Introduction

Building on the knowledge that atoms have a positively charged nucleus at their center (from Chapter 12), which is much smaller than the atom itself (by a factor of about $10^4$ in radius, meaning $10^{12}$ times smaller in volume), this chapter delves into the structure and properties of the nucleus itself. Despite occupying a tiny fraction of the atom's volume, the nucleus contains most of the atom's mass (over 99.9%).

Key questions explored include: Does the nucleus have an internal structure? What are its constituents? How are they held together? The chapter discusses nuclear size, mass, and stability, as well as associated phenomena like radioactivity, nuclear fission, and nuclear fusion.

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Atomic Masses And Composition Of Nucleus

Atomic masses are extremely small. To conveniently express them, the **atomic mass unit (u)** is used. It is defined as one-twelfth of the mass of a carbon-12 ($^{12}$C) atom.

$\mathbf{1 \text{ u} = \frac{1}{12} \times \text{Mass of one } ^{12}\text{C atom} \approx 1.660539 \times 10^{-27} \text{ kg}}$.

Atomic masses in 'u' are often close to integers but have exceptions (e.g., chlorine $\approx 35.46$ u). Accurate mass measurements using devices like the mass spectrometer revealed the existence of **isotopes** – atoms of the same element (same chemical properties, same atomic number $Z$) that differ in mass.

Every element is a mixture of isotopes with varying abundances. For example, chlorine has isotopes with masses near 35 u and 37 u. The atomic mass of chlorine is the weighted average of its isotopes' masses based on their natural abundances.

Even hydrogen has three isotopes: $^1$H (proton, $\approx 1$ u, most abundant), $^2$H (deuterium, $\approx 2$ u), and $^3$H (tritium, $\approx 3$ u, unstable). The nucleus of ordinary hydrogen ($^1$H) is a single particle called a **proton**. Its mass is $m_p \approx 1.00727$ u $\approx 1.67262 \times 10^{-27}$ kg. Protons are stable and carry a single positive fundamental charge ($+e$).

The positive charge of the nucleus is due to protons. The number of protons in a nucleus is equal to the **atomic number ($Z$)** of the element. Since an atom is neutral, it has $Z$ electrons outside the nucleus. Early ideas of electrons inside the nucleus were later disproved.

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Discovery Of Neutron

The masses of deuterium ($^2$H) and tritium ($^3$H) nuclei are approximately twice and three times the mass of a proton, respectively, yet they are isotopes of hydrogen ($Z=1$, one proton). This suggested the presence of neutral matter within the nucleus. In 1932, James Chadwick discovered the **neutron** by observing neutral radiation emitted when beryllium was bombarded with alpha particles. These neutral particles had a mass very close to that of a proton.

The mass of a neutron ($m_n$) is approximately $1.00866$ u $\approx 1.6749 \times 10^{-27}$ kg, slightly greater than a proton's mass. A free neutron is unstable, decaying into a proton, electron, and antineutrino, but is stable inside a nucleus.

The composition of a nucleus is defined by:

• **Atomic number (Z):** Number of protons.
• **Neutron number (N):** Number of neutrons.
• **Mass number (A):** Total number of protons and neutrons ($A = Z + N$). Protons and neutrons together are called **nucleons**.

A nucleus is represented as $\mathbf{^A_Z X}$, where X is the element's chemical symbol (e.g., $^{197}_{79}$Au has $Z=79, A=197, N=197-79=118$).

• Isotopes: Same Z, different N (and thus different A). E.g., $^1_1$H, $^2_1$H, $^3_1$H.
• Isobars: Same A, different Z and N. E.g., $^3_1$H, $^3_2$He.
• Isotones: Same N, different Z and A. E.g., $^{198}_{80}$Hg ($N=118$) and $^{197}_{79}$Au ($N=118$).

Chemical properties are determined by Z (electron structure), so isotopes have identical chemical behaviour.

Size Of The Nucleus

Rutherford's alpha-scattering experiments provided initial estimates of nuclear size. The distance of closest approach of alpha particles suggested the gold nucleus radius was less than about $4 \times 10^{-14}$ m. More precise measurements using electron scattering experiments show that the radius ($R$) of a nucleus with mass number $A$ is given by:

$\mathbf{R = R_0 A^{1/3}}$

where $R_0$ is a constant approximately equal to $1.2 \times 10^{-15}$ m (1.2 fm, where $1 \text{ fm} = 10^{-15}$ m, also called a fermi). This relation shows that the volume of the nucleus (proportional to $R^3$) is directly proportional to the mass number $A$ (total number of nucleons). Since mass is also proportional to $A$, this implies that the **density of nuclear matter is constant** for all nuclei, regardless of their size.

The density of nuclear matter is extremely high, approximately $2.3 \times 10^{17} \text{ kg/m}^3$, far greater than the density of ordinary matter like water ($10^3 \text{ kg/m}^3$). This confirms that atoms are mostly empty space, with nearly all mass concentrated in the tiny nucleus.

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Mass-Energy And Nuclear Binding Energy

Einstein's theory of special relativity revealed that mass and energy are interconvertible. Mass itself is a form of energy. The famous mass-energy equivalence relation is:

$\mathbf{E = mc^2}$

where $E$ is the energy equivalent of mass $m$, and $c$ is the speed of light in vacuum. This principle is fundamental to understanding nuclear reactions and nuclear binding energy. In any reaction, the total mass-energy is conserved.

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Mass – Energy

The equivalence of mass and energy is experimentally verified in nuclear processes. A small amount of mass can be converted into a large amount of energy due to the factor $c^2$ (a very large number).

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Nuclear Binding Energy

The mass of a nucleus ($M$) is always found to be less than the sum of the masses of its individual constituent protons and neutrons (in their free, unbound state). This difference in mass is called the **mass defect ($\Delta M$)**.

For a nucleus with $Z$ protons and $N=(A-Z)$ neutrons:

$\mathbf{\Delta M = [Z m_p + N m_n] - M}$

where $m_p$ and $m_n$ are the masses of a free proton and neutron, and $M$ is the actual measured mass of the nucleus.

This mass defect $\Delta M$ is equivalent to energy $\Delta M c^2$. This energy is the **binding energy ($E_b$)** of the nucleus.

$\mathbf{E_b = \Delta M c^2 = [Z m_p + (A - Z) m_n - M]c^2}$

The binding energy $E_b$ is the energy that would be released if the nucleus were formed by bringing together its constituent nucleons from rest at infinite separation. Conversely, it is the minimum energy required to break a nucleus into its individual nucleons.

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A more useful measure of nuclear stability is the **binding energy per nucleon ($E_{bn}$)**, which is the total binding energy $E_b$ divided by the total number of nucleons $A$:

$\mathbf{E_{bn} = \frac{E_b}{A} = \frac{[Z m_p + (A - Z) m_n - M]c^2}{A}}$

$E_{bn}$ represents the average energy required to remove one nucleon from the nucleus. Figure 13.1 shows the plot of $E_{bn}$ versus mass number $A$.

Key features of the $E_{bn}$ curve:

• $E_{bn}$ is nearly constant ($\approx 8$ MeV) for intermediate mass nuclei ($30 < A < 170$). This indicates that nuclear forces are short-ranged and saturate; a nucleon interacts only with its nearest neighbours.
• $E_{bn}$ is lower for light nuclei ($A < 30$) and heavy nuclei ($A > 170$).
• The peak around $A=56$ (iron) means these nuclei are the most stable.
• Heavy nuclei (high A) have lower $E_{bn}$ than intermediate nuclei. Fission (breaking heavy nucleus into intermediate ones) releases energy.
• Light nuclei (low A) have lower $E_{bn}$ than heavier fused nuclei. Fusion (combining light nuclei) releases energy.

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Nuclear Force

The **nuclear force** is the strong attractive force that binds protons and neutrons together in the nucleus, overcoming the electrostatic repulsion between protons. It is much stronger than Coulomb and gravitational forces.

Properties of the nuclear force:

• Much stronger than electromagnetic and gravitational forces.
• It is a **short-range force**, falling rapidly to zero beyond a few femtometres ($\sim 0.8$ fm). This explains the saturation of binding energy per nucleon for larger nuclei.
• It is attractive at distances greater than about 0.8 fm, but becomes strongly repulsive at very short distances (less than 0.8 fm), preventing the nucleons from collapsing into each other.
• It is approximately the same between any pair of nucleons (neutron-neutron, proton-neutron, proton-proton). It does not depend on the electric charge.

Unlike Coulomb's law, there is no simple mathematical formula for the nuclear force.

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Radioactivity

**Radioactivity**, discovered by A. H. Becquerel in 1896, is a nuclear phenomenon where unstable nuclei spontaneously transform by emitting particles and/or energy. This process is called **radioactive decay**. Three main types occur naturally:

• (i) $\alpha$-decay: Emission of an alpha particle, which is a helium nucleus ($^4_2$He). Mass number $A$ decreases by 4, atomic number $Z$ decreases by 2. E.g., $^{238}_{92}$U $\to$ $^{234}_{90}$Th + $^4_2$He. Occurs only if the total mass of products is less than the parent nucleus mass (positive Q-value).
• (ii) $\beta$-decay: Emission of an electron ($\beta^-$-decay) or a positron ($\beta^+$-decay). Mass number $A$ remains unchanged. In $\beta^-$-decay, a neutron converts to a proton ($n \to p + e^- + \bar{\nu}$), so $Z$ increases by 1. In $\beta^+$-decay, a proton converts to a neutron ($p \to n + e^+ + \nu$), so $Z$ decreases by 1. $\beta$-decay involves the weak nuclear force and is accompanied by emission of antineutrino ($\bar{\nu}$) in $\beta^-$ decay and neutrino ($\nu$) in $\beta^+$ decay.
• (iii) $\gamma$-decay: Emission of high-energy photons ($\gamma$-rays) from an excited nucleus transitioning to a lower energy state. Nuclear energy level spacings are in MeV range. Typically follows $\alpha$ or $\beta$ decay if the daughter nucleus is in an excited state.

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Law Of Radioactive Decay

The number of nuclei undergoing radioactive decay per unit time is proportional to the total number of radioactive nuclei ($N$) in the sample. This is the law of radioactive decay. If $dN$ is the change in $N$ in time $dt$, then $dN/dt \propto N$. Since $N$ decreases, $dN/dt = -\lambda N$, where $\lambda$ is the **decay constant** or disintegration constant.

Integrating this gives the number of undecayed nuclei at time $t$:

$\mathbf{N(t) = N_0 e^{-\lambda t}}$

where $N_0$ is the initial number of nuclei at $t=0$.

The total **decay rate ($R$)** or **activity** of the sample is the number of decays per unit time: $R = -dN/dt = \lambda N$.

$\mathbf{R(t) = \lambda N(t) = \lambda N_0 e^{-\lambda t} = R_0 e^{-\lambda t}}$

where $R_0 = \lambda N_0$ is the initial activity. Activity is the experimentally measurable quantity.

The SI unit for activity is the **becquerel (Bq)**: $1 \text{ Bq} = 1$ decay/second. Another unit is the **curie (Ci)**: $1 \text{ Ci} = 3.7 \times 10^{10}$ decays/second.

Characteristic time scales for decay:

• **Half-life ($T_{1/2}$):** Time for half of the initial nuclei to decay. $N(T_{1/2}) = N_0/2$. Using $N(t)=N_0e^{-\lambda t}$: $N_0/2 = N_0 e^{-\lambda T_{1/2}} \implies 1/2 = e^{-\lambda T_{1/2}} \implies e^{\lambda T_{1/2}} = 2 \implies \lambda T_{1/2} = \ln 2$.

$\mathbf{T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}}$.

• **Mean life ($\tau$):** Average lifetime of a nucleus. $\mathbf{\tau = 1/\lambda}$.

Relationship: $T_{1/2} = \tau \ln 2 \approx 0.693 \tau$.

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Alpha Decay

**Alpha decay ($\alpha$-decay)** is a type of radioactive decay where an unstable nucleus emits an alpha particle ($^4_2$He nucleus). This transforms the parent nucleus into a daughter nucleus with a mass number decreased by 4 and an atomic number decreased by 2.

General $\alpha$-decay equation: $\mathbf{^A_Z X \to ^{A-4}_{Z-2} Y + ^4_2 He}$.

Example: $^{238}_{92}$U $\to$ $^{234}_{90}$Th + $^4_2$He.

$\alpha$-decay is energetically possible only if the mass of the parent nucleus is greater than the sum of the masses of the daughter nucleus and the alpha particle. The energy released in the decay is called the **disintegration energy** or **Q-value**, given by $\mathbf{Q = (m_X - m_Y - m_{He})c^2}$. If the parent nucleus is at rest, the products' kinetic energy equals the Q-value. $Q > 0$ for spontaneous decay.

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Beta Decay

**Beta decay ($\beta$-decay)** is a type of radioactive decay where a nucleus emits an electron ($\beta^-$-decay) or a positron ($\beta^+$-decay) and a neutrino ($\nu$) or antineutrino ($\bar{\nu}$).

• $\beta^-$-decay: A neutron transforms into a proton, emitting an electron ($e^-$) and an antineutrino ($\bar{\nu}$). The atomic number $Z$ increases by 1, mass number $A$ stays the same. $\mathbf{^A_Z X \to ^A_{Z+1} Y + e^- + \bar{\nu}}$.
• $\beta^+$-decay: A proton transforms into a neutron, emitting a positron ($e^+$) and a neutrino ($\nu$). The atomic number $Z$ decreases by 1, mass number $A$ stays the same. $\mathbf{^A_Z X \to ^A_{Z-1} Y + e^+ + \nu}$.

Neutrinos and antineutrinos are nearly massless, neutral particles with very weak interactions, making them hard to detect. They are emitted to conserve energy, momentum, and other quantities (like lepton number) in beta decay.

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Gamma Decay

**Gamma decay ($\gamma$-decay)** occurs when a nucleus in an excited energy state transitions to a lower energy state, emitting a high-energy photon ($\gamma$-ray). Nuclear energy levels are much higher (MeV range) than atomic levels (eV range). $\gamma$-decay does not change the mass number ($A$) or atomic number ($Z$) of the nucleus, only its energy state.

$\gamma$-emission typically follows $\alpha$ or $\beta$ decay when the daughter nucleus is left in an excited state. The excited nucleus then decays to the ground state through one or more $\gamma$-ray emissions.

Nuclear Energy

The binding energy curve (Fig. 13.1) shows that nuclei with intermediate mass numbers are more stable (higher $E_{bn}$) than very light or very heavy nuclei. This implies that nuclear reactions can release energy if they transform less tightly bound nuclei into more tightly bound nuclei.

• **Fission:** A heavy nucleus breaks into lighter fragments, where the fragments have higher $E_{bn}$ than the original nucleus.
• **Fusion:** Two light nuclei combine to form a heavier nucleus, where the fused nucleus has higher $E_{bn}$ than the original nuclei.

Nuclear reactions involve energies in the MeV range, about a million times larger than the eV energies in chemical reactions. This makes nuclear processes extremely powerful energy sources.

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Fission

**Nuclear fission** is a process where a heavy nucleus splits into two or more lighter nuclei, releasing a large amount of energy and often a few neutrons. Neutron-induced fission is particularly important. For example, $^{235}_{92}$U undergoes fission when bombarded by a slow neutron, producing intermediate fragments and typically 2 or 3 neutrons.

$\mathbf{^1_0 n + ^{235}_{92}U \to ^{236}_{92}U^* \to \text{Fragments} + \text{Neutrons} + \text{Energy}}$

Example: $^1_0 n + ^{235}_{92}U \to ^{144}_{56}Ba + ^{89}_{36}Kr + 3^1_0 n + \text{Energy}$.

The Q-value (energy released) in uranium fission is around 200 MeV per fission event. This energy comes from the increase in binding energy per nucleon when nucleons in the less stable heavy nucleus rearrange into more stable intermediate nuclei. The fragments are usually radioactive and undergo further $\beta$ decays.

The release of multiple neutrons per fission is crucial for a **chain reaction**, where neutrons from one fission trigger further fissions. This is the basis for nuclear reactors (controlled chain reaction) and nuclear bombs (uncontrolled, rapid chain reaction).

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Nuclear Reactor

A **nuclear reactor** is a device that sustains and controls a nuclear chain reaction, typically fission, to produce energy (usually heat, which is then converted to electricity). Key components:

• **Fuel:** Fissionable material (e.g., enriched uranium, plutonium) in fabricated form.
• **Moderator:** Material (water, heavy water, graphite) to slow down fast neutrons released in fission to thermal neutrons, which are more effective at causing fission in $^{235}$U.
• **Control rods:** Made of neutron-absorbing material (cadmium, boron) to regulate the chain reaction rate by controlling the number of neutrons available. Inserting or withdrawing rods changes the multiplication factor $K$ (ratio of fissions in one generation to the previous). $K=1$ for critical (steady power), $K>1$ is supercritical (power increases), $K<1$ is subcritical (reaction dies out). Safety rods can shut down the reactor quickly.
• **Coolant:** Fluid (water, gas, liquid metal) to remove the heat generated by fission.
• **Reflector:** Material surrounding the core to reflect neutrons back into the core, reducing leakage.
• **Shielding:** Protective layers to absorb harmful radiation (neutrons, gamma rays) from the core.
• **Containment vessel:** Strong structure to prevent release of radioactive materials.

Heat removed by the coolant is used to generate steam, which drives turbines to produce electricity. Nuclear reactors produce radioactive waste requiring careful handling and disposal. India's atomic energy program focuses on using abundant thorium alongside uranium via a three-stage strategy.

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Nuclear Fusion – Energy Generation In Stars

**Nuclear fusion** is a process where two or more light nuclei combine (fuse) to form a heavier nucleus, releasing a large amount of energy. This is energetically favourable for light nuclei because the heavier fused nucleus is more tightly bound (higher $E_{bn}$) than the original light nuclei.

Examples of fusion reactions involving hydrogen isotopes:

• $^1_1$H + $^1_1$H $\to$ $^2_1$H + $e^+$ + $\nu_e$ + 0.42 MeV
• $^2_1$H + $^2_1$H $\to$ $^3_2$He + $n$ + 3.27 MeV
• $^2_1$H + $^2_1$H $\to$ $^3_1$H + $^1_1$H + 4.03 MeV

Fusion requires the participating nuclei to come extremely close for the short-range strong nuclear force to overcome their electrostatic (Coulomb) repulsion. This requires high kinetic energies, meaning extremely high temperatures. This is called **thermonuclear fusion**. Temperatures on the order of $10^8$ K are needed.

Thermonuclear fusion is the source of energy in stars, including the Sun. The process in the Sun is a multi-step **proton-proton (p-p) cycle**, effectively converting 4 hydrogen nuclei into one helium nucleus with a release of about 26.7 MeV of energy.

Stars can fuse elements up to those near the peak of the binding energy curve (around iron). Fusion of heavier elements is generally endothermic.

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Controlled Thermonuclear Fusion

Replicating thermonuclear fusion on Earth in a controlled manner to generate power is a major scientific and technological challenge. It requires heating fuel (like deuterium and tritium) to extremely high temperatures ($> 10^8$ K) to create a plasma, and then confining this plasma using strong magnetic fields (since no material container can withstand such temperatures). Research on controlled fusion is ongoing worldwide (e.g., the ITER project). Success would provide a clean and virtually unlimited energy source.

Uncontrolled fusion, triggered by a fission bomb, is the principle behind hydrogen bombs.

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SUMMARY

This chapter delves into the properties and behaviour of atomic nuclei.

• Atom has a nucleus; radius $\sim 10^{-4}$ times atom radius. Nucleus contains $>99.9\%$ of atom's mass.
• Atomic mass unit (u): $1\text{u} = 1/12$ mass of $^{12}$C atom $\approx 1.66 \times 10^{-27}$ kg.
• Nucleus composition: Protons ($Z$) and neutrons ($N$). $A = Z+N$ (mass number). Protons have charge $+e$, neutrons are neutral. Nucleons (protons/neutrons) are bound by strong nuclear force.
• Isotopes: Same $Z$, different $N$. Isobars: Same $A$. Isotones: Same $N$.
• Nuclear radius $R = R_0 A^{1/3}$, $R_0 \approx 1.2$ fm. Nuclear density is constant $\sim 10^{17}$ kg/m$^3$.
• Mass-energy equivalence: $E=mc^2$. Used to define binding energy.
• Mass defect $\Delta M = [Zm_p + (A-Z)m_n] - M_{nucleus}$. Binding energy $E_b = \Delta M c^2$. Energy to break nucleus.
• Binding energy per nucleon $E_{bn} = E_b/A$. Measure of nuclear stability. Peaks at $A \approx 56$. Nearly constant (saturation) for $30 < A < 170$.
• Nuclear force: Strong, short-range, attractive (at $\sim 0.8$ fm), repulsive at very short distances. Independent of charge.
• Radioactivity: Unstable nuclei decay spontaneously. $\alpha$-decay ($^4_2$He), $\beta$-decay ($e^-$ or $e^+$), $\gamma$-decay (photon).
• Radioactive decay law: $N(t) = N_0 e^{-\lambda t}$. Activity $R = \lambda N = R_0 e^{-\lambda t}$. $T_{1/2} = \frac{\ln 2}{\lambda} = \tau \ln 2$. $R$ is decays/second (Bq or Ci).
• Nuclear energy: Released from transmutation to more tightly bound nuclei. $Q = \Delta M c^2 > 0$.
• Fission: Heavy nucleus splits ($A > 170 \to 30 < A < 170$), releases energy and neutrons. Used in reactors (controlled chain reaction, $K=1$) and bombs (uncontrolled).
• Fusion: Light nuclei combine ($A \le 10 \to > 10$), releases energy. Requires high temperature (thermonuclear). Source of energy in stars. Controlled fusion is research area.

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Points to Ponder

Further thoughts on concepts:

• Nuclear density is constant, unlike atomic density.
• Electron scattering senses charge distribution; $\alpha$-scattering senses nuclear matter.
• Mass and energy conservation are unified by $E=mc^2$. Key evidence from nuclear physics.
• $E_{bn}$ curve shows possibility of exothermic fission and fusion.
• Fusion needs high temp to overcome Coulomb repulsion.
• Peaks in $E_{bn}$ suggest nuclear shell structure.
• $e^-$, $e^+$ are particle-antiparticle pair. Annihilation. Neutrino/antineutrino in $\beta$ decay.
• Free neutron unstable; free proton stable (in nucleus).
• $\gamma$-emission from nucleus deexcitation shows discrete nuclear energy levels.
• Radioactivity indicates nuclear instability (neutron/proton ratio deviation from stability).

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Summary of Physical Quantities:

Physical Quantity	Symbol	Dimensions	Unit	Remarks
Atomic mass unit	u	[M]	u	1/12th mass of $^{12}$C atom
Disintegration constant	$\lambda$	[T$^{-1}$]	s$^{-1}$	Rate of decay
Half-life	$T_{1/2}$	[T]	s	Time for half decay
Mean life	$\tau$	[T]	s	Time for $1/e$ decay
Activity	$R$	[T$^{-1}$]	Bq (Becquerel)	Decays per second

You may find the following data useful in solving the exercises:

$e = 1.6 \times 10^{–19}C$

$N = 6.023 \times 10^{23}$ per mole

$1/(4\pi\epsilon_0) = 9 \times 10^9 N m^2/C^2$

$k = 1.381 \times 10^{–23}J K^{–1}$

$1 \text{ MeV} = 1.6 \times 10^{–13}J$

$1 \text{ u} = 931.5 \text{ MeV}/c^2$

$1 \text{ year} = 3.154 \times 10^7 s$

$m_H = 1.007825 \text{ u}$

$m_n = 1.008665 \text{ u}$

$m(^4_2\text{He}) = 4.002603 \text{ u}$

$m_e = 0.000548 \text{ u}$

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ADDITIONAL EXERCISES

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