(a) The daughter nucleus formed in radioactive decay is often radioactive. Let N10 represent the number of parent nuclei at time t = 0, N1(t) the number of parent nuclei at time t, and λ1 the decay constant of the parent. Suppose the number of daughter nuclei at time t = 0 is zero. Let N2(t)be the number of daughter nuclei at time t and let λ2 be the decay constant of the daughter. Show that N2(t) satisfies the differential equation (dN2)/(dt) = λ1N1 - λ2N2(b) Verify by substitution that this differential equation has the solution N2(t) = (N10λ1)/(λ1 - λ2)(e-λ2t - e-λ1t)This equation is the law of successive radioactive decays. (c) 218Po decays into 214Pb with a half-life of 3.10 min, and 214Pb decays into 214Bi with a half-life of 26.8 min. On the same axes, plot graphs of N1(t) for 218Po and N2(t) for 214Pb. Let N10 = 1 000 nuclei and choose values of t from 0 to36 min in 2-min intervals. (d) The curve for 214Pb obtained in part (c) at first rises to a maximum and then starts to decay. At what instant tm is the number of 214Pb nuclei a maximum? (e) By applying the condition for a maximum dN2/dt = 0, derive a symbolic equation for tm in terms of λ1and λ2. (f) Explain whether the value obtained in part (c) agrees with this equation.
Radioactive decay
The emission of energy to produce ionizing radiation is known as radioactive decay. Alpha, beta particles, and gamma rays are examples of ionizing radiation that could be released. Radioactive decay happens in radionuclides, which are imbalanced atoms. This periodic table's elements come in a variety of shapes and sizes. Several of these kinds are stable like nitrogen-14, hydrogen-2, and potassium-40, whereas others are not like uranium-238. In nature, one of the most stable phases of an element is usually the most prevalent. Every element, meanwhile, has an unstable state. Unstable variants are radioactive and release ionizing radiation. Certain elements, including uranium, have no stable forms and are constantly radioactive. Radionuclides are elements that release ionizing radiation.
Artificial Radioactivity
The radioactivity can be simply referred to as particle emission from nuclei due to the nuclear instability. There are different types of radiation such as alpha, beta and gamma radiation. Along with these there are different types of decay as well.
(a) The daughter nucleus formed in radioactive decay is often radioactive. Let N10 represent the number of parent nuclei at time t = 0, N1(t) the number of parent nuclei at time t, and λ1 the decay constant of the parent. Suppose the number of daughter nuclei at time t = 0 is zero. Let N2(t)
be the number of daughter nuclei at time t and let λ2 be the decay constant of the daughter. Show that N2(t) satisfies the differential equation
(dN2)/(dt) = λ1N1 - λ2N2
(b) Verify by substitution that this differential equation has the solution
N2(t) = (N10λ1)/(λ1 - λ2)(e-λ2t - e-λ1t)
This equation is the law of successive radioactive decays. (c) 218Po decays into 214Pb with a half-life of 3.10 min, and 214Pb decays into 214Bi with a half-life of 26.8 min. On the same axes, plot graphs of N1(t) for 218Po and N2(t) for 214Pb. Let N10 = 1 000 nuclei and choose values of t from 0 to
36 min in 2-min intervals. (d) The curve for 214Pb obtained in part (c) at first rises to a maximum and then starts to decay. At what instant tm is the number of 214Pb nuclei a maximum? (e) By applying the condition for a maximum dN2/dt = 0, derive a symbolic equation for tm in terms of λ1
and λ2. (f) Explain whether the value obtained in part (c) agrees with this equation.
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 14 images
