{"id":10237,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=10237"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"radioactive-decay-and-energy-in-modern-physics","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/radioactive-decay-and-energy-in-modern-physics\/","title":{"rendered":"Radioactive Decay And Energy In Modern Physics"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Unit: <\/strong>Quantum, Atomic, and Nuclear Physics<\/strong><\/h2>\n

Chapter: <\/strong>Radioactive Decay and Energy in Modern Physics <\/strong><\/h3>\n

Reference: AP Physics Algebra, <\/em>Quantum, Atomic, and Nuclear Physics, Radioactive Decay and Energy in Modern Physics, Energy and Radioactive Decay, Atomic Masses and Composition of Nucleus, Size of the Nucleus<\/p>\n

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After studying this chapter, you should be able to,<\/strong><\/p>\n

    \n
  • state the Radioactivity and Decay Law<\/li>\n
  • explain the concepts of Mass-Energy Equivalence<\/li>\n
  • state the concept of Atomic Masses and the Composition of the Nucleus<\/li>\n<\/ul>\n

    Systems and fundamental forces<\/strong><\/p>\n

    There are four fundamental forces in nature: gravitational, electromagnetic, strong nuclear, and weak nuclear forces.<\/p>\n

    The gravitational force is responsible for the attraction between masses.<\/p>\n

    The electromagnetic force is responsible for interactions between charged particles and is described by Maxwell’s equations.<\/p>\n

    The strong nuclear force binds protons and neutrons together in the atomic nucleus and is the strongest fundamental force.<\/p>\n

    The weak nuclear force is responsible for certain types of radioactive decay and plays a role in nuclear reactions.<\/p>\n

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    Radioactivity and Decay Law<\/strong><\/p>\n

    \u2022 Nucleus: <\/strong>The nucleus can be defined as the central part of an atom, made up of neutrons, protons, and other elementary particles. The nucleus has<\/p>\n

    protons and neutrons inside it. They are called nucleons.<\/p>\n

    \u2022 Mass Number: <\/strong>The total number of protons and neutrons present inside the nucleus of an atom of an element is referred to as mass number (A) of<\/p>\n

    the element.<\/p>\n

    \u2022 Atomic Number: <\/strong>The number of protons present in the nucleus of an atom of an element is known as the atomic number (Z) of the element.<\/p>\n

    \u2022 Nuclear Size: <\/strong>The radius of the nucleus R\u221d<\/em><\/p>\n

    \"\"<\/p>\n

    Source: Kapdec.com<\/p>\n<\/div>\n

    \u00a0A1\/3<\/sup><\/p>\n

    R= R0<\/sub>A1\/3<\/sup> where R0<\/sub> = 1.2 \u00d7 10\u201315<\/sup> m is an empirical constant.<\/p>\n

    \u00a0<\/p>\n

    Nuclear Density: <\/strong>Nuclear density is independent of mass number and is therefore same for all nuclei.<\/p>\n

    \u00a0<\/p>\n

    Atomic Mass Unit: <\/strong>Abbreviated as amu and is defined as one-twelfth of the mass of a carbon nucleus. It is also denoted by u.<\/p>\n

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    Therefore,<\/p>\n

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    Source: Kapdec.com<\/p>\n<\/div>\n

    \u2022 Isomers: <\/strong>The atoms that have the same mass number, and atomic number but different radioactive properties are known isomers.<\/p>\n

    \u2022 Isotones: <\/strong>Atoms of elements that have different mass numbers, and atomic numbers but the same number of neutrons are known as isotones. e.g., 1<\/sub>H3<\/sup>, 2<\/sub>H4<\/sup> and 6<\/sub>C14<\/sup>, 8<\/sub>O16 <\/sup>are isotones.<\/p>\n

    \u2022 Isobars<\/strong>: <\/em>The atoms of an element having different atomic numbers but the same mass numbers are known as isobars. e.g., 1<\/sub>H3<\/sup>, 2<\/sub>H3<\/sup> and 10<\/sub>Na22<\/sup>, 10<\/sub>Ne22<\/sup> are isobars.<\/p>\n

    \u00a0<\/p>\n

    \u2022 Isotopes<\/strong>: <\/em>Atoms of an element that have different mass numbers but the same atomic number are known as isotopes. e.g., 1<\/sub>H1<\/sup>, 1<\/sub>H2<\/sup>, 1<\/sub>H3<\/sup> is an example of isotopes.<\/p>\n

    \u2022 Nuclear Force<\/strong>: <\/em>Nuclear force can be referred to as the force that acts inside the nucleus or between nucleons. These forces are neither electrostatic nor gravitational in nature. They have a very short range and are independent of any charge. They are a hundred times that of electrostatic force and 1038<\/sup> times that of gravitational force.<\/p>\n

    \u00a0<\/p>\n

    \u2022 Radioactivity<\/strong>: <\/em>Radioactivity refers to the breakdown of heavy elements into comparably lighter elements by the emission of radiations. This phenomenon was discovered by Henry Becquerel in 1896.<\/p>\n

    \u00a0<\/p>\n

    \"\"<\/p>\n

    Source: Kapdec.com<\/p>\n<\/div>\n

    \u00a0<\/p>\n

    \u00a0<\/p>\n

    \u00a0<\/p>\n

    \u00a0<\/p>\n

    \u2022 Radioactive Decay law<\/em><\/strong><\/p>\n

    The Radioactive law states that the rate of disintegration of radioactive atoms at any instance is directly proportional to the number of radioactive atoms present in the given sample at that instant.<\/p>\n

    Rate of disintegration – dN\/<\/em>dt<\/em>\u00a0\u221d<\/em>\u00a0N<\/p>\n

    – dN\/<\/em>dt<\/em>\u00a0=<\/em><\/p>\n

    \"\"<\/p>\n

    Source: Kapdec.com<\/p>\n<\/div>\n

    \u00a0\u03bb<\/em>\/N where l is the decay constant. The number of undecayed atoms present in the sample at any instance N <\/em>= N<\/em>0<\/sub> e<\/em>\u2212\u03bb<\/sup>t<\/sup><\/em> <\/em>where N0<\/sub> is the number of atoms at time t = 0 and N is the number of atoms at time t.<\/p>\n

    \u00a0<\/p>\n

    \u2022 Activity of a radioactive element<\/strong><\/p>\n

    The activity of a radioactive element is equal to its rate of disintegration.<\/p>\n

    Activity R = – dN\/<\/em>dt<\/em><\/p>\n

    The activity of the sample after time t, R <\/em>= R<\/em>0<\/sub> e\u2013<\/em>\u03bb<\/sup>t<\/sup><\/em> <\/em>Its SI unit is Becquerel (Bq). Curie and Rutherford <\/em>are its other units.<\/p>\n

    1 Curie = 3.7 \u00d7 1010<\/sup> decay\/s and 1 Rutherford = 106<\/sup> decay\/s<\/p>\n

    \u2022 Half-life of a radioactive element<\/strong><\/p>\n

    Half-life (T) of a radioactive element is the time taken for the radioactivity of an isotope to fall to half its original value. The relation between disintegration constant and the half-life is given by<\/p>\n

    \"\"<\/p>\n

    Source: Kapdec.com<\/p>\n<\/div>\n

    \u2022 Average Life or Mean Life (<\/strong>\u03c4<\/em><\/strong><\/p>\n

    \"\"<\/p>\n

    Source: Kapdec.com<\/p>\n<\/div>\n

    )<\/strong><\/p>\n

    Average life or mean life (\u03c4<\/em><\/p>\n

    \"\"<\/p>\n

    Source: Kapdec.com<\/p>\n<\/div>\n

    ) of a radioactive element can be defined as the ratio of the total life time of all the atoms and the total number of atoms present, initially in the sample.<\/p>\n

    Relation between half-life and average life \u03c4<\/em><\/p>\n

    \"\"<\/p>\n

    Source: Kapdec.com<\/p>\n<\/div>\n

    \u00a0= 1.44T Relation between an average life and decay constant<\/p>\n

    \u03c4<\/em><\/p>\n

    \"\"<\/p>\n

    Source: Kapdec.com<\/p>\n<\/div>\n

    \u00a0= 1<\/em>\u03bb<\/sup><\/p>\n

    \"\"<\/p>\n

    Source: Kapdec.com<\/p>\n<\/div>\n

    \u2022 Alpha decay: <\/strong>In alpha decay, a nucleus gets transformed into a different nucleus and a<\/p>\n

    particle is emitted. The general form can be expressed as:<\/p>\n

    \"\"<\/p>\n

    Source: Kapdec.com<\/p>\n<\/div>\n

    and the Q value:<\/p>\n

    \"\"<\/p>\n

    Source: Kapdec.com<\/p>\n<\/div>\n

    \u2022 Beta decay: <\/strong>When a nucleus undoes beta decay, it emits an electron or a positron. When an electron is emitted, it is said to be beta minus decay while in beta plus decay, a positron is emitted.<\/p>\n

    \u2022 Gamma decay: <\/strong>In gamma decay, the photons are emitted from the nuclei having MeV energy and thus, the gamma rays are emitted. This is called gamma decay.<\/p>\n

    Energy plays a fundamental role in modern physics and is a central concept in various areas, including radioactive decay and Einstein’s famous equation E=mc\u00b2. Here are some key notes on energy in modern physics:<\/p>\n

    Energy and Radioactive Decay:<\/strong><\/p>\n

      \n
    • Radioactive decay is a spontaneous process in which unstable atomic nuclei transform into more stable configurations.<\/li>\n
    • During radioactive decay, energy is released in the form of radiation, such as alpha particles, beta particles, or gamma rays.<\/li>\n
    • The energy released in radioactive decay comes from the conversion of the mass defect into energy.<\/li>\n<\/ul>\n

      Mass-Energy Equivalence (E=mc\u00b2):<\/strong><\/p>\n

        \n
      • Einstein’s mass-energy equivalence principle states that mass (m) and energy (E) are interchangeable and are two different forms of the same underlying quantity.<\/li>\n
      • The equation E=mc\u00b2 expresses the relationship between energy and mass, where c represents the speed of light in a vacuum (approximately 3 x 108<\/sup>\u00a0meters per second).<\/li>\n
      • According to this equation, even a small amount of mass contains a large amount of energy. The energy content is given by multiplying the mass by the square of the speed of light.<\/li>\n<\/ul>\n

        Nuclear Reactions and Energy:<\/strong><\/p>\n

          \n
        • Nuclear reactions involve changes in the nucleus of an atom and can release or absorb significant amounts of energy.<\/li>\n
        • In nuclear reactions, the total mass before the reaction is not always equal to the total mass after the reaction due to mass-energy equivalence.<\/li>\n
        • Energy is released when the total mass after the reaction is less than the total mass before the reaction. This energy release is known as a mass defect.<\/li>\n<\/ul>\n

          Conservation of Energy:<\/strong><\/p>\n

            \n
          • In modern physics, energy is subject to the law of conservation of energy, which states that energy cannot be created or destroyed but can only be transformed from one form to another.<\/li>\n
          • In various processes, such as radioactive decay or nuclear reactions, the total energy before and after the process remains constant, although it may change its form.<\/li>\n<\/ul>\n

            Energy Units:<\/strong><\/p>\n

              \n
            • In the International System of Units (SI), energy is measured in joules (J). One joule is equal to the energy transferred when one newton of force acts over a distance of one meter.<\/li>\n
            • In nuclear and particle physics, the electron volt (eV) is often used as a unit of energy. One electron-volt is defined as the energy gained or lost by an electron when it is accelerated by an electric potential difference of one volt.<\/li>\n<\/ul>\n

              Atomic Masses and Composition of Nucleus:<\/strong><\/p>\n

                \n
              • The mass of an atom is very small, compared to a kilogram. The kilogram is not a very convenient unit to measure such small quantities. Therefore, a different mass unit is used for expressing atomic masses.<\/li>\n
              • This unit is the atomic mass unit (u), defined as 1\/12th of the mass of the carbon (12<\/sup>C) atom. According to this definition<\/li>\n
              • The atomic masses of various elements expressed in an atomic mass unit (u) are close to being integral multiples of the mass of a hydrogen atom.<\/li>\n
              • The measurement of atomic masses reveals the existence of different types of atoms of the same element, which exhibit the same chemical properties but differ in mass. Such atomic species of the same element differing in mass are called isotopes.<\/li>\n
              • The relative abundance of different isotopes differs from element to element. Chlorine, for example, has two isotopes having masses 34.98 u and 36.98 u.<\/li>\n<\/ul>\n

                Size of the Nucleus<\/strong><\/p>\n

                It has been found that a nucleus of mass number A has a radius<\/p>\n

                The density of nuclear matter is approximately 2.3 \u00d7 1017<\/sup> kg m\u20133<\/sup>.<\/p>\n

                Example:<\/strong> Calculate the energy equivalent of 1 g of substance.<\/p>\n

                Solution:<\/strong> Energy,<\/p>\n

                E = 10\u20133<\/sup> \u00d7 (3 \u00d7 108<\/sup>) 2 <\/sup>J<\/p>\n

                E = 10\u20133<\/sup> \u00d7 9 \u00d7 1016<\/sup> = 9 \u00d7 1013<\/sup> J<\/p>\n

                Thus, if one gram of matter is converted to energy, there is a release of an enormous amount of energy.<\/p>\n

                This concept is important in understanding nuclear masses and the interaction of nuclei with one another.<\/p>\n

                Key points<\/strong><\/p>\n

                \u00a0<\/p>\n

                  \n
                • Radioactive decay is the spontaneous disintegration of atomic nuclei, resulting in the emission of radiation and the formation of different elements.<\/li>\n
                • There are three common types of radioactive decay: alpha decay, beta decay, and gamma decay. In alpha decay, an alpha particle (two protons and two neutrons) is emitted. In beta decay, either an electron (beta-minus decay) or a positron (beta-plus decay) is emitted. Gamma decay involves the emission of high-energy gamma rays.<\/li>\n
                • Radioactive decay is a random process that cannot be predicted for individual atoms. However, for a large number of atoms, the decay process follows a statistical pattern described by a decay constant.<\/li>\n
                • The decay of radioactive isotopes is characterized by their half-life, which is the time it takes for half of the original sample to decay. Different isotopes have different half-lives, ranging from fractions of a second to billions of years.<\/li>\n
                • Radioactive decay plays a crucial role in various fields, including nuclear power, medicine (e.g., radiation therapy and medical imaging), and carbon dating for determining the age of ancient artefacts and fossils.<\/li>\n
                • Energy is a fundamental concept in modern physics that describes the ability of a system to do work or transfer heat. It is a conserved quantity, meaning it cannot be created or destroyed, only transformed from one form to another.<\/li>\n<\/ul>\n

                  \u00a0<\/p>\n

                    \n
                  • In the context of modern physics, energy is related to the fundamental forces and particles that govern the behaviour of matter and radiation.<\/li>\n<\/ul>\n

                    \u00a0<\/p>\n

                      \n
                    • Albert Einstein’s theory of relativity, particularly his mass-energy equivalence equation (E=mc\u00b2), revolutionized the understanding of energy. It states that mass and energy are interchangeable, and a small amount of mass can be converted into a large amount of energy.<\/li>\n<\/ul>\n

                      \u00a0<\/p>\n

                        \n
                      • The study of subatomic particles and their interactions, as described by quantum mechanics, revealed the existence of various energy states and energy quantization. Particles such as electrons can only occupy discrete energy levels within an atom.<\/li>\n<\/ul>\n

                        \u00a0<\/p>\n

                          \n
                        • In particle physics, the concept of energy is essential in understanding particle interactions, such as the creation and annihilation of particles in particle accelerators.<\/li>\n<\/ul>\n

                          \u00a0<\/p>\n

                            \n
                          • The discovery of new energy phenomena, such as dark energy and dark matter, has expanded our understanding of the universe but also posed new challenges in explaining the nature of these mysterious components.<\/li>\n<\/ul>\n

                            \u00a0<\/p>\n

                              \n
                            • Energy conservation is a fundamental principle in modern physics and is closely related to the laws of thermodynamics. It states that the total energy of a closed system remains constant, even though it can be converted between different forms (e.g., mechanical, thermal, electromagnetic).<\/li>\n<\/ul>\n

                              \u00a0<\/p>\n

                              <\/p>\n