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The half-life of radium is \(1.62 \times 10^{3}\) years. How many radium atoms decay in \(1.00 \mathrm{~s}\) in a \(1.00 \mathrm{~g}\) sample of radium? The atomic weight of radium is \(226 \mathrm{~kg} / \mathrm{kmol}\). A \(1.00-\mathrm{g}\) sample is \(0.00100 \mathrm{~kg}\), which for radium of atomic number 226 is \((0.00100 / 226) \mathrm{kmol}\). Since each kilomole contains \(6.02 \times 10^{26}\) atoms, $$ N=\left(\frac{0.00100}{226} \mathrm{kmol}\right)\left(6.02 \times 10^{26} \frac{\text { atoms }}{\mathrm{kmol}}\right)=2.66 \times 10^{21} \text { atoms } $$ The decay constant is $$ \lambda=\frac{0.693}{t_{1 / 2}}=\frac{0.693}{(1620 \mathrm{y})\left(3.156 \times 10^{7} \mathrm{~s} / \mathrm{y}\right)}=1.36 \times 10^{-11} \mathrm{~s}^{-1} $$ Then \(\quad \frac{\Delta N}{\Delta t}=\lambda N=\left(1.36 \times 10^{-11} \mathrm{~s}^{-1}\right)\left(2.66 \times 10^{21}\right)=3.61 \times 10^{10} \mathrm{~s}^{-1}\) is the number of disintegrations per second in \(1.00 \mathrm{~g}\) of radium. The above result leads to the definition of the curie (Ci) as a unit of activity: $$ 1 \mathrm{Ci}=3.7 \times 10^{10} \text { disintegrations } / \mathrm{s} $$ Because of its convenient size, we shall sometimes use the curie in subsequent problems, even though the official SI unit of activity is the becquerel.

Step by step solution

01

Determine the Number of Radium Atoms in the Sample

Convert the mass of the radium sample from grams to kilograms: the sample is 1.00 g, which is 0.00100 kg. Calculate how many kilomoles this is by dividing by the atomic weight of radium, 226 kg/kmol: \[\text{Number of kilomoles} = \frac{0.00100 \, \text{kg}}{226 \, \text{kg/kmol}} = 4.42 \times 10^{-6} \, \text{kmol}\]Multiply by Avogadro's number to find the total number of atoms:\[N = \left(4.42 \times 10^{-6} \, \text{kmol}\right) \times \left(6.02 \times 10^{26} \, \text{atoms/kmol}\right) = 2.66 \times 10^{21} \, \text{atoms}\]

02

Calculate the Decay Constant

Use the half-life formula for exponential decay to calculate the decay constant (\(\lambda\)):\[\lambda = \frac{0.693}{t_{1/2}} = \frac{0.693}{1620 \, \text{years}} = \frac{0.693}{1620 \, \text{years} \times 3.156 \times 10^7 \, \text{s/year}} = 1.36 \times 10^{-11} \, \text{s}^{-1}\]This decay constant is used to calculate the number of decays per second.

03

Determine How Many Atoms Decay per Second

Calculate the rate of decay (\(\Delta N/\Delta t\)) by multiplying the decay constant by the number of atoms:\[\frac{\Delta N}{\Delta t} = \lambda N = \left(1.36 \times 10^{-11} \, \text{s}^{-1}\right) \times \left(2.66 \times 10^{21}\right) = 3.61 \times 10^{10} \, \text{decays/second}\]This value represents the number of radium atoms decaying per second in the given sample.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Half-life

Half-life is a crucial concept in nuclear physics when studying radioactive decay. It represents the time required for half of a sample of radioactive material to decay, resulting in the transformation of unstable isotopes into stable ones or into different isotopes. Understanding half-life is essential for calculating how much of a substance remains after a certain period.
For example, the half-life of radium is 1620 years, meaning after 1620 years, half of any radium sample will have decayed. Calculating the amount left using half-life helps scientists understand the stability and longevity of elements.
In calculations, the formula used is:\[\text{Half-life} (t_{1/2}) = \frac{0.693}{\lambda}\] where \(\lambda\) is the decay constant.

Decay Constant

The decay constant, represented by the Greek letter \(\lambda\), is a fundamental factor in understanding how quickly a radioactive isotope will decay. It represents the probability per unit time that a single atom will decay. In other words, it serves as a measure of how fast an isotope decays.
The formula for the decay constant is derived from the half-life equation:\[\lambda = \frac{0.693}{t_{1/2}}\] where \(t_{1/2}\) is the half-life of the radioactive substance.
A higher decay constant means the substance decays more rapidly, while a lower value indicates a slower rate of decay. This concept is pivotal in calculating the rate of decay and understanding the longevity of isotopes like radium.

Avogadro's Number

Avogadro's number is one of the cornerstones in chemistry and physics, providing the bridge between the atomic scale and the world we can measure. It defines the number of particles (atoms, molecules, or ions) in one mole of substance, typically:

• Avogadro's number is \(6.02 \times 10^{23} \) particles/mole.

In the context of the exercise, Avogadro's number allows us to determine the number of radium atoms in 1 gram of the element.
Radium has an atomic weight of 226 g/mol, and when split into kilometers and multiplied by Avogadro's number, it reveals how many atoms are present in a given mass. Understanding Avogadro's number is indispensable for demonstrating the scale of atomic mass to observable mass quantities in laboratory settings.

Atomic Weight

Atomic weight is a fundamental concept essential for understanding the mass of an atom relative to one twelfth of the mass of a carbon-12 atom. It is typically measured in daltons or unified atomic mass units (u), but for calculations involving larger samples, such as kilometers, is used.
For radium in the example provided, its atomic weight is 226 kg/kmol.
With this knowledge, one can convert between the mass in grams or kilograms and the amount of substance in moles or kilomole. This conversion allows calculations of the exact number of atoms in a given mass, essential in figuring out decay rates and other isotope properties.

Curie Unit

The curie (Ci) is a unit of radioactivity used to describe the intensity of radioactive decay. It is defined based on the rate at which a sample of radioactive material decays. Specifically:

• 1 Ci equals \(3.7 \times 10^{10}\) disintegrations per second.

This unit is not part of the International System of Units (SI) because the SI unit for activity is the becquerel (Bq), where 1 Bq equals one disintegration per second. However, the curie is still commonly used in certain applications due to its practical size.
In the described exercise, the disintegration rate of radium in the curie unit serves to illustrate how rapidly the radioactive substance decays, providing crucial insight into its stability and behavior.