You want to find out how many atoms of the isotope 65 Cu are in a small sample of material. You

You want to find out how many atoms of the isotope 65 Cu are...

You want to find out how many atoms of the isotope 65 $\mathrm{Cu}$ are in a small sample of material. You bombard the sample with neutrons to ensure that on the order of 1$\%$ of these copper nuclei absorb a neutron. After activation, you turn off the neutron flux and then use a highly efficient detector to monitor the gamma radiation that comes out of the sample. Assume half of the 66 $\mathrm{Cu}$ nuclei emit a 1.04 -MeV gamma ray in their decay. (The other half of the activated nuclei decay directly to the ground state of $66 \mathrm{Ni} . )$ If after 10 $\mathrm{min}$ (two half-lives) you have detected $1.00 \times 10^{4} \mathrm{MeV}$ gamma ray in their decay. (The other half of the activated nuclei decay directly to the ground state of 66 $\mathrm{Ni}$ . If after 10 min (two half-lives) you have detected $1.00 \times 10^{4} \mathrm{MeV}$ of photon energy at $1.04 \mathrm{MeV},(\mathrm{a})$ approximately how many 65 $\mathrm{Cu}$ atoms are in the sample? (b) Assume the sample contains natural copper. Refer to the isotopic abundances listed in Table 44.2 and estimate the total mass of copper in the sample.

Key Concepts

Atom-to-Mass Conversion

Once the number of atoms is determined, converting that number into mass involves using the atomic mass of the element and Avogadro's number, which relates the number of atoms to the amount of substance in moles. This conversion is fundamental in chemistry and physics, allowing quantitative comparisons between measured atomic-level events and macroscopic quantities like mass.

Isotopic Abundance

Isotopic abundance refers to the relative amount of each isotope present in a naturally occurring element. When a sample consists of multiple isotopes, knowing their natural abundances allows for conversion between measurements made on a specific isotope (for example, one that was activated) and the total mass or number of atoms of the element in the sample. This concept is essential in determining the overall sample mass from the measured subset of radioactive decays.

Gamma-ray Detection and Energy Conversion

Gamma-ray spectroscopy involves detecting gamma photons emitted during nuclear decay and measuring their energies. By knowing the energy per photon and the total measured energy, one can estimate the number of decay events. This conversion from energy detected to event count, accounting for detector efficiency and branching ratios, is crucial for back-calculating the initial number of atoms present in the sample.

Radioactive Decay and Half-Life

Radioactive decay is a random process where unstable nuclei transform into more stable configurations by emitting radiation. The half-life of a radioactive isotope is the time it takes for half of the nuclei in a sample to decay. This concept enables the determination of the remaining quantity of radioactive nuclei over a given time period, and is used in the problem to relate observed radiation after a set time to the original number of activated atoms.

Radioactive Activation Analysis

This concept involves converting stable nuclei into radioactive ones by bombarding them with particles (such as neutrons). In the context of the problem, a small fraction of the nuclei absorb a neutron to become radioactive, allowing measurements of their decay to inform on the original number of target atoms. It is a powerful analytical technique for determining elemental or isotopic concentrations in a sample.

Branching Ratio in Nuclear Decay

In many nuclear decays, an unstable nucleus can decay through more than one pathway, with a specific probability assigned to each. The branching ratio specifies the fraction of decays that result in a particular emission (such as a gamma ray). In the problem, only a fraction of decays produce the gamma rays measured, and knowing this ratio is essential to accurately connect the detected radiation to the number of radioactive decays, and hence to the number of atoms initially present.

Transcript

00:01 So to start this question of, let us write on what are the different processes that actually occur.

00:08 So first off you have some amounts of copper 65 within the usual copper sample.

00:21 1 % of this is activated by bombarding with neutrons.

00:30 And this gives us copper 66.

00:35 Copper 66 will decay via two different different.

00:41 Pathways right one goes directly to the ground states of nickel and the other half actually emits a gamma ray in the decay and the gamma ray has an energy of 4 .0 4m ev and you detect right over two half -lives that the total gamma energy is given as shown.

01:25 So this is the total amount of gamma energy that was detected in two half -lives.

01:30 And we want to work backwards to find out what is the amount of copper 65 that we initially have.

01:40 So first off you will need to find the number of gamma, number of gamma rays that were actually detected.

01:53 We divide the total energy by the energy of each gamma ray.

01:56 Get 9 .62 times that about 3 to find out how much was actually decayed.

02:18 Right? how much decay that occurs in this process? number of decay would be equals to number of copper 66 multiplied by 1 minus half to the power of 2.

02:41 Two half -life has occurred.

02:45 So two half -lives will be multiplying the original amounts of copper 66 by half twice, right? we've got to multiply half by twice because two half -lives...

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