would allow for the atmospheric Argon to be developed in a very short time, less than 10,000 years?
a) Could the early earth have had a much higher level of 40K?
To generate the required amount of 40AR in less than 10,000 years would require levels of 40K at over 10,000 times the present levels. And then the short time allowed for decay would result in better than 99.8 percent of these high levels remaining at present. Therefore this option can be ruled out!
b) Possibly during the early stages of creation the decay mode was different and most all, or a very high percentage, of the 40K became 40AR.
This could possibly provide up to about an eight fold acceleration in generation of Argon, but well short of the required 10,000 fold increase.
c) Could the early earth being in a molten state allowed a very rapid outgassing rate, fast enough to allow the majority of the Argon to be generated on the first day?
As mentioned there is an indication of an early "burp", but the time required for a considerable amount of 40Ar to be generated in the earth limits the magnitude of this initial outgassing phase. And also "And the Spirit of God moved upon the face of the waters." of Genesis 1:2 indicates a earth during day one that had a surface cool enough for liquid water to be present. So if the earth was initially in a molten state it could only be for part of day one. Thus requiring a very long day one or ruling out this proposal.
d) Some have proposed that the impacts of gaseous comets provided the main source for the early earth atmosphere.
The interpretation of the early Genesis record to require creation within 6 literal days would not allow time for these proposed turbulent conditions on the earth to settle down to a reasonable steady state following such catastrophic collisions. For this to have been the source of much of the early atmosphere would again require a long day one.
e) Some have proposed a very rapid burst of radioactive decay at the beginning of creation, a condition that does not fit the laws of physics as we presently know them.
Such a burst could only be present during day one and day two of the creation or it would seem to subject all of the early life forms to a greater than "Chernobyl" radioactive condition which would seem to be incompatible with the development of life. "That variations in the ratio of 129Xe to other Xe isotopes as observed suggests (1) the Earth formed shortly (within 170 Ma) after a nucleosynthetic event (probably a supernova), and (2) a substantial fractionation of I from Xe must have occurred early in Earth's history. So 129Xe variations provide clues to the early degassing history of the Earth." These conclusions are contrary to this proposal and is the best science available today. Scientists have experimented and theorized about some means of increasing the isotope decay rate, for to do so might be advantageous say in energy generation, but to date no such method or possible theory has been discovered.
Conclusion: The total amount of 40Ar in the atmosphere today and the results of studying the concentrations of the other noble gas levels in the atmosphere and in the earth is another strong indicator of a old age earth.