Question

A radioactive decay series that begins with ${}_{90}^{232}$ Th ends with formation of the stable nuclide ${}_{82}^{208}\mathrm{Pb}$. How many alphaparticle emissions and how many beta-particle emissions are involved in the sequence of radioactive decays?

Short answer

In the radioactive decay series of Thorium-232 (${}_{90}^{232}$Th) to Lead-208 (${}_{82}^{208}$Pb), there are 6 alpha emissions and 4 beta emissions involved.

Step-by-step solution

Count the decrease in atomic number and mass number

First, we need to determine the change in atomic number and atomic weight between the initial and final nuclides. Change in atomic number (Z): 90 (Thorium) - 82 (Lead) = 8 Change in atomic weight (A): 232 (Thorium) - 208 (Lead) = 24

Calculate the number of alpha emissions

Alpha emission decreases the mass number by 4 and the atomic number by 2. Since the decrease in mass number is 24, we can calculate the number of alpha emissions by dividing the change in mass number by 4: Number of alpha emissions = $\frac{24}{4}=6$ Thus, there are 6 alpha emissions involved in this radioactive decay series.

Calculate the number of beta emissions

After 6 alpha emissions, the atomic number will have decreased by 12 (6 emissions * 2 protons lost per emission). To calculate the number of beta (-) emissions, we have to determine how many more protons are needed to reach the atomic number 82. Change in atomic number after alpha emissions: 90 (Thorium) - 12 = 78 Now, we need to see how many beta emissions (increase in atomic number by 1 per emission) are needed to reach 82: Number of beta emissions = 82 (Lead) - 78 = 4 Thus, there are 4 beta emissions involved in this radioactive decay series. In conclusion, the radioactive decay series involves 6 alpha emissions and 4 beta emissions to transform Thorium-232 into Lead-208.

Key concepts

Alpha Particle Emissions

Alpha particle emissions play a crucial role in the radioactive decay process of certain elements. An alpha particle is composed of two protons and two neutrons, identical to a helium nucleus. During alpha decay, an unstable nucleus emits an alpha particle, thus reducing the mass number ($A$) of the original atom by 4 units, and the atomic number ($Z$), which represents the number of protons in the nucleus, by 2 units.

For example, in our exercise, each alpha decay event transforms the nucleus by decreasing both the atomic and mass numbers continuously until stability is achieved. As alpha particles are hefty and carry a double positive charge, their emission significantly changes the characteristics of the radioactive material, often resulting in a new element with more stable properties.

Beta Particle Emissions

While alpha emissions reduce the number of protons in a nucleus, beta particle emissions have an entirely different effect. A beta particle is either an electron or positron that is emitted from a nucleus during radioactive decay. This process occurs when a neutron in the nucleus transforms into a proton and an electron (\beta^{-} decay).The emitted electron, called a beta particle, carries away excess energy. The key aspect to remember is that beta decay increases the atomic number ($Z$) of an element by 1 because of the additional proton, while the mass number ($A$) remains unchanged. In our textbook example, beta particle emissions are necessary after alpha emissions to increase the atomic number, thus transforming the intermediate element into its stable form, which in this case is lead.

Atomic Number

The atomic number, represented as ($Z$), is a fundamental characteristic that defines an element. It is the number of protons present in the nucleus of an atom. The atomic number is critically important in determining the identity of an element and its position in the periodic table. When an alpha or beta particle is emitted, the atomic number will change, leading to the transformation of the original element into a different element.

An increase in the atomic number indicates the element is moving up the periodic table (transmutation into an element with more protons), while a decrease suggests a movement down the periodic table (an element with fewer protons). Our exercise demonstrates how the atomic number changes during radioactive decay, with the atomic number of thorium decreasing from 90 to 82 to become lead.

Mass Number

The mass number ($A$) is another important characteristic of an atom. It is the total number of protons and neutrons in an atom's nucleus. Unlike the atomic number, the mass number is not conserved in radioactive decay involving alpha or beta emissions. Instead, it changes depending on the type of particle emitted.

In the case of alpha decay, the mass number decreases by 4 because the emitted alpha particle consists of 2 protons and 2 neutrons. However, for beta decay, the mass number remains unchanged as a neutron is converted to a proton with the emission of a beta particle (electron), which has a negligible mass. In the exercise, the mass number plays a pivotal role in determining the number of alpha particle emissions by examining the difference in mass numbers between the starting and the ending isotopes, showing a clear reduction from 232 to 208.