Question

A particular nuclide is found to have lost 3 neutrons and 1 proton after a decay chain. What combination of \(\alpha\) and \(\beta\) decays could account for this result?

Short answer

Answer: The combination of decays that can result in the loss of 3 neutrons and 1 proton is 1 alpha decay and 1 beta decay.

Step-by-step solution

Determine the decay chain equation

Let the combination of alpha decays be represented by 'a' and the combination of beta decays be represented by 'b'. Since in an alpha decay, a nucleus loses 2 protons and 2 neutrons, the total loss of protons in 'a' alpha decays is 2a. In a beta decay, a neutron turns into a proton, so the total gain of protons in 'b' beta decays is b. For the entire decay chain, the loss of 1 proton can be represented by the following equation: 2a - b = 1 Similarly, for neutrons, the total loss of neutrons in 'a' alpha decays is 2a, and the total loss of neutrons in 'b' beta decays is b. The total loss of 3 neutrons in the entire decay chain is represented by the following equation: 2a + b = 3

Solve the system of equations

Now, we have a system of two linear equations with two variables, a and b: 1) 2a - b = 1 2) 2a + b = 3 We can solve these equations using different methods, but let's use the elimination method. Add equation (1) and equation (2) to eliminate the 'b' variable: 2a - b + 2a + b = 1 + 3 4a = 4 Now, let's solve for 'a': a = 4/4 a = 1 To find the value of 'b', substitute the value of 'a' into either equation, such as equation (1): 2(1) - b = 1 2 - b = 1 b = 1

Final result

The combination of alpha (a) and beta (b) decays that can account for the loss of 3 neutrons and 1 proton after a decay chain is: a = 1 (one alpha decay) b = 1 (one beta decay)

Key concepts

Alpha Decay

Alpha decay is a type of radioactive decay where an unstable nucleus emits an alpha particle. This particle consists of 2 protons and 2 neutrons, which is essentially the same as a helium-4 nucleus. As a result, the original element loses mass and changes its atomic number.

• When an atom undergoes alpha decay, it loses 2 protons, causing a decrease in the element's atomic number by 2.
• The loss of 2 neutrons decreases the neutron number in the nucleus.
• This form of decay reduces the mass number by 4.

Alpha decay typically occurs in heavy elements—those with a high mass number—because the large nucleus is more unstable. After losing an alpha particle, the nucleus becomes a different element with properties dependent on the atomic number.

Beta Decay

Beta decay is another form of radioactive decay where a neutron in an unstable nucleus changes into a proton. This process emits a beta particle, which is a high-energy electron. Unlike alpha decay, beta decay affects the atomic number but does not significantly change the mass number.

• In beta decay, an emitted electron (beta particle) results in a neutron depleting.
• The transformation increases the number of protons by one.
• As a result, the atomic number increases by 1.

Beta decay can happen as beta-minus decay, where an electron is emitted, or beta-plus decay, which emits a positron. Beta-plus decay is not mentioned directly here but is crucial in general understanding. This process helps in reaching a more stable ratio of protons to neutrons within the nucleus.

Decay Chain

A decay chain refers to the series of radioactive decays that certain isotopes undergo before reaching a stable state. Each decay results in the transformation into different elements or isotopes until a stable, non-radioactive element is formed.

• Each step in the chain involves a different type of decay, such as alpha or beta.
• The result is progressively lighter and more stable elements.
• Decay chains help explain how complex decay processes evolve and lead to a final stable element.

In the exercise's context, recognizing the sequence and type of the decays—like one alpha and one beta decay—allows students to predict the changes in atomic and mass numbers over a decay chain.

Nuclear Reactions

Nuclear reactions are processes that involve changes in an atom's nucleus. These reactions can occur spontaneously, such as in decay, or be induced, such as in nuclear fission or fusion. They are characterized by the alteration of one element into another, often releasing energy.

• Alpha and beta decay are examples of natural nuclear reactions leading to different elements.
• These reactions help stabilize the original unstable nucleus.
• In nuclear reactions, the conservation of mass-energy and charge is crucial.

Understanding nuclear reactions is vital for comprehending various phenomena, ranging from energy generation in stars to the decay of radioactive elements. They also play a role in practical applications, including medical treatments and energy production.