(II) A sample of (238)/(92) U is decaying at a rate of 3.70 ×10^2 decays/s. What is the mass of

(II) A sample of (238)/(92) U is decaying at a rate of 3.70...

Key Concepts

Exponential Decay Law in Quantitative Analysis

This law, given as N(t) = N?e^(-?t), links the initial number of nuclei to the quantity remaining at any point in time. In quantitative analysis, it allows for the calculation of either the remaining number of radioactive nuclei or the initial quantity based on measurements of decay rate and known decay constants.

Decay Constant

The decay constant is a parameter that characterizes the rate of radioactive decay of a particular isotope. It represents the probability per unit time that a single nucleus will decay and is related to the half-life of the substance by the equation ? = ln(2)/T?/?.

Avogadro's Number and Molar Mass

Avogadro's Number is the number of atoms or molecules in one mole of a substance, enabling conversion between the number of atoms and the amount of substance in moles. Molar mass is used to convert between the number of moles and the actual mass of a sample. These concepts are essential for determining the mass of a radioactive sample once the number of radioactive atoms is known.

Radioactive Decay

Radioactive decay describes the process by which unstable atomic nuclei lose energy by emitting radiation. This decay happens at a rate that is statistically predictable and is generally modeled by an exponential decay law, where the number of atoms decaying per unit time is proportional to the number of undecayed nuclei present.

Activity

Activity is a measure of the number of decays per unit time in a radioactive sample, typically expressed in decays per second (Becquerels) or in curies. It provides a direct quantification of the disintegration rate of the radioactive material.

Transcript

00:01 Okay, so for this problem, we're given the decay rate of a sample of uranium, and we want to know the mass of the sample.

00:09 So this is very similar to the previous problems.

00:13 So we're given, let's go ahead and start this out by writing down the given.

00:19 So r is 3 .70 times 10 squared decays for a second.

00:27 And the molar mass of uranium, which i had to look up, was 238 grams per mole.

00:45 And the half -life also looked up.

00:49 So it was 4 .5 billion years, which is 1 .4 times 10 to the 17 seconds.

01:00 And yeah so basically what we want to do is find the number actually yeah put these in a box is our little kind of ingredient prep area now i'll start the real math so we know that n not is the rate divided by lambda just based on the fact that r is to find us a derivative of n at with respect to t at time equals zero and n is this n n n n n n n n n n gives you this equation which i quickly went through because i've seen it in other situations um so it's it's quite frequently used and yeah so then mass is going to be this n knot um times the molar mass so the grams per mole and then divided by avagadro's number.

01:57 I'm not sure if there's a common symbol.

01:58 I read that it was like n -not, but we're already using that for the number.

02:02 So i'm just going to say a -v -number for avagadro's number.

02:07 So that's particles per mole, n is particles, and then m is grams per mole...

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