Question

An isotope of rutherfordium, \({ }_{10}^{257} \mathrm{Rf}\), is formed by the bombardment of californium- 249 by carbon-12. In the process, neutrons are emitted. The new isotope formed decays rapidly, emitting an alpha particle. (a) How many neutrons are emitted for every Cf-249 bombarded? (b) Write the nuclear symbol for the isotope formed by the decay of Rf-257.

Short answer

4 neutrons are emitted for every Californium-249 bombarded. (b) What is the nuclear symbol for the isotope formed by the decay of Rutherfordium-257? The nuclear symbol for the isotope formed by the decay of Rutherfordium-257 is ${}_{102}^{253}\mathrm{X}$.

Step-by-step solution

Identify the conservation of nucleons in a nuclear reaction

In a nuclear reaction, the total number of nucleons (protons + neutrons) remains constant. This means, the sum of the nucleons before the reaction equals the sum of nucleons after the reaction.

Write the given nuclear reaction

In this problem, Californium-249 (\({ }_{98}^{249}\mathrm{Cf}\)) is bombarded by Carbon-12 (\({ }_{6}^{12}\mathrm{C}\)) forming an isotope of Rutherfordium (\({ }_{10}^{257}\mathrm{Rf}\)) and emitting neutrons (\(n\)). The nuclear equation for this reaction is: \({ }_{98}^{249}\mathrm{Cf} + { }_{6}^{12}\mathrm{C} \rightarrow { }_{104}^{257}\mathrm{Rf} + x\cdot n\) where `x` is the number of neutrons emitted during the reaction.

Use conservation of nucleons to find the number of emitted neutrons

According to the conservation of nucleons, the total number of nucleons before the reaction equals the total number of nucleons after the reaction: 249 + 12 = 257 + x Now, we solve for x to find the number of neutrons emitted: x = 261 - 257 = 4 Thus, 4 neutrons are emitted for every Californium-249 bombarded. #Part (b): Writing the nuclear symbol for the isotope formed by the decay of Rutherfordium-257#

Identify the process of alpha decay

Alpha decay is a process in which an unstable nucleus emits an alpha particle, leading to a decrease in the atomic number by 2 and the mass number by 4. An alpha particle consists of 2 protons and 2 neutrons (\({ }_{2}^{4}\mathrm{He}\)).

Write the alpha decay nuclear equation for Rf-257

The alpha decay of Rutherfordium-257 can be represented as follows: \({ }_{104}^{257}\mathrm{Rf} \rightarrow { }_{Z}^{A}\mathrm{X} + { }_{2}^{4}\mathrm{He}\) where `X` is the nucleus formed after the decay, `Z` is its atomic number, and `A` is its mass number.

Calculate the new atomic number and mass number

To find the new atomic number and mass number, we use the conservation of nucleons. The sum of nucleons before the decay should be equal to the sum of nucleons after the decay. For the atomic numbers, we have: 104 = Z + 2 So, the new atomic number Z is 102. For the mass numbers, we have: 257 = A + 4 So, the new mass number A is 253. Thus, the nuclear symbol for the isotope formed by the decay of Rutherfordium-257 is \({ }_{102}^{253}\mathrm{X}\).

Key concepts

Isotope Decay

Isotope decay is a fascinating phenomenon where an unstable atomic nucleus loses energy by emitting radiation. In simpler terms, it's like a transformation that allows the atom to reach a more stable state. When an isotope decays, it can change into a different element altogether. This process is unpredictable and occurs at an atomic level.
A crucial aspect of isotope decay is its half-life, which is the time it takes for half of a sample of the isotope to decay. This concept is important because it tells us about the stability of an isotope and how quickly it tends to decay.
Isotope decay is not just a topic in textbooks; it's something that has practical applications in dating materials and in medical treatments.

Neutron Emission

During certain nuclear reactions, like the bombardment of californium-249 with carbon-12 in our problem, neutrons are emitted. Neutron emission is a type of radioactive decay where a neutron is ejected from an atomic nucleus.
This emission plays a significant role in nuclear reactions since it affects the composition and properties of the resulting isotopes. Neutrons are crucial because they don't have an electric charge. This makes them very effective at penetrating other nuclei, which is why they are often used in experiments to trigger nuclear reactions.
When it comes to nuclear reactions, neutrons may be emitted as a byproduct, and their number can be determined using the conservation of nucleons. For instance, in the exercise, we found that 4 neutrons were emitted as a result of the reaction.

Alpha Decay

Alpha decay is a type of radioactive decay where an atomic nucleus releases an alpha particle. An alpha particle consists of 2 protons and 2 neutrons, essentially a helium nucleus. This kind of decay results in the reduction of the atomic number by 2 and the mass number by 4.
Alpha decay is characteristic of heavy elements, where the nucleus is too large to be stable. Through alpha decay, the nucleus attempts to become more stable. For example, the isotope \({ }_{104}^{257}\mathrm{Rf}\) undergoes alpha decay to form \({ }_{102}^{253}\mathrm{X}\).
Alpha particles, due to their composition, are relatively heavy and carry a positive charge, which makes them less penetrative compared to other types of radiation. They are usually stopped by sheets of paper or even human skin.

Conservation of Nucleons

Conservation of nucleons is a fundamental principle in nuclear reactions. It states that the total number of nucleons (protons plus neutrons) remains constant throughout the reaction.
This principle is crucial for predicting the outcomes of nuclear reactions. It helps balances nuclear equations, ensuring that the total mass and atomic numbers on both sides of the equation are equal. For our exercise, using this principle allowed us to calculate that 4 neutrons were emitted.
The conservation of nucleons provides insight into the formation of new isotopes and helps in estimating the resultant elements after processes like alpha decay. It ensures that despite the transformations that occur within a nuclear reaction, the fundamental components, the nucleons themselves, are conserved.