The common isotope of uranium, ^238 U, has a half-life of 4.47 ×10^9 years, decaying to ^234 Th

The common isotope of uranium, ^238 U, has a half-life of...

The common isotope of uranium, $^{238} \mathrm{U},$ has a half-life of $4.47 \times 10^{9}$ years, decaying to $^{234} \mathrm{Th}$ by alpha emission. (a) What is the decay constant? (b) What mass of uranium is required for an activity of 1.00 curie? (c) How many alpha particles are emitted per second by 10.0 g of uranium?

Key Concepts

Alpha Emission

Alpha emission is a type of radioactive decay in which an unstable nucleus releases an alpha particle, composed of two protons and two neutrons. This process not only decreases the mass number but also transforms the element into a different isotope or even a different element, depending on the nuclear reaction.

Radioactive Decay

Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable configuration by emitting particles or energy. This process is inherently probabilistic and is described by exponential decay laws, forming the basis for understanding how unstable isotopes evolve over time.

Exponential Decay Law

The exponential decay law mathematically models the decrease in the number of undecayed nuclei over time. It expresses that the rate of decay is proportional to the number of remaining nuclei, using the decay constant to set the pace of the decay process.

Half-life

Half-life is the period required for half of a sample of a radioactive substance to decay. It is intrinsic to each radionuclide and is vital for determining the decay constant, as well as for predicting the behavior and longevity of the radioactive material.

Decay Constant

The decay constant is a parameter that quantifies the probability of a single nucleus decaying per unit time. It is directly related to the half-life by the relation ? = ln(2) / (half-life), serving as a fundamental element in calculations involving radioactive decay.

Activity

Activity refers to the number of nuclear decays occurring per unit time in a radioactive sample. It is an essential concept in nuclear science that links the microscopic decay events to a measurable macroscopic quantity, often expressed in units like becquerels or curies.

Curie

The curie is a unit of radioactivity defined as 3.7 x 10^10 decays per second. It provides a scale for expressing the activity of a radioactive substance and is particularly useful in situations where large amounts of radioactivity are involved.

Transcript

00:01 So in this question, we have a common isotope of peribrand experiencing a decay into this element of 234 thm i'm actually not sure how to pronounce this, but it's a different element and it is it decays by an unframed emission because we know we can see that it's atomic, it's a mass number decrease by four.

00:26 So in part a, when given it's hard five and now defines the decay constant.

00:32 So the decay the relationship of half -life and decay constant is decay constant, is going to two divided by some half -life.

00:41 So we all we need to do is put in all the numbers.

00:48 The half -life is years, but we need to convert it to seconds.

00:57 Years times years per second, i think.

01:04 It's also in the index of how many years is the second.

01:13 So 3 .146 times 10 to the seventh, second is a year.

01:24 Okay, and this gives us around 4 .92 times 10 to the negative 18.

01:35 Second in first.

01:37 Okay, so that is d .k .c.

01:39 Lepta.

01:40 What mass of uranium is required? when activity of 1 query.