DOS 514  Week 1 Discussion
Writing Prompt
Initial Post: Transient vs. Secular Equilibrium
According to Khan, the ratio between the activity of one isotope and the activity of its decay product (called the parent and daughter respectively) will reach a constant if the halflife of the parent is longer than the halflife of the daughter.^{1} When this happens, the system is said to be in equilibrium.
Khan explains that there are two kinds of equilibrium.^{1} If the half life of the parent is not much longer than the half life of its daughter, then the system will reach transient equilibrium as both isotopes decay. If the half life of the parent is much longer than the half life of the daughter, then the system will reach a secular equilibrium which is sustained over a long period of time.
The formula that governs the ratios of the activities is:
${A}_{2}={A}_{1}\frac{{\lambda}_{2}}{{\lambda}_{2}{\lambda}_{1}}(1{e}^{({\lambda}_{2}{\lambda}_{1})t})$
If λ_{1} is near in size to λ_{2}, then after a short time, the exponent part gets very small and the formula approaches:
${A}_{2}={A}_{1}\frac{{\lambda}_{2}}{{\lambda}_{2}{\lambda}_{1}}(10)$
This lets the formula be simplified after the equilibrium time to:
${A}_{2}={A}_{1}\frac{{\lambda}_{2}}{{\lambda}_{2}{\lambda}_{1}}$
If λ_{1} is extremely small compared to λ_{2}, then it essentially disappears from the equation, letting it be simplified to:
${A}_{2}={A}_{1}\frac{{\lambda}_{2}}{{\lambda}_{2}0}$
and then:
${A}_{2}={A}_{1}$
These formula variants yield very different results, but this is the basis of differentiating the two kinds of equilibria.
Since the relative activity of two members of a decay chain is predictable in a closed system, radioactive equilibrium can be used for measurement of the age of samples in the process of radiometric dating. Carbon14 dating is one commonlyknown form of radiometric dating, but other elements and isotopes can be used. Rubin, MacDougall, & Perfit have proposed the use of ^{210}Po  ^{210}Pb equilibrium as a way to calculate the age of recent glassy volcanic eruption products to within a few years.^{2} They note that in hot lava eruptions, polonium's relatively high volatility means that it is 95% degassed from fresh lava, while lead is only 1% degassed. This basically resets the clock while the lava is liquid. Once the lava cools, it may form a glassy deposit, which will immobilize everything including gases. This creates a situation much like Khan's example of a radium source inside a sealed vial to keep in the daughter radon gas.^{1}
Rubin et al point out that while ^{210}Po is also a decay product of ^{210}Bi, that parent has only a 5 day halflife, and after 2530 days the amount of ^{210}Po in the glassy deposit will be almost entirely dependent on ^{210}Pb. The halflife of ^{210}Po (138.4 days^{2}) and the much longer half life of ^{210}Pb (22.2 years^{3}) makes this an example of secular equilibrium. Both half lives are too short for this a suitable technique for ancient eruptions, but Rubin et al were interested in a finding a way to measure the age of relatively young eruptions that occurred underwater where they might not be discovered until several years later.
Grayden, Seattle
 Khan FM, Gibbons JP. The physics of radiation therapy. 5th ed. Philadelphia, PA: Lippincott, Williams, and Wilkins; 2014:18.
 Rubin KH, Macdougall JD, Perfit MR. ^{210}Po^{210}Pb dating of recent volcanic eruptions on the sea floor. Nature. 1994;368(6474):841844.
 Gray T. ^{210}Pb. Periodic Table Website. http://periodictable.com/Isotopes/082.210/index.p.full.html. Published 2007. Accessed September 3, 2014.
Followup Post Directed to Another Student
Kristy,
I was struggling a bit with the math too, since Khan does a bit of magic handwaving "trust me, it works" explanation without actually writing out a lot of the intermediate steps in the formula reduction or explaining what each piece really means.
One thing that I had a really hard time with was the exponent part and why it disappears. It's been quite a while since I've dealt with exotic exponents (fractional or negative) so I had to remind myself of what they mean. Khan (Sal Khan from Khan Academy, not Faiz M. Khan) explains that when you have a negative exponent, instead of multiplying by that number a certain number of times, you should do the opposite and divide by that number a certain number of times^{1}. This means you can rewrite the equation with a positive exponent by throwing the whole thing under a "1".
${e}^{blah}=\frac{1}{{e}^{blah}}$
Using the actual example...
${e}^{({\lambda}_{2}{\lambda}_{1})t}=\frac{1}{{e}^{({\lambda}_{2}{\lambda}_{1})t}}$
The exponent of "e" grows as time "t" increases, making the result of the fraction get closer to zero as time goes on.
$\frac{1}{exponentiallygrowingnumber}\to approaches0quickly$
In my initial post, I completely failed to make the connection that the equilibrium we're achieving occurs when the exponent part starts to disappear. Khan (our Khan) talks about things that happen after equilibrium is achieved, but as far as I can tell he doesn't actually come out and say that equilibrium is all about the exponent disappearing.^{2}
Because I didn't understand this point, I did a bit of magic handwaving when I said "If λ_{1} is near in size to λ_{2}, then after a short time, the exponent part gets very small". The formula I posted right after that statement (the one showing the exponent getting zeroed out) actually has nothing to do with the relative magnitudes of λ_{1} and λ_{2}; the exponent will eventually get zeroed out either way. I should have talked more about how λ_{1} and λ_{2} influence the rate at which that zeroing happens, rather than jumping straight to talking about what happens AFTER equilibrium.
In the exponent term, if λ_{1} is tiny compared to λ_{2}, as in a secular equilibrium, then it can basically be ignored and the time to reach equilibrium will be dependent only on the size of λ_{2}. If λ_{2} and λ_{1} are on the same order of magnitude, as in transient equilibrium, subtracting one from the other will reduce the ability of "t" to grow the exponent quickly. In other words, it will take longer for the whole fraction to resolve itself to 0, but it will still get there eventually.
Here's another part that made my brain hurt:
It will take longer to resolve, but relative to what? In Khan's example, he chose two parent/daughter pairs with very different sets of halflives^{2}. In his examples the transient equilibrium is actually achieved FASTER in terms of absolute time elapsed (maybe 30 hours) than the secular equilibrium, which took about 2025 days. If we instead measure in terms of halflives, his transient equilibrium is achieved in around 4 halflives of the daughter, while the secular equilibrium took around 6 halflives of the daughter. Not much difference there. What if we look at it in terms of the halflives of the parents? In that case, transient equilibrium was achieved in about 1/3 of a half life, while secular equilibrium was achieved in about 0.00004 halflives of the parent.
That right there is the big difference between transient and secular equilibrium. In order for λ_{1} to be tiny relative to λ_{2}, halflife_{1} must be MASSIVE relative to halflife_{2}, because λ and halflife are inversely related. Regardless of how much absolute time has passed, a secular equilibrium graph will always show the activity of the parent as a pretty much straight line across the top because the equilibrium is achieved pretty much instantly from the parent's perspective and barely any of the parent material will have decayed. In a transient equilibrium system, the slowness of the exponent disappearing means that the parent material will show significant decay by the time equilibrium is reached, and that is why both curves are pointed downwards. It's all about the shape of the graph, and not the actual numbers on the time scale.
I would like to point out that I puzzled most of this out on the fly, so everyone PLEASE check my assumptions to make sure I'm not glossing over important details again.
 Khan S. Negative exponents. Khan Academy Website. https://www.khanacademy.org/math/prealgebra/exponentsradicals/negativeexponentstutorial/v/negativeexponents. Published August 2, 2013. Accessed September 4, 2014.
 Khan FM, Gibbons JP. The physics of radiation therapy. 5th ed. Philadelphia, PA: Lippincott, Williams, and Wilkins; 2014:18.
Academic Courses > DOS 514 > Transient vs. Secular Equilibrium

Written September 3, 2014
First Semester, PreInternship 