# 16.1: Essential equations (2023)

1. Last update
2. save as pdf
• ID of the page
304001
• $$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}}}$$ $$\newcommand{\vecd}{\overset{-\!- \!\rightharpoonup}{\vphantom{a}\smash{#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{ span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{rango}\,}$$ $$\newcommand{\RealPart }{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\ norma}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm {span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\ mathrm{nulo}\,}$$ $$\newcommand{\rango}{\mathrm{rango}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{ \ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argumento}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{s p an}}$$$$\nuevocomando{\AA}{\unicode[.8,0]{x212B}}$$

##### learning objectives
• Identify common particles and energies involved in nuclear reactions.
• Write and balance nuclear equations.

Changes in nuclei that result in changes in their atomic numbers, mass numbers, or energy states arenuclear reactions. To describe a nuclear reaction, we use an equation that identifies the nuclides involved in the reaction, their mass numbers and atomic numbers, and the other particles involved in the reaction.

## Types of particles in nuclear reactions

Many entities can be involved in nuclear reactions. The most common are protons, neutrons, alpha particles, beta particles, positrons and gamma rays, as shown in Table $$\PageIndex{1}$$.

Table $$\PageIndex{1}$$ A summary of the names, symbols, representations and descriptions of the most common particles in nuclear reactions. ## equilibrium of nuclear reactions

A balanced chemical reaction equation reflects the fact that during a chemical reaction, bonds are broken and formed and atoms rearrange themselves, but the total number of atoms of each element is conserved and does not change. A balanced nuclear reaction equation indicates that there is rearrangement during a nuclear reaction, but of subatomic particles rather than atoms. Nuclear reactions also follow conservation laws and balance in two ways:

1. The sum of the mass numbers of the reactants is equal to the sum of the mass numbers of the products.
2. The sum of the charges on the reactants is equal to the sum of the charges on the products.

If the atomic number and mass number of all but one of the particles in a nuclear reaction are known, we can identify the particle by balancing the reaction. For example, we can determine that $$\ce{^{17}_8O}$$ is a product of the nuclear reaction of $$\ce{^{14}_7N}$$ and $$\ce{^4_2He} \ ) if we knew that a proton, \(\ce{^1_1H}$$, is one of two products. The example $$\PageIndex{1}$$ shows how we can identify a nuclide by balancing the nuclear reaction.

## Nuclear decay processes

Radioactive decay involves the emission of a particle and/or energy when one atom transforms into another. In most cases, the atom changes its identity to become a new element. There are four different types of emissions that occur.

### alpha emission

Alpha Decomposition $$\left( \alpha \right)$$It involves the release of helium ions from the nucleus of an atom. This ion consists of two protons and two neutrons and has a $$2+$$ charge. The release of a $$\alpha$$ particle produces a new atom that has an atomic number two less than the original atom and an atomic weight four less. A typical alpha decay reaction is the conversion of uranium-238 to thorium:

$\ce{^{238}_{92}U} \rightarrow \ce{^{234}_{90}Th} + \ce{^4_2 \alpha}^+ \nonumber$

We see a decrease of two in atomic number (uranium to thorium) and a decrease of four in atomic weight (238 to 234). Emission is usually not written with the indicated atomic number and weight, as it is a common particle whose properties must be memorized. Alpha emission is often accompanied by gamma radiation $$\left( \gamma \right)$$, a form of energy release. Many of the largest elements in the periodic table are alpha emitters.

Chemists often use the namesparent isotopeydaughter isotopeto represent the original atom and the distinct product of the alpha particle. In the example above, $_{92}^{238}\textrm{U} \nonumber$ is the major isotope and $_{90}^{234}\textrm{Th} \nonumber$ is the isotope son. When one element changes into another in this way, it suffersradioactive decay.

##### Example $$\PageIndex{1}$$

Write the nuclear equation that represents the radioactive decay of radon-222 by the emission of alpha particles and identify the daughter isotope.

###### Solution

Radon has an atomic number of 86, so the parent isotope is represented as $_{86}^{222}\textrm{Rn} \nonumber$

We represent alpha particle as

$_{2}^{4}\textrm{He} \nonumber$

Use subtraction (222 − 4 = 218 and 86 − 2 = 84) to identify the child isotope as polonium:

$_{86}^{222}\textrm{Rn}\rightarrow \; _ {2}^{4}\textrm{Él}+\: _{84}^{218}\textrm{Th} \nonumber$

##### Exercise $$\PageIndex{1}$$

Write the nuclear equation that represents the radioactive decay of polonium-208 by the emission of alpha particles and identify the daughter isotope.

(Video) 16.1 - Functions to Model Linear Relationships

Responder

$_{80}^{208}\textrm{Po}\rightarrow \; _ {2}^{4}\textrm{Ele}+\:_{82}^{204}\textrm{Pb} \nonumber$

$_{82}^{204}\textrm{Pb} \nonumber$

(Video) 16.1. Nondimensionalization of Governing Equations

### beta emission

Beta $$\left( \beta \right)$$ decaimentoit's a more complicated process. Unlike $$\alpha$$ emission, which simply ejects a particle, $$\beta$$ emission involves the transformation of a neutron in the nucleus into a proton and an electron. Then the electron is ejected from the nucleus. In the process, the atomic number increases by one while the atomic weight remains the same. As in the case of $$\alpha$$ emissions, $$\beta$$ emissions are often accompanied by $$\gamma$$ radiation.

A typical beta decay process involves carbon-14, which is often used in radioactive dating techniques. The reaction forms nitrogen-14 and one electron:

$\ce{^{14}_6C} \rightarrow \ce{^{14}_7N} + \ce{^0_{-1}e} \nonúmero$

Again, beta emission is usually denoted simply by the Greek letter $$\beta$$; memorization of the process is necessary to follow nuclear calculations where the Greek letter $$\beta$$ appears without other notation.

##### Example $$\PageIndex{2}$$

Write the nuclear equation that represents the radioactive decay of boron-12 by the emission of beta particles and identify the daughter isotope. A gamma ray is emitted simultaneously with the beta particle.

###### Solution

The parent isotope is $B512," id="MathJax-Element-16-Frame" role="presentation" style="position:relative;" tabindex="0">_ {2}^{4}\textrm{He} \nonumber$

B512," role="presentation" style="position:relative;" tabindex="0">while one of the products isB512," role="presentation" style="position:relative;" tabindex="0">$_{-1}^{0}\textrm{e} \nonumber$

For the mass and atomic numbers to have the same value on both sides, the mass number of the child isotope must be 12 and its atomic number must be 6. The element that has atomic number 6 is carbon. Therefore, the complete nuclear equation is as follows:

$_{5}^{12}\textrm{B}\rightarrow \;_{6}^{12}\textrm{C}+_{-1}^{0}\textrm{e}+\gamma \no number$

(Video) Lesson 16.1 Solving Exponential Equations

The daughter isotope is carbon-12.

##### Exercise $$\PageIndex{2}$$

Write the nuclear equation that represents the radioactive decay of rubidium-87 by the emission of beta particles and identify the daughter isotope.

Responder

$_{37}^{87}\textrm{Rb}\rightarrow \;_{38}^{87}\textrm{Sr}+_{-1}^{0}\textrm{e} \nonumber \ ] \[_{38}^{87}\textrm{Sr} \nonúmero$

### gamma emission

Gamma radiation $$\left( \gamma \right)$$it is simply energy. It can be released by itself or, more commonly, in association with other radiation events. There is no change in atomic number or atomic weight in a single emission of $$\gamma$$. Often, an isotope can produce $$\gamma$$ radiation as a result of a transition to a metastable isotope. This type of isotope can simply "settle", with a displacement of particles in the nucleus. The composition of the atom is not changed, but the nucleus can be considered more "comfortable" after the change. This change increases the stability of the isotope from the energetically unstable (or "metastable") isotope to a more stable form of the nucleus. range ($$\range$$) emission can occur virtually instantaneously, as occurs in the alpha decay of uranium-238 to thorium-234, where the asterisk denotes an excited state:

$^{238}_{92}\textrm{U}\rightarrow \, \underset{\textrm{excited} \\ \textrm{nuclear} \\ \textrm{state}}{^{234}_{90 }\textrm{Th*}} + ^{4}_{2}\alpha\xrightarrow {\textrm{relaxation}\,}\,^{234}_{90}\textrm{Th}+^{0} _{0}\gamma\label{Eq13}$

If we ignore the decay event that created the excited core, then

$^{234}_{88}\textrm{Th*} \rightarrow\, ^{234}_{88}\textrm{Th}+^{0}_{0}\gamma\label{Eq14} \ ] or more generally, \[^{A}_{Z}\textrm{X*} \rightarrow\, ^{A}_{Z}\textrm{X}+^{0}_{0}\gamma\label{Eq15} \ ] Gamma emission can also occur after a significant delay. For example, technetium-99metrohas a half-life of approximately 6 hours before emitting a γ-ray to form technetium-99 (themetrois for metastable). Since γ-rays are energy, their emission does not affect the mass number or atomic number of the daughter nuclide. Therefore, gamma-ray emission is the only type of radiation that does not necessarily involve the conversion of one element into another, although it is almost always observed in conjunction with some other nuclear decay reaction. ### positron emission Apositronit is a positive electron (a form of antimatter). This rare type of emission occurs when a proton becomes a neutron and a positron in the nucleus, with ejection of the positron. The atomic number will decrease by one as long as the atomic weight does not change. A positron is often referred to as $$\beta^+$$. Carbon-11 emits a positron to become boron-11: \[\ce{^{11}_6C} \rightarrow \ce{^{11}_5B} + \ce{^0_{+1} \beta} \sin number$

(Video) Elementary Algebra Lesson 16.1: Equations of Lines and Slopes Part 1

### electronic capture

An alternative way for a nuclide to increase its neutron to proton ratio is through a phenomenon called electron capture. In electron capture, the nucleus of the atom captures an electron from an inner orbital and combines it with a proton to form a neutron. For example, silver-106 undergoes electron capture to become palladium-106.

$\ce{^{106}_{47}Ag} + \ce{^0_{-1}e} \rightarrow \ce{^{106}_{46}Pd} \nonumber$

Note that the overall result of electron capture is identical to positron emission. The atomic number decreases by one while the mass number remains the same.

Table $$\PageIndex{2}$$ Different types of decomposition and changes in atomic and mass numbers. The following are equations for several nuclear reactions that have played important roles in the history of nuclear chemistry:

• The first naturally occurring unstable element to be isolated, polonium, was discovered by Polish scientist Marie Curie and her husband Pierre in 1898. It decays and emits α particles: $\ce{^{212}_{84}Po⟶ ^ {208}_{82}Pb + ^4_2He} \unnumbered$
• The first nuclide to be prepared by artificial means was an isotope of oxygen,17O. It was made by Ernest Rutherford in 1919 by bombarding nitrogen atoms with α particles: $\ce{^{14}_7N + ^4_2α⟶ ^{17}_8O + ^1_1H} \nonumber$
• James Chadwick discovered the neutron in 1932, as a previously unknown neutral particle produced together with12C by the nuclear reaction between9be and4They: $\ce{^9_4Be + ^4_2He⟶ ^{12}_6C + ^1_0n} \nonumber$
• The first element to be prepared that does not exist naturally on Earth, technetium, was created by the bombardment of molybdenum by deuterons (heavy hydrogen, $$\ce{^2_1H}$$), by Emilio Segre and Carlo Perrier in 1937. : $\ce{^2_1H + ^{97}_{42}Mo⟶2^1_0n + ^{97}_{43}Tc} \nonumber$
• The first controlled nuclear chain reaction was performed in a reactor at the University of Chicago in 1942. One of the many reactions involved was: $\ce{^{235}_{92}U + ^1_0n⟶ ^{ 87} _ { 35}Br + ^{146}_{57}La + 3^1_0n} \number$

## Summary

• Nuclei can undergo reactions that change their number of protons, number of neutrons or energy state.
• Many different particles can be involved in nuclear reactions. The most common are protons, neutrons, positrons (which are positively charged electrons), electrons, alpha (α) particles (which are high-energy helium nuclei), beta (β) particles (which are high-energy electrons), and particles. .gamma. (γ) rays (which make up high-energy electromagnetic radiation).
• As with chemical reactions, nuclear reactions are always balanced. When a nuclear reaction occurs, the total mass (number) and total charge remain unchanged.

## Collaborators and Assignments

• TextMap: Initial Chemistry (Ball et al.)
• TextMap: Chemistry Central Science (Brown et al.)
• (Video) 16.1 Inductions (Basic Mathematics)

## Videos

1. 16.1 Review of Maclaurin Series and Separable Diff Eqns
(Thomas Gibbons)
2. Algebra 2 Chapter 16.1 Properties of Logarithms
(MathGives YouPower)
3. BIOE 3310 Lec 16.1 - "Micro" conservation of mass, aka continuity equation
(Timothy Lannin)
4. Algebra 2 Chapter 16.1 Exercises 1-12 Properties of Logarithms Problems
(MathGives YouPower)
5. Alg2H lesson 16.1- Matrix Order
(Katie Maloney)
6. Math 099 - 16.1 Solving Quadratic Equations Using the Square Root Method
(Mrs. Igo's Classroom)
Top Articles
Latest Posts
Article information

Author: Jeremiah Abshire

Last Updated: 03/05/2023

Views: 6632

Rating: 4.3 / 5 (54 voted)

Author information

Name: Jeremiah Abshire

Birthday: 1993-09-14

Address: Apt. 425 92748 Jannie Centers, Port Nikitaville, VT 82110

Phone: +8096210939894