Hubbry Logo
E-foldingE-foldingMain
Open search
E-folding
Community hub
E-folding
logo
7 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
E-folding
E-folding
E-folding
View on Wikipedia
from Wikipedia

In science, e-folding is the time interval in which an exponentially growing quantity increases or decreases by a factor of e;[1] it is the base-e analog of doubling time. This term is often used in many areas of science, such as in atmospheric chemistry, medicine, theoretical physics, and cosmology.

In cosmology the e-folding time scale is the proper time in which the length of a patch of space or spacetime increases by the factor e.

In finance, the logarithmic return or continuously compounded return, also known as force of interest, is the reciprocal of the e-folding time.

The process of evolving to equilibrium is often characterized by a time scale called the e-folding time, τ. This time is used for processes which evolve exponentially toward a final state (equilibrium). In other words, if we examine an observable, X, associated with a system, (temperature or density, for example) then after a time, τ, the initial difference between the initial value of the observable and the equilibrium value, ΔXi, will have decreased to ΔXi /e where the number e ≈ 2.71828.

  • Te e-folding time
  • N(t) amount at time t
  • N(0) initial amount
  • Td doubling time
  • ln(2) ≈ 0.693 natural logarithm of 2
  • r% growth rate in time t

Example of lifetime as e-folding time

[edit]

The concept of e-folding time may be used in the analysis of kinetics. Consider a chemical species A, which decays into another chemical species, B. We could depict this as an equation:

Let us assume that this reaction follows first order kinetics, meaning that the conversion of A into B depends only on the concentration of A, and the rate constant which dictates the velocity at which this happens, k. We could write the following reaction to describe this first order kinetic process:

This ordinary differential equation states that a change (in this case the disappearance) of the concentration of A, d[A]/dt, is equal to the rate constant k multiplied by the concentration of A. Consider what the units of k would be. On the left hand side, we have a concentration divided by a unit of time. The units for k would need to allow for these to be replicated on the right hand side. For this reason, the units of k, here, would be 1/time.

Because this is a linear, homogeneous and separable differential equation, we may separate the terms such that the equation becomes:

We may then integrate both sides of this equation, which results in the inclusion of the constant e:

where [A]f and [A]i are the final and initial concentrations of A. Upon comparing the ratio on the left hand side to the equation on the right hand side, we conclude that the ratio between the final and initial concentrations follows an exponential function, of which e is the base.

As mentioned above, the units for k are inverse time. If we were to take the reciprocal of this, we would be left with units of time. For this reason, we often state that the lifetime of a species that undergoes first order decay is equal to the reciprocal of k. Consider, now, what would happen if we were to set the time, t, to the reciprocal of the rate constant, k, i.e. t = 1/k. This would yield

This states that after one lifetime (1/k), the ratio of final to initial concentrations is equal to about 0.37. Stated another way, after one lifetime, we have

which means that we have lost (1 − 0.37 = 0.63) 63% of A, with only 37% left. With this, we now know that if we have 1 lifetime passed, we have gone through 1 "e-folding". What would 2 "e-foldings" look like? After two lifetimes, we would have t = 1/k + 1/k = 2/k, which would result in

which says that only about 14% of A remains. It is in this manner that e-folding lends us an easy way to describe the number of lifetimes that have passed. After 1 lifetime, we have 1/e remaining. After 2 lifetimes, we have 1/e2 remaining. One lifetime, therefore, is one e-folding time, which is the most descriptive way of stating the decay.

See also

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

E-folding

View on Grokipedia
from Grokipedia
e-Folding is a concept in mathematics and physics that describes the characteristic time scale for exponential growth or decay processes, specifically the time interval over which a quantity increases or decreases by a factor of e (Euler's number, approximately 2.71828). This measure, known as the e-folding time τ, quantifies the rate of change in systems governed by exponential functions of the form y(t)=y0ekty(t) = y_0 e^{kt}, where kk is the growth or decay constant and τ = 1/|k|. It serves as the natural analog to the doubling time (for growth by a factor of 2) or halving time (for decay by a factor of 1/2), but is particularly convenient in continuous models due to the properties of the natural exponential function. The e-folding time arises naturally in the solution to linear differential equations modeling phenomena like , , and thermal relaxation, where the system's behavior is dominated by a single exponential term. For instance, in decay processes, τ represents the time for the quantity to reach 1/e (about 36.8%) of its initial value, providing a standardized metric for comparing rates across different systems. In growth scenarios, such as bacterial proliferation or interest compounding continuously, it indicates the time to multiply by e, facilitating precise calculations in fields requiring logarithmic scales. e-Folding times find broad applications in scientific disciplines, including for estimating lifetimes (e.g., global burdens decaying with τ ≈ 14 days for certain radionuclides), for modeling water flushing in estuaries, and for assessing moisture memory persistence. In climate and environmental modeling, they help characterize the response times of systems to perturbations, such as accumulation or pollutant dispersion. These uses underscore e-folding's role as a fundamental tool for interpreting and predicting exponential behaviors in natural and engineered processes.

Definition and Basic Concepts

Core Definition

The e-folding time, denoted as τe\tau_e, is the characteristic duration over which an exponentially growing or decaying quantity changes by a factor of ee (approximately 2.71828), where ee is Euler's number, the base of the natural logarithm. In growth scenarios, the quantity increases to ee times its initial value; in decay scenarios, it decreases to 1/e1/e (about 36.8%) of its initial value. This scale serves as the natural time constant in models described by base-ee exponentials, providing a standardized measure for the rate of exponential processes without relying on arbitrary bases like 2 for doubling times. Euler's number ee emerges as the fundamental constant in continuous and natural growth processes, defined as the limit lim⁡n→∞(1+1/n)n\lim_{n \to \infty} (1 + 1/n)^n, and it underlies the exe^x whose equals itself, making it ideal for describing rates of change proportional to the quantity itself. The e-folding time is particularly useful because it aligns directly with this property, offering a dimensionally consistent timescale tied to the exponential rate constant λ\lambda. The general formula for the e-folding time is τe=1/∣λ∣\tau_e = 1 / |\lambda|, where λ\lambda is the continuous growth or decay rate from the dN/dt=λNdN/dt = \lambda N, with N(t)=N0eλtN(t) = N_0 e^{\lambda t}. The concept of e-folding originates from 18th-century developments in calculus, particularly the solutions to first-order linear differential equations modeling population growth or radioactive decay, as advanced by Leonhard Euler, who formalized the use of ee in exponential expressions around 1727–1728. Although no single inventor is credited with the term "e-folding time" itself, it stems from Euler's foundational work on the exponential function and its applications in solving such equations. Numerically, after one e-folding time in decay, approximately 36.8% of the original quantity remains, while in growth, it reaches about 271.8% of the initial value, illustrating the symmetric yet inverse nature of these processes.

Relation to Exponential Processes

The e-folding time serves as the characteristic time scale in the general exponential model for a N(t)N(t) evolving as N(t)=N0eλtN(t) = N_0 e^{\lambda t}, where λ\lambda is the growth rate (positive for growth, negative for decay), and the e-folding time is defined as τe=1/∣λ∣\tau_e = 1/|\lambda|, representing the duration over which the exponent changes by 1, thereby multiplying or dividing the by e≈2.718e \approx 2.718. This formulation arises naturally from the solution to the first-order linear differential equation dNdt=λN\frac{dN}{dt} = \lambda N, making τe\tau_e the reciprocal of the rate constant and providing a direct measure of the process's speed. In contrast to discrete compounding, where growth occurs in finite steps (e.g., annual interest), the e-folding time aligns seamlessly with continuous processes, as the continuous limit of discrete exponential models yields the base-ee form through the of the natural logarithm. For instance, repeated intervals approaching zero results in the effective growth factor eλte^{\lambda t}, underscoring why e-folding is intrinsic to models without artificial . After nn e-foldings, the quantity has changed by a factor of ene^n, offering a scalable way to quantify cumulative exponential effects over multiple s. This property facilitates tracking long-term behavior in systems where the rate remains constant. E-folding is favored in scientific modeling because it corresponds directly to logarithm's , streamlining analytical solutions and numerical integrations in differential equations by normalizing the exponent to unity per . For example, in a with λ=0.1\lambda = 0.1 per unit time, τe=10\tau_e = 10 units, resulting in a factor-of-ee change after that interval.

Mathematical Formulation

Exponential Growth

In exponential growth, a quantity N(t)N(t) increases according to the model N(t)=N0ertN(t) = N_0 e^{rt}, where N0N_0 is the initial value, tt is time, and r>0r > 0 is the continuous growth rate with units of inverse time. This formulation arises from the dNdt=rN\frac{dN}{dt} = r N, whose solution describes processes where the rate of change is proportional to the current size, such as in unconstrained population expansion. The e-folding time τe\tau_e is defined as the duration required for N(t)N(t) to increase by a factor of e≈2.718e \approx 2.718. To derive it, set N(τe)=eN0N(\tau_e) = e N_0, yielding eN0=N0erτee N_0 = N_0 e^{r \tau_e}, so e=erτee = e^{r \tau_e}. Taking the natural logarithm gives ln⁡(e)=rτe\ln(e) = r \tau_e, hence τe=1r\tau_e = \frac{1}{r}. This time scale provides a natural measure for growth rates in base-ee exponential models. Each e-folding interval multiplies the quantity by ee, allowing cumulative growth to be tracked as successive factors of ee; this is particularly useful in scenarios of unbounded exponential increase, where the focus is on relative rather than absolute changes. The total number of e-foldings over time tt is given by n=rt=tτen = r t = \frac{t}{\tau_e}, representing how many such multiplicative steps have occurred. For instance, in modeled continuously, a growth rate of r=0.05r = 0.05 per year yields τe≈20\tau_e \approx 20 years for the to e-fold.

Exponential Decay

In , a quantity decreases over time according to the model N(t)=N0e−t/τN(t) = N_0 e^{-t/\tau}, where N0N_0 is the initial value, tt is time, and τ>0\tau > 0 is the , which directly corresponds to the e-folding time τe=τ\tau_e = \tau. This formulation arises from the dN/dt=−(1/τ)NdN/dt = - (1/\tau) N, indicating that the rate of decrease is proportional to the current value of NN. The e-folding time τe\tau_e is derived by identifying the interval over which the quantity reduces by a factor of ee: substituting t=τet = \tau_e yields N(τe)=N0/eN(\tau_e) = N_0 / e, meaning the remaining is 1/e≈0.3681/e \approx 0.368 or 36.8%. After nn e-foldings, the remaining is e−ne^{-n}, providing a natural scale for quantifying the persistence of the decaying quantity. This e-folding time represents the mean lifetime in decay processes, where the probability of survival up to time tt follows e−t/τe^{-t/\tau}; thus, τ\tau is the expected time before decay occurs. In , for a reaction with rate constant kk, the e-folding time is τe=1/k\tau_e = 1/k, after which approximately 37% of the reactant remains undecayed. Unlike , which involves a positive exponent leading to multiplication by ee, the negative exponent here ensures division by ee per e-folding interval.

Applications

In Physical Sciences

In physical sciences, the e-folding time, denoted as τe\tau_e or simply τ\tau, represents the characteristic timescale over which exponentially decaying processes reduce a quantity to 1/e1/e (approximately 37%) of its initial value. This concept is central to modeling decay phenomena in , chemical reactions, and electrical systems, where the underlying dynamics follow kinetics governed by a rate constant. In , the e-folding time corresponds to the mean lifetime τ=1/λ\tau = 1/\lambda of an , where λ\lambda is the decay constant related to the by λ=ln⁡(2)/t1/2\lambda = \ln(2)/t_{1/2}. For , used in , the half-life is 5730 years, yielding a mean lifetime τ≈8267\tau \approx 8267 years calculated as t1/2/ln⁡(2)t_{1/2}/\ln(2). This timescale quantifies the average time an unstable nucleus persists before decaying, essential for predicting isotope abundance in nuclear reactions and . Chemical kinetics employs the e-folding time for first-order reactions, where the rate law is d[A]/dt=−k[A]d[A]/dt = -k[A] and τe=1/k\tau_e = 1/k, the time for reactant concentration to drop to 1/e1/e of its initial value. In , drug elimination often follows this model; after one e-folding time, approximately 37% of the drug remains in the body, influencing dosing intervals for medications like antibiotics. This framework also applies to unimolecular decompositions in gas-phase reactions, providing a measure of reaction progress independent of initial concentrations. In , the e-folding time manifests as the τ=RC\tau = RC in resistor- (RC) circuits, governing the exponential discharge of a through a . During discharge, the voltage across the falls to 1/e1/e of its initial value after time τ\tau, a principle used in timing circuits and filters; for instance, with R=1R = 1 kΩ\Omega and C=1C = 1 μ\muF, τ=1\tau = 1 ms, establishing the circuit's response speed. This behavior underscores the ubiquity of exponential transients in transient analysis of linear circuits. Nuclear physics provides a concrete example with the charged pion (π±\pi^\pm), whose mean lifetime τe≈2.6×10−8\tau_e \approx 2.6 \times 10^{-8} seconds reflects its decay primarily to muons and neutrinos, as measured in particle accelerators. This short e-folding time highlights the particle's instability, informing models of strong interaction dynamics in high-energy collisions. In atmospheric chemistry, aerosol lifetimes are often expressed as e-folding times, capturing removal processes like wet deposition and coagulation. For tropospheric particles, such as sulfate aerosols, typical lifetimes range from 1 to 2 weeks (e.g., τe≈14\tau_e \approx 14 days), influencing air quality and radiative forcing by determining how long pollutants persist before scavenging.

In Cosmology and Astrophysics

In cosmology, the concept of e-folding is pivotal to the theory of cosmic inflation, which posits a phase of rapid exponential expansion in the early universe. The number of e-folds, denoted NN, quantifies this expansion and is defined as N=ln⁡(afinalainitial),N = \ln\left(\frac{a_\mathrm{final}}{a_\mathrm{initial}}\right), where aa is the scale factor of the universe. This measure arises from the integration of the Hubble parameter over time, N=∫H dtN = \int H \, dt, with H=a˙/aH = \dot{a}/a. During inflation, driven by a scalar field (the inflaton), the universe undergoes quasi-exponential growth, and NN determines the extent to which initial irregularities are smoothed out. Each e-fold corresponds to the scale factor increasing by a factor of e≈2.718e \approx 2.718, effectively multiplying spatial distances by this amount and roughly doubling them in linear scale. To resolve key puzzles such as the (why distant regions appear homogeneous) and the (why the universe's density is near-critical), inflationary models typically require N≈50N \approx 50 to 6060 over the scales. In the limit of a de Sitter-like spacetime approximation during , where HH is constant, the e-folding time τe=1/H\tau_e = 1/H defines the characteristic timescale for one e-fold of expansion; at grand unified theory energy scales (∼1015\sim 10^{15}--101610^{16} GeV), this yields τe∼10−36\tau_e \sim 10^{-36} seconds. In (BBN), e-folding times of the universe's expansion play a crucial role in determining the freeze-out of nuclear reactions. Around T∼0.8T \sim 0.8 MeV (corresponding to ∼1\sim 1 second after the ), weak interactions maintaining neutron-proton equilibrium fall out of balance when their rates drop below the expansion rate HH. Here, the e-folding time 1/H1/H sets the timescale for this freeze-out, fixing the neutron-to-proton ratio at approximately 1:6 before subsequent decays and reactions produce light elements like helium-4. This sensitivity to expansion highlights how e-folding governs the brief window for primordial . In , e-folding also characterizes exponential phases in stellar explosions, notably the late-time light curves of core-collapse supernovae. For Type II supernovae, the post-peak decline often follows an powered by the of 56^{56}Co (produced from initial 56^{56}Ni), with the light curve tail exhibiting an e-folding time matching the isotope's decay constant of approximately 111 days. This alignment provides insights into the explosion energetics and nickel yields, typically 0.07--0.2 solar masses.

In Environmental and Atmospheric Science

In environmental and , the e-folding time serves as a key metric for quantifying the timescales of exponential accumulation and dispersion processes within the system, particularly in dynamics, pollutant transport, and hydrological cycles. It provides insight into how rapidly concentrations of gases, tracers, or burdens evolve under continuous sources or sinks, aiding in the prediction of environmental impacts and the design of mitigation strategies. In the context of , the e-folding time describes the duration required for atmospheric CO₂ concentrations to increase by a factor of (approximately 2.718) at a constant fractional growth rate r, calculated as τ_e = 1/r. As of , observations indicate an annual growth rate of about 0.7% (derived from an increase of ~3 ppm against a baseline of ~423 ppm), yielding an e-folding time of approximately 143 years; this timescale highlights the persistent buildup of CO₂ and its long-term influence on global warming. Historical average fractional growth rate of approximately 0.15% per year from pre-industrial levels of 280 ppm to 425 ppm over ~275 years (1750–2025), corresponding to an e-folding time of around 667 years, underscoring the accelerating nature of anthropogenic emissions. Atmospheric flushing processes, such as the washout of by or ventilation, are modeled using e-folding times to estimate removal rates in idealized models of the atmosphere or hydrological systems. The e-folding time is given by τ_e = V / Q, where V is the volume of the or and Q is the effective outflow rate (e.g., due to or rainfall scavenging); this represents the time for pollutant concentrations to decline to 1/e of their initial value following an emission pulse. For instance, in regional air quality assessments, this metric helps evaluate how quickly harmful substances like are cleared from the , typically on timescales of hours to days depending on meteorological conditions. Soil moisture memory, which influences land-atmosphere interactions and seasonal forecasting, is characterized by the e-folding time of its function in land surface models. This timescale, often ranging from a few days in arid regions to 2–4 weeks in humid or vegetated areas, reflects how long soil moisture anomalies persist before decaying exponentially due to , infiltration, and recharge processes. Studies using observations and model simulations confirm these short- to medium-term memories, emphasizing their role in modulating predictability and propagation. In circulation, the e-folding time for tracer mixing quantifies ventilation rates—the rate at which masses exchange with the surface and incorporate dissolved substances like nutrients or carbon. In the upper , these timescales are on the order of years, as inferred from chlorofluorocarbon (CFC) tracer distributions and global circulation models, providing critical constraints on the 's role in and heat uptake. For aerosol burdens, the global e-folding lifetime of tropospheric particles, formed from emissions, is approximately one week, as estimated from pulse-response models that simulate their wet and dry deposition. This short timescale governs the of aerosols, which scatter sunlight and cool the climate regionally, but limits their global persistence compared to longer-lived gases.

In Finance and Economics

In finance, the e-folding time finds application in continuous , where an investment's value grows exponentially according to the formula W(t)=W0ertW(t) = W_0 e^{rt}, with rr as the continuous return rate and tt as time. The e-folding time τe=1/r\tau_e = 1/r represents the period over which the investment multiplies by a factor of e≈2.718e \approx 2.718, providing a natural measure of growth pace in models assuming infinitely frequent . Logarithmic returns, or continuously compounded returns, directly relate to this timescale, as the annualized return equals r=1/τer = 1/\tau_e. For instance, an e-folding time of 10 years corresponds to a 10% continuous annual return, facilitating additive analysis of multi-period performance since log returns sum over time intervals. In economic growth models like the Solow framework, e-folding times approximate the transitional dynamics of output, which converges exponentially to its steady-state level at rate λ=(1−α)(n+g+δ)\lambda = (1 - \alpha)(n + g + \delta), where α\alpha is capital's output share, nn , gg growth, and δ\delta ; thus, τe=1/λ\tau_e = 1/\lambda quantifies the time for output to approach equilibrium by factor ee. As of the end of 2024, the S&P 500's historical real annual geometric return of approximately 7.07% from 1928 implies an -folding time of about 14.1 years for the index value to increase by a factor of ee, assuming continuous . This e-folding metric contrasts with the for , which estimates td≈72/(100r)t_d \approx 72 / (100r) years for discrete (deriving from ln⁡2/r≈0.693/r\ln 2 / r \approx 0.693 / r), whereas e-folding uses the exact 1/r1/r for base-ee growth, offering precision in continuous models without approximation.

Versus Doubling Time

The doubling time, denoted as τd\tau_d, is the duration required for an exponentially growing quantity to increase by a factor of 2.\) In the continuous exponential model \(N(t) = N_0 e^{\lambda t}, where λ>0\lambda > 0 is the growth rate, the doubling time relates to the e-folding time τe=1/λ\tau_e = 1/\lambda by τd=ln⁡(2)⋅τe≈0.693τe\tau_d = \ln(2) \cdot \tau_e \approx 0.693 \tau_e.() This relationship derives from setting N(τd)=2N0N(\tau_d) = 2 N_0, yielding eλτd=2e^{\lambda \tau_d} = 2, so λτd=ln⁡2\lambda \tau_d = \ln 2 and τd=ln⁡2/λ\tau_d = \ln 2 / \lambda.\) Rearranging gives \(\tau_e = \tau_d / \ln 2 \approx 1.443 \tau_d.() Doubling time is typically employed in scenarios where base-2 increments provide intuitive benchmarks, such as bacterial population growth or resource doubling in , whereas e-folding time facilitates analytical convenience in equations involving natural logarithms.\) For instance, if the e-folding time is 10 years, the doubling time is approximately 6.93 years.\( In , for continuous at an annual growth rate of r%r\%, the doubling time is estimated by the as approximately 72/r72 / r years, while the e-folding time approximates 100/r100 / r years.\) The [rule of 72](/page/Rule_of_72) provides a close practical [approximation](/page/Approximation) to the exact continuous doubling time \(\ln 2 / (r/100) \approx 69.3 / r years.()

Versus Half-Life

The half-life, denoted t1/2t_{1/2}, is the time required for a quantity undergoing exponential decay to reduce to half its initial value, corresponding to a multiplicative factor of 1/21/2. In relation to the e-folding time τe\tau_e, the half-life is given by t1/2=ln⁡(2) τe≈0.693 τet_{1/2} = \ln(2) \, \tau_e \approx 0.693 \, \tau_e. This relationship derives from the equation N(t)=N0e−t/τeN(t) = N_0 e^{-t / \tau_e}, where N(t)N(t) is the at time tt and N0N_0 is the . Setting N(t)=N0/2N(t) = N_0 / 2 at t=t1/2t = t_{1/2} yields N0/2=N0e−t1/2/τeN_0 / 2 = N_0 e^{-t_{1/2} / \tau_e}, which simplifies to 1/2=e−t1/2/τe1/2 = e^{-t_{1/2} / \tau_e}. Taking the natural logarithm of both sides gives ln⁡(1/2)=−t1/2/τe\ln(1/2) = -t_{1/2} / \tau_e, or −ln⁡(2)=−t1/2/τe-\ln(2) = -t_{1/2} / \tau_e, so t1/2=ln⁡(2) τet_{1/2} = \ln(2) \, \tau_e. The half-life is often preferred for intuitive reporting in contexts like radioactivity, where it provides a straightforward measure of decay progress—after one half-life, half remains; after two, one-quarter remains—facilitating practical predictions without delving into the underlying exponential model. In contrast, the e-folding time τe\tau_e is more commonly used in theoretical modeling, as it directly aligns with the decay constant in the exponential formulation. For example, if the e-folding time (mean life) of a radioactive isotope is τe=10\tau_e = 10 days, the half-life is approximately t1/2≈6.93t_{1/2} \approx 6.93 days. In , the is a standard metric for describing drug elimination in , allowing clinicians to estimate dosing intervals based on how long it takes for plasma concentrations to halve. However, the e-folding time underlies the core pharmacokinetic equations, as drug concentrations follow the same form C(t)=C0e−t/τeC(t) = C_0 e^{-t / \tau_e}, where τe\tau_e relates to clearance and .

References

  1. https://e.math.cornell.edu/people/belk/writing/LinearODEs2.pdf
  2. https://atmos.uw.edu/~dennis/211_Exponential_Growth.pdf
  3. https://acp.copernicus.org/articles/14/4313/2014/acp-14-4313-2014.pdf
  4. https://www.mdpi.com/2073-4441/12/10/2717
  5. https://journals.ametsoc.org/view/journals/hydr/20/6/jhm-d-18-0141_1.pdf
  6. https://projects.iq.harvard.edu/files/acmg/files/intro_atmo_chem_bookchap3.pdf
  7. https://hplgit.github.io/scaling-book/doc/pub/book/html/sphinx-cbc/._book003.html
  8. https://mathresearch.utsa.edu/wiki/index.php?title=Euler%27s_Number
  9. https://www.scirp.org/journal/paperinformation?paperid=120075
  10. https://mathresearch.utsa.edu/wiki/index.php?title=Exponential_Growth_and_Decay
  11. https://home.csulb.edu/~acarter3/PDFs/carter-ABT1999.pdf
  12. http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/meanlif.html
  13. https://faculty.gvsu.edu/mcbaneg/chm358wi02.pdf
  14. https://www.britannica.com/science/carbon-14
  15. https://gml.noaa.gov/education/isotopes/decay.html
  16. https://www.sciencesnail.com/uploads/1/9/9/3/19934321/first_order_drug_elimination_kinetics.pdf
  17. https://openstax.org/books/university-physics-volume-2/pages/10-6-rc-circuits
  18. https://pdg.lbl.gov/2024/listings/rpp2024-list-pi-plus-minus.pdf
  19. https://www.mdpi.com/2218-1997/10/7/284
  20. https://pdg.lbl.gov/2024/reviews/rpp2024-rev-inflation.pdf
  21. https://background.uchicago.edu/~whu/Courses/Ast408_19/ast321_1.pdf
  22. https://link.aps.org/doi/10.1103/RevModPhys.88.015004
  23. https://iopscience.iop.org/article/10.1086/523258/fulltext/
  24. https://www.carbonbrief.org/analysis-fossil-fuel-co2-emissions-to-set-new-record-in-2025-as-land-sink-recovers/
  25. https://nepis.epa.gov/Exe/ZyPURL.cgi?Dockey=20015LPD.TXT
  26. https://www.ipcc.ch/site/assets/uploads/2018/02/WG1AR5_Chapter08_FINAL.pdf
  27. https://corporatefinanceinstitute.com/resources/wealth-management/continuously-compounded-return/
  28. https://www.investopedia.com/terms/c/continuouscompounding.asp
  29. https://365financialanalyst.com/knowledge-hub/corporate-finance/log-return/
  30. https://pages.stern.nyu.edu/~adamodar/New_Home_Page/datafile/histretSP.html
  31. https://www.investopedia.com/ask/answers/051815/what-difference-between-rule-70-and-rule-72.asp
  32. https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Nuclear_Chemistry/Nuclear_Kinetics/Radioactive_Decay_Rates
  33. http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/halfli2.html
  34. http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/halfli.html
  35. https://www.cambridge.org/core/books/physics-pharmacology-and-physiology-for-anaesthetists/time-constant-and-half-life/95D329C5F710700E64D8809182E9F89D
  36. https://www.merckvetmanual.com/pharmacology/pharmacology-introduction/pharmacokinetics
Add your contribution
Related Hubs
User Avatar
No comments yet.