I just encountered a student who said that according to his A-level definition, the radioactive decay constant is defined as the probability of a radioactive nuclide decaying per unit time. I dont know which idiot devised this erroneous definition. If the decay constant, in s^-1, is greater than 1, does this imply that the nuclide has more than a 100% of decaying in one second? Nonsense. An isotope's decay constant value is intrinsic. It is related to probability by this formula: \(P(s)=2^{\frac{-\lambda \Delta t}{ln2}}\) where P(s) is the probabillity of a certain nuclide surviving after a given time period, λ is the decay constant, and Δt is the length of the time period. The probability of the nuclide decaying is simply 1-P(s).
The decay constant can be greater than one. \( N_f = N_o e^{-\lambda t}\) So lets assume that the starting amount is one mole, lambda is 2/d and t is 4 days. The amount of atoms after 4 days would be 2.0 X 10^20.
Yes it's wrong. The probability changes over time but your student's definition is literally a constant. Here's the best definition I've found: From this you could come up with the probability of a specific radionuclide decaying over a specific time period but you'd have to have the number of starting atoms.
yeah the formula is based on a binomial expectation. but i can't believe that exam setters want students to write out the orange text.
If \(N(t) = N_{\tiny 0} \; e^{-\lambda t}\) then the instantaneous rate of number of atoms decaying per unit time is \(-\frac{d N}{dt} = \lambda N_{\tiny 0} \; e^{- \lambda t} = \lambda N(t)\) so the instantaneous rate of proportional decay per unit time is \( - \frac{1}{N(t)} \frac{d N(t)}{dt} = - \frac{d \, \ln N(t)}{dt} = \lambda\) which is a macroscopic law equivalent to the microscopic statement that independently, for each undecayed atom, \(\lambda\) is the instantaneous probability density of decay per unit time. The actual probability of a particular atom's decay over a finite time T is the solution to the differential equation, \(P'(T) = \left( 1 - P(T) \right) \lambda, \; P(0) = 0\) with solution \(P(T) = 1 - e^{-\lambda T}\).
Just an observation, but the "instantaneous rate of proportional decay per unit time" threw me for a loop because it can legitimately be greater than one (it simply won't do so for the duration of the unit time).
The "rate" isn't the problem, it's "proportional rate" that gave me pause. A proportion greater than one implies that material would decay a number of atoms exceeding its content per unit time; the reason this is permissible is because its an instantaneous rate which changes in absolute terms as soon as any decay has occurred.
Lambda is a rate constant for a given isotope, it's only units are \(s^{-1}\). When you plug lambda into that equation you will know the number of isotope atoms after a certain amount of time based on the rate at which the atoms decay, it does not mater if you have 1000 atoms or \( 10^{23}\) atoms
The decay constant is related to the half life by: \(\lambda = \ln 2/\tau_{1/2}\) The half life is the average time taken for half of something to decay, so the decay constant is a kind of average of number of decays per unit time.
: "The fact that the radioactive decay follows the exponential law implies that this phenomenon is statistical in nature. Every nucleus in a sample of the radionuclide has a certain probability of decaying, but there is no way to know in advance which nuclei will actually decay in a particular time span. If the sample is large enough-that is, if many nuclei are present-the actual fraction of it that decays in a certain time span will be very close to the probability for many individual nucleus to decay" (Concepts of Modern Physics, Arthur Beiser2010 McGraw Hill) : decay constant can be interpreted as the probability per unit time that any individual nucleus will decay" ( 2nd para, page 1486Sears& Zemanasky's University Physics, HughD. Young, Pearson 2008 : ...,decay constant is the probability of decay per nucleus per second. (page 1452, Physics for Scientists and Engineers, Serway, 2000) "What determines when an unstable nucleus decays? Radioactive decay is a quantum-mechanical process that can only be described in terms of probability. Given a collection of identical nuclides, they do not decay at the same time, and there is no way to predict which one decays when. The decay probability for one nucleus is independent of its past history and of the other nuclei. Each radioactive nuclide has a certain decay probability per unit tim. The decay probability per unit time is also called decay constant. Since probability is a pure number the unit is per.second. (page 1063, Fundemntals of Physics,Alan & Betty, McGraw Hill 2008)
