The isotope 79^198 Au (atomic mass =197.968 u ) of gold has...

The isotope ${ }_{79}^{198} \mathrm{Au}$ (atomic mass $=197.968 \mathrm{u}$ ) of gold has a half-life of $2.69$ days and is used in cancer therapy. What mass (in grams) of this isotope is required to produce an activity of 315 Ci?

Key Concepts

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This spontaneous process occurs at a rate characterized by statistical probability, and it forms the basis for calculating the number of atomic disintegrations per unit time—an essential concept when linking the amount of a radioactive substance to its activity.

Half-life

The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. It is a fundamental parameter that governs the decay process and is critical for determining the decay constant, which in turn is used to calculate the activity of the substance over time.

Decay Constant

The decay constant, often represented by lambda (?), is a measure of the probability per unit time that a single nucleus will decay. It is mathematically related to the half-life of the substance via the relation ? = ln(2) / (half-life), providing a bridge between the measurable half-life and the rate at which decays occur.

Nuclear Activity and Its Units

Nuclear activity is defined as the rate at which a radioactive sample undergoes decay and is measured in terms of disintegrations per second. In practical applications, activity is often expressed in Curies (Ci), where one Curie corresponds to 3.7 x 10^10 disintegrations per second. This unit conversion is essential for relating laboratory measurements to standardized units used in medical and industrial applications.

Molar Mass and Avogadro's Number

Molar mass and Avogadro's number are used to convert between the macroscopic mass of a substance and the microscopic number of atoms present. By knowing the atomic mass in grams per mole and using Avogadro's constant (approximately 6.022 x 10^23 atoms/mole), one can calculate the number of atoms corresponding to a specific mass, thus linking the sample’s mass to its activity through the number of radioactive atoms present.

Transcript

00:01 Hello, and in this question here, we want to determine the amount of mass we require of the isotope of gold 198.

00:10 That produces an activity of 115 curie.

00:16 Okay? so, first of all, one curie, so one curie is equal to 3 .7 times 10 to the 10 backer -rel.

00:26 And backrail is the si units of activity, so we're just going to work in si units.

00:31 Whole time.

00:33 So this implies 315 curie is equal to 100 and 1 .1655 times 10 to the 13 akerell.

00:46 Okay.

00:47 So we want to determine how much mass of gold we need in order to achieve this activity.

00:53 Okay.

00:54 So well the activity, the activity is defined as the decay constant times the number of nuclei.

01:01 And the decay constant is equal to the natural log of 2 divided by the half -life.

01:08 Okay? so if we return back up here, the half -life is equal to 2 .69 days.

01:14 And we convert this to seconds by multiplying by how many seconds there in a day.

01:19 So that's 86 ,000.

01:21 So just go back it down here.

01:23 One day is equal to 86 ,400 seconds.

01:29 Means the half life in seconds is equal to 232 ,416 seconds.

01:38 So, oops, so this means we can sub in for lambda and this gives a decay constant of 2 .982...

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