CHM 101 German Objectives- Combined-3.docx

# Atomic structure and electron configuration This section delves into the fundamental building blocks of matter, exploring the composition of atoms, the properties of their constituent particles, and the arrangement of electrons within an atom. ### 1.1 The atom: fundamental particles and properties * An atom is the smallest particle of an element that can participate in any chemical change. * Atoms consist principally of three fundamental particles: protons, neutrons, and electrons. * The maximum mass of an atom is concentrated in its nucleus. * The nucleus of an atom contains protons and neutrons. * Particles that revolve around the nucleus are called electrons. * The mass of a proton is almost equal to the mass of a neutron, and both are significantly larger than the mass of an electron. Specifically, the mass of a proton is approximately 1836 times that of an electron. #### 1.1.1 Atomic number * The atomic number ($Z$) of an atom is defined as the number of protons in the nucleus. * The atomic number determines the element. * The number of protons and the number of electrons are always equal in a neutral atom. * If an atom has a given atomic number, it has that number of protons. For example, beryllium, which has 4 protons, has an atomic number of 4. #### 1.1.2 Mass number * The mass number ($A$) of an atom is the sum of the number of protons and neutrons in its nucleus. * It represents the total number of nucleons (protons and neutrons) in the nucleus. * The formula for the mass number is $A = Z + N$, where $Z$ is the atomic number and $N$ is the number of neutrons. #### 1.1.3 Isotopes * Isotopes are atoms of the same element that contain the same number of protons but different numbers of neutrons. * This means isotopes of an element have the same atomic number but different mass numbers. * Consequently, isotopes differ in the number of neutrons. * Example: Chlorine has two common isotopes, chlorine-35 ($^{35}$Cl) and chlorine-37 ($^{37}$Cl). * The number of neutrons in $^{238}_{94}$Pu is calculated as mass number - atomic number = $238 - 94 = 144$. #### 1.1.4 Atomic mass and relative atomic mass * The term for the average mass of isotopes of an element, taking into account their relative abundances, is the atomic weight or relative atomic mass. * The average atomic mass of an element depends on the abundance of its isotopes. * **Example:** Bromine is composed of $^{79}_{35}$Br with a mass of 78.9183 amu and $^{81}_{35}$Br with a mass of 80.9163 amu. A sample is 50.69% Br-79 and 49.31% Br-81. The atomic weight of bromine is calculated as: $(0.5069 \times 78.9183) + (0.4931 \times 80.9163) = 39.9963 + 39.9031 = 79.90$ amu. * The mass of 0.5 mol of NaCl (molar mass of NaCl ≈ 58.44 g/mol) is $0.5 \times 58.44 = 29.22$ grams. * The molar mass of H₂SO₄ is approximately $(2 \times 1.008) + 32.06 + (4 \times 15.999) = 98.07$ g/mol. * The molar mass of water (H₂O) is approximately $(2 \times 1.008) + 15.999 = 18.015$ g/mol. * The molar mass of NH₄NO₃ is approximately $(14.007) + (4 \times 1.008) + (14.007) + (3 \times 15.999) = 80.04$ g/mol. * The molar mass of CO₂ is approximately $12.011 + (2 \times 15.999) = 44.009$ g/mol. ### 1.2 Electron configuration The electron configuration of an element describes the arrangement of electrons in its atomic orbitals. #### 1.2.1 Quantum numbers * The arrangement and energy of electrons in an atom are described by a set of quantum numbers. * The four quantum numbers are: 1. **Principal quantum number ($n$):** Designates the main shell or energy level in which an electron resides. Values are positive integers (1, 2, 3, ...). It determines the size of the orbital. 2. **Angular momentum (azimuthal) quantum number ($l$):** Defines the shape of the subshell. Its values range from 0 to $n-1$. * $l=0$ corresponds to an s orbital (spherical shape). * $l=1$ corresponds to a p orbital (dumbbell shape). * $l=2$ corresponds to a d orbital. * $l=3$ corresponds to an f orbital. 3. **Magnetic quantum number ($m_l$):** Determines the orbital orientation in space within a subshell. Its values range from $-l$ to $+l$, including 0. 4. **Spin quantum number ($m_s$):** Describes the intrinsic angular momentum of an electron, which has two possible values: $+1/2$ or $-1/2$ (spin up or spin down). * Quantum numbers of an atom can be defined on the basis of the Schrödinger equation. * The angular momentum quantum number ($l$) determines the shape of the subshell. * The principal quantum number ($n$) indicates the energy level of an electron. For a 3p orbital, the principal quantum number is 3. #### 1.2.2 Atomic orbitals and subshells * A region of space in which the probability of finding an electron is high is called an atomic orbital. * The shape of an s orbital is spherical. * A p orbital has a dumbbell shape. * The p-orbital is further subdivided into three sub-orbitals, otherwise called p orbitals. * The number of orbitals in a d-subshell is 5. * The number of orbitals in the second energy level ($n=2$) is $1 (2s) + 3 (2p) = 4$. * The number of orbitals in a d-subshell is 5. * The maximum number of electrons that can be accommodated in a sublevel for which $l=3$ (an f subshell) is $2(2l+1) = 2(2 \times 3 + 1) = 2(7) = 14$. * The maximum number of electrons that can hold an f-orbital is 14. * A single orbital can hold a maximum of 2 electrons. #### 1.2.3 Electron filling principles * **Aufbau Principle:** Electrons fill the lowest energy orbitals first. According to Aufbau’s principle, 5s will be filled before 4d and 5p. The order of filling is generally: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p. * **Hund's Rule of Maximum Multiplicity:** Electrons fill into orbitals singly before pairing begins within a subshell. All orbitals of the same energy level are occupied by single electrons with parallel spins before any two electrons occupy the same orbital. * **Pauli Exclusion Principle:** No two electrons in an atom can have the same set of four quantum numbers. This implies that an orbital can hold a maximum of two electrons, and these electrons must have opposite spins. #### 1.2.4 Electron configurations of elements * **spdf notation:** This notation represents the arrangement of electrons in shells and subshells. * **Fluorine (A = 9):** $1s^2 2s^2 2p^5$ * **Sulphur (A = 16):** $1s^2 2s^2 2p^6 3s^2 3p^4$ * **Chromium (A = 24):** Chromium is an exception to the Aufbau principle. It has the configuration $1s^2 2s^2 2p^6 3s^2 3p^6 4s^1 3d^5$. This is because a half-filled d-subshell is more stable. * **Nickel (Z = 28):** $1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^8$ * **Oxygen (A = 16):** $1s^2 2s^2 2p^4$ * **Silicon (Z = 14):** $1s^2 2s^2 2p^6 3s^2 3p^2$ * **Argon:** $1s^2 2s^2 2p^6 3s^2 3p^6$ * **Potassium (Z = 19):** $1s^2 2s^2 2p^6 3s^2 3p^6 4s^1$ * **Calcium (Z = 20):** $1s^2 2s^2 2p^6 3s^2 3p^6 4s^2$ * **Ions:** * **Sodium ion, Na²⁺:** To form Na²⁺, two electrons are removed from neutral sodium (Na, Z=11, configuration $1s^2 2s^2 2p^6 3s^1$). The configuration becomes $1s^2 2s^2 2p^6$. * **P³⁻:** Phosphorus (Z=15) has configuration $1s^2 2s^2 2p^6 3s^2 3p^3$. Gaining 3 electrons to form P³⁻ results in $1s^2 2s^2 2p^6 3s^2 3p^6$. * An ion differs from an atom because it has an unequal number of protons and electrons, resulting in a net charge. * An ion with more protons than electrons is called a cation. * When a non-metal gains an electron, it forms an anion. * The chemical reactivity of an element is determined by its valence electrons. * The subatomic particles that determine chemical properties are electrons, particularly valence electrons. * The number of valence electrons often corresponds to the group number (for main group elements). * The group number of an element gives the number of valence electrons. * Elements that possess the same number of electrons have the same chemical properties. * The electron configuration of oxygen in spdf notation is $1s^2 2s^2 2p^4$. * The electronic configuration of an element with atomic number 20 is $1s^2 2s^2 2p^6 3s^2 3p^6 4s^2$. * The element with configuration ending in 3d⁶ is Iron (Fe, Z=26). * The element with configuration ending in 3p³ is Phosphorus (P, Z=15). * The element with configuration ending in 4s¹ is Potassium (K, Z=19). * The spdf configuration for silicon (Z = 14) is $1s^2 2s^2 2p^6 3s^2 3p^2$. * The electronic configuration of an element with atomic number 20 is $1s^2 2s^2 2p^6 3s^2 3p^6 4s^2$. * The neutral atom that has 2 electrons in the first shell, 8 in the second, and 8 in the third has an atomic number of $2 + 8 + 8 = 18$. This element is Argon. * The electronic configuration of Nickel (Z = 28) is $1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^8$. * The orbital that is filled after 3p in the Aufbau principle is 4s. * The 4d orbital belongs to the shell with principal quantum number 5. * The principal quantum number for the 3p orbital is 3. #### 1.2.5 Properties related to electron configuration * **Valency:** The valency of Carbon (C) is typically 4. * **Electronegativity:** The most electronegative element is Fluorine. * **Atomic radius:** Of the species P³⁻, Cl⁻, Ar, K⁺, the one with the largest radius is P³⁻ because it has the most electrons relative to its nuclear charge, leading to weaker attraction. * **Ionization energy:** The Period 3 element with the lowest first ionization energy is Potassium (K). * **Metallic character:** Between Na and Al, Sodium (Na) is more metallic. * **Full K and L shells:** An element with full K ($n=1$) and L ($n=2$) shells has an atomic number of $2 (\text{in K}) + 8 (\text{in L}) = 10$. This element is Neon. * **L shell capacity:** The L shell ($n=2$) can hold up to $2n^2 = 2(2^2) = 8$ electrons. ### 1.3 Historical context of atomic theory * Dalton proposed the atomic theory. * **Dalton's Laws (implied from the document):** While not explicitly listed as "laws," Dalton's atomic theory is based on several postulates, including: * Elements are composed of atoms, which are indivisible and indestructible particles. * Atoms of the same element are identical in mass and properties. * Atoms of different elements differ in mass and properties. * Atoms combine in simple whole-number ratios to form compounds. * Atoms are not created or destroyed in chemical reactions, only rearranged. * By chemical means, chemical elements cannot be broken down into simpler substances. * The electron was discovered by J.J. Thomson. * Charge to mass ratio was first calculated by J.J. Thomson. * The nucleus of an atom is where the protons and neutrons are concentrated. ### 1.4 Calculations involving atomic structure * **Empirical and Molecular Formulas:** * **Example:** A compound contains 14.4% hydrogen and 85.6% carbon by mass. To find the empirical formula, assume 100 g of the compound. This gives 14.4 g H and 85.6 g C. * Moles of H = $14.4 \text{ g} / 1.01 \text{ g/mol} \approx 14.26$ mol * Moles of C = $85.6 \text{ g} / 12.01 \text{ g/mol} \approx 7.13$ mol * Ratio H:C = $14.26 / 7.13 : 7.13 / 7.13 \approx 2:1$. * Empirical formula is CH₂. * **Example:** Determine the empirical formula for chrysotile asbestos with 28.03% Mg, 21.60% Si, 1.16% H, and 49.21% O. * Moles Mg = $28.03 / 24.305 \approx 1.153$ * Moles Si = $21.60 / 28.085 \approx 0.769$ * Moles H = $1.16 / 1.008 \approx 1.151$ * Moles O = $49.21 / 15.999 \approx 3.076$ * Divide by the smallest (0.769): Mg ≈ 1.5, Si = 1, H ≈ 1.5, O ≈ 4. * Multiply by 2 to get whole numbers: Mg = 3, Si = 2, H = 3, O = 8. * Empirical formula: Mg₃Si₂H₃O₈. * If the empirical formula is CH₂O and the molar mass is 180 g/mol, the molecular formula is found by calculating the empirical formula mass (12.01 + 2*1.008 + 15.999 ≈ 30.026 g/mol). The multiple is $180 / 30.026 \approx 6$. Molecular formula = $(CH_2O)_6 = C_6H_{12}O_6$. * **Percent Composition by Weight:** * **Example:** Calculate the percent by weight of carbon in 154 g of C₄H₈O₃. * Molar mass of C₄H₈O₃ = $(4 \times 12.01) + (8 \times 1.008) + (3 \times 15.999) = 48.04 + 8.064 + 47.997 = 104.1$ g/mol. * Mass of carbon in one mole = $4 \times 12.01 = 48.04$ g. * Percent by weight of carbon = $(48.04 / 104.1) \times 100\% \approx 46.15\%$. * **Example:** Find the percent composition S in N₂S₂. * Molar mass of N₂S₂ = $(2 \times 14.01) + (2 \times 32.06) = 28.02 + 64.12 = 92.14$ g/mol. * Mass of sulfur in one mole = $2 \times 32.06 = 64.12$ g. * Percent composition of S = $(64.12 / 92.14) \times 100\% \approx 69.60\%$. * **Atomic Weight Calculation:** * The atomic weight of magnesium is calculated using the abundance of its isotopes. * The relative atomic mass of copper can be determined from its mass spectrum. * **Number of particles:** * The number of atoms in 2 moles of CO₂ is $2 \times \text{Avogadro's number} = 2 \times 6.022 \times 10^{23} = 1.2044 \times 10^{24}$ molecules. * The number of moles in 88 g of CO₂ (molar mass ≈ 44 g/mol) is $88 \text{ g} / 44 \text{ g/mol} = 2$ mol. * **Number of protons, neutrons, and electrons:** * In an oxygen atom, the number of nucleons is 16. Since oxygen has atomic number 8 (8 protons), its mass number is $8 + (\text{number of neutrons})$. If the number of nucleons (mass number) is 16, then the number of neutrons is $16 - 8 = 8$. The atomic mass is generally taken as the mass number for isotopes in introductory contexts. * The number of protons and neutrons in $^{14}_6$C are 6 and $14-6=8$, respectively. * The number of protons, electrons, and neutrons in $^{35}_{17}$Cl⁻ are: Protons = 17, Electrons = $17+1 = 18$, Neutrons = $35-17 = 18$. * The number of electrons contained in $^{54}_{24}$Cr is 24, as it is a neutral atom. ### 1.5 Advanced concepts (briefly touched upon in the document) * **Atomic orbital is spherical in shape:** This refers to an s orbital. * **Quantum number that distinguishes s orbitals from p orbitals:** This is the angular momentum quantum number ($l$). For s orbitals, $l=0$, and for p orbitals, $l=1$. * **The spin quantum number of the last electron in oxygen:** Oxygen's configuration is $1s^2 2s^2 2p^4$. The last electron is in a 2p orbital. The electrons in the 2p subshell are paired as follows: 2px¹ 2py¹ 2pz¹. If the last electron added is in 2pz, its spin could be $+1/2$ or $-1/2$. * **The 3d orbital belongs to the shell with principal quantum number 4.** * **The energy required to remove one mole of electrons from one mole of gaseous atoms is called ionization energy.** * **The 4d orbital belongs to the shell with principal quantum number 5.** ### 1.6 Summary of key particles and properties * **Positively charged particle:** Proton. * **Neutral atom with K, L, and M shells full:** This would be an atom with 2 electrons in the first shell, 8 in the second, and 8 in the third, totaling 18 electrons. This neutral atom is Argon (Z=18). * **Neutral atom with full K and L shells:** Atomic number is 10 (Neon). * **Particles responsible for the mass of an atom:** Protons and neutrons. * **Particle with a negative charge:** Electron. * **Particle carrying a positive charge:** Proton. * **Subatomic particles in an atom:** Protons, neutrons, and electrons. * **The pathway of an electron:** This is generally referred to as an orbital. * **The fundamental particles of an atom:** Protons, neutrons, and electrons. --- # Chemical bonding and molecular structure This topic explores the nature of chemical bonds, including ionic and covalent bonding, and how these bonds determine the structure and properties of molecules. ### 2.1 Fundamental particles and atomic structure Atoms are the fundamental building blocks of all matter. They are composed of three primary subatomic particles: protons, neutrons, and electrons. * **Protons:** Positively charged particles located in the nucleus. The number of protons defines the atomic number of an element and determines its chemical identity. The symbol for proton number is $Z$. * **Neutrons:** Neutrally charged particles also found in the nucleus. Protons and neutrons are collectively referred to as nucleons and are responsible for the mass of an atom. * **Electrons:** Negatively charged particles that revolve around the nucleus in specific pathways called orbitals. The discovery of the electron is attributed to J.J. Thomson. The mass number ($A$) of an atom is the sum of its protons and neutrons: $$ A = \text{number of protons} + \text{number of neutrons} $$ An ion is an atom that has a different number of electrons than protons, resulting in a net electrical charge. A neutral atom has an equal number of protons and electrons. Elements with the same number of protons but different numbers of neutrons are called isotopes. ### 2.2 Electron configuration and quantum numbers The arrangement of electrons in an atom is known as its electron configuration, which can be described using spdf notation. This arrangement is governed by several principles and quantum numbers: * **Aufbau Principle:** Electrons fill the lowest energy orbitals first. * **Pauli Exclusion Principle:** No two electrons in an atom can have the same set of four quantum numbers. This also implies that a single orbital can hold a maximum of two electrons, and these electrons must have opposite spins. * **Hund's Rule of Maximum Multiplicity:** Electrons fill orbitals of the same energy singly before pairing begins. The quantum numbers describe the properties of atomic orbitals and electrons: * **Principal quantum number ($n$):** Designates the main energy level or shell an electron resides in. * **Angular momentum quantum number ($l$):** Defines the shape of the subshell and orbital. * $l=0$ corresponds to an s orbital, which is spherical in shape. * $l=1$ corresponds to a p orbital, which has a dumbbell shape. * $l=2$ corresponds to a d orbital, with more complex shapes. * $l=3$ corresponds to an f orbital, with even more complex shapes. The number of orbitals in a given energy level ($n$) is $n^2$. The L shell (n=2) can hold up to 8 electrons. The maximum number of electrons that can be accommodated in a sublevel for which $l = 3$ is $2(2l+1) = 2(2 \times 3 + 1) = 14$ electrons. * **Magnetic quantum number ($m_l$):** Determines the orientation of the orbital in space. * **Spin quantum number ($m_s$):** Describes the intrinsic angular momentum of an electron, which can be either spin-up ($+\frac{1}{2}$) or spin-down ($-\frac{1}{2}$). **Example:** The electronic configuration for oxygen (atomic number 8) in spdf notation is $1s^22s^22p^4$. The chemical properties of an element are determined by its valence electrons, which are the outermost electrons. ### 2.3 Chemical bonding Chemical bonding refers to the attractive forces that hold atoms together in compounds. These bonds typically involve the valence electrons. The electronegativity difference between two elements plays a crucial role in determining the type of bond formed. #### 2.3.1 Ionic bonding Ionic bonding occurs when there is a large electronegativity difference between two elements. One atom (typically a metal) loses electrons to form a positive ion (cation), and another atom (typically a non-metal) gains these electrons to form a negative ion (anion). The electrostatic attraction between these oppositely charged ions forms the ionic bond. **Example:** Sodium (Na) loses one electron to form a $\text{Na}^+$ ion, which is isoelectronic with Neon (Ne). Fluorine (F) gains an electron to form an $\text{F}^-$ ion. The compound formed is sodium fluoride (NaF). #### 2.3.2 Covalent bonding Covalent bonding occurs when atoms share electrons to achieve a stable electron configuration, typically a full outer shell. This type of bonding is common between non-metal elements where the electronegativity difference is smaller. * A single covalent bond involves sharing one pair of electrons. * A double covalent bond involves sharing two pairs of electrons. * A triple covalent bond involves sharing three pairs of electrons. **Example:** In hydrogen fluoride (HF), hydrogen shares one electron with fluorine, and fluorine shares one electron with hydrogen. In carbon dioxide ($\text{CO}_2$), carbon forms double bonds with each oxygen atom, sharing two pairs of electrons with each. ### 2.4 Molecular structure and empirical formulas The way atoms are arranged in a molecule, determined by the types and number of chemical bonds, is its molecular structure. * **Empirical Formula:** The simplest whole-number ratio of atoms of each element in a compound. **Example:** Analysis of a covalent compound showed it contained 14.4% hydrogen and 85.6% carbon by mass. To find the empirical formula, assume 100 g of the compound: 14.4 g H and 85.6 g C. Convert to moles: $14.4 \text{ g H} / 1.01 \text{ g/mol H} \approx 14.3 \text{ mol H}$ and $85.6 \text{ g C} / 12.01 \text{ g/mol C} \approx 7.13 \text{ mol C}$. The ratio of C to H is approximately 1:2. The empirical formula is $\text{CH}_2$. * **Molecular Formula:** The actual number of atoms of each element in a molecule. It is a multiple of the empirical formula. **Example:** If the empirical formula is $\text{CH}_2\text{O}$ and the molar mass is 180 g/mol, the molecular formula is calculated by finding the ratio of the molar mass to the empirical formula mass ($12.01 + 2(1.01) + 16.00 = 30.03$ g/mol). The ratio is $180 / 30.03 \approx 6$. Therefore, the molecular formula is $(\text{CH}_2\text{O})_6$, which is $\text{C}_6\text{H}_{12}\text{O}_6$. ### 2.5 Other relevant concepts * **Valency:** The combining power of an element, often related to the number of electrons it can gain, lose, or share. The valency of Carbon (C) is typically 4. * **Period Number:** In the periodic table, the period number indicates the principal energy level of the valence electrons. * **Atomic Radius:** The size of an atom. Species with the same number of electrons but different nuclear charges will have different radii; for example, among $\text{P}^{3-}$, $\text{Cl}^-$, Ar, and $\text{K}^+$, the one with the largest radius is $\text{P}^{3-}$ due to having the most electron shells and the least effective nuclear charge. * **First Ionization Energy:** The energy required to remove one mole of electrons from one mole of gaseous atoms. Elements in the same period generally show an increase in first ionization energy across the period. The Period 3 element with the lowest first ionization energy is Sodium (Na). * **Metallic Character:** The tendency of an element to lose electrons. Sodium (Na) is more metallic than Aluminum (Al). * **Electronegativity:** The ability of an atom to attract electrons in a covalent bond. Fluorine is the most electronegative element. * **Inner Electrons:** Electrons in shells closer to the nucleus reduce the pull of the nucleus on outer electrons due to the shielding effect. * **Noble Gases:** Found in Group 18 of the periodic table, these elements have full valence electron shells and are generally unreactive. * **Catalyst:** A substance that increases the rate of a chemical reaction without being consumed. In biological systems, these are called enzymes. * **Exothermic Reaction:** A reaction that releases energy, often in the form of heat, into the surroundings. The sign of $q$ for an exothermic process is negative, as is its enthalpy change ($\Delta H$). * **Endothermic Reaction:** A reaction that absorbs energy from the surroundings. * **Radioactivity:** The spontaneous emission of particles or energy from an unstable atomic nucleus. Products of radioactive decay include alpha particles ($\alpha$), beta particles ($\beta$), and gamma rays ($\gamma$). The rate of radioactive decay is not influenced by external factors like temperature or pressure but depends on the stability of the nucleus. The half-life of a radioactive isotope is the time required for half of the nuclide to decay. * **Nuclear Fission:** The splitting of a heavy nucleus into two or more lighter nuclei. * **Nuclear Fusion:** The merging of two light nuclei to form a heavier nucleus, releasing a large amount of energy. * **Mass Defect:** The difference in mass between a nucleus and its constituent nucleons, which is converted into energy according to Einstein's equation $E=mc^2$. * **Binding Energy:** The energy required to decompose a nucleus into its constituent nucleons. --- # Gas laws and kinetic theory This topic explores the macroscopic behavior of gases as described by empirical gas laws and explains these behaviors through the microscopic model of the kinetic theory of gases. ### 3.1 The kinetic theory of gases The kinetic theory of gases provides a microscopic explanation for the behavior of gases, based on several key assumptions about the nature of gas particles. #### 3.1.1 Assumptions of the kinetic theory The kinetic theory of gases is built upon the following postulates: * **Particles in motion:** Gases are composed of a large number of tiny particles (atoms or molecules) that are in constant, random, and straight-line motion. * **Negligible particle volume:** The volume occupied by the gas particles themselves is negligible compared to the total volume of the container. In essence, gas molecules are considered point masses. * **No intermolecular forces:** There are no attractive or repulsive forces between gas particles. They only interact during collisions. * **Elastic collisions:** Collisions between gas particles and between particles and the walls of the container are perfectly elastic. This means that kinetic energy is conserved during these collisions; no energy is lost or gained. * **Kinetic energy and temperature:** The average kinetic energy of the gas particles is directly proportional to the absolute temperature of the gas. #### 3.1.2 Macroscopic implications of kinetic theory The kinetic theory helps explain observable gas properties: * **Pressure:** Pressure exerted by a gas is due to the collisions of gas molecules with the walls of the container. * **Temperature:** Temperature is a measure of the average kinetic energy of the gas molecules. At a given temperature, all gases have the same average kinetic energy, though molecules of lighter gases move faster than those of heavier gases. * **Volume and shape:** Gases have neither a defined volume nor a defined shape. They expand to fill their container. * **Compressibility:** Gases are highly compressible because the interatomic distances are large, and the molecules rarely interact. ### 3.2 Empirical gas laws Empirical gas laws describe the relationships between macroscopic properties of gases: pressure ($P$), volume ($V$), temperature ($T$), and the amount of gas (in moles, $n$). #### 3.2.1 Boyle's Law Boyle's Law states that at a constant temperature and a fixed amount of gas, the volume of the gas is inversely proportional to its pressure. Mathematically, this can be expressed as: $$ P \propto \frac{1}{V} $$ Or, $$ PV = k_1 $$ where $k_1$ is a constant. For a given amount of gas at constant temperature, if the initial conditions are $P_1$ and $V_1$, and the final conditions are $P_2$ and $V_2$, then: $$ P_1V_1 = P_2V_2 $$ > **Tip:** Imagine a helium balloon. If you take it deep underwater, the increased external pressure will cause it to shrivel as the volume of the helium inside decreases. #### 3.2.2 Charles's Law Charles's Law states that at a constant pressure and a fixed amount of gas, the volume of the gas is directly proportional to its absolute temperature. Mathematically, this can be expressed as: $$ V \propto T $$ Or, $$ \frac{V}{T} = k_2 $$ where $k_2$ is a constant and $T$ is in Kelvin. For a given amount of gas at constant pressure, if the initial conditions are $V_1$ and $T_1$, and the final conditions are $V_2$ and $T_2$, then: $$ \frac{V_1}{T_1} = \frac{V_2}{T_2} $$ > **Example:** If you take a balloon to a cold place like the North Pole, it will shrink as the helium cools and decreases in volume. Conversely, on a hot tropical island, the balloon will expand as the helium heats up and increases in volume. #### 3.2.3 Avogadro's Law Avogadro's Law states that at a constant pressure and temperature, the volume of a gas is directly proportional to the number of moles of gas. Mathematically, this can be expressed as: $$ V \propto n $$ Or, $$ \frac{V}{n} = k_3 $$ where $k_3$ is a constant. This law implies that equal volumes of all gases at the same temperature and pressure contain the same number of molecules. ### 3.3 The Ideal Gas Law The Ideal Gas Law combines the relationships described by Boyle's Law, Charles's Law, and Avogadro's Law into a single equation that describes the behavior of an ideal gas. It relates pressure ($P$), volume ($V$), the number of moles ($n$), and absolute temperature ($T$). The Ideal Gas Law is given by: $$ PV = nRT $$ where $R$ is the ideal gas constant. The value of $R$ depends on the units used for $P$, $V$, and $T$. Common values include: * $0.0821 \text{ L} \cdot \text{atm} / (\text{mol} \cdot \text{K})$ * $8.314 \text{ J} / (\text{mol} \cdot \text{K})$ * $62.36 \text{ L} \cdot \text{Torr} / (\text{mol} \cdot \text{K})$ The Ideal Gas Law assumes that gas molecules have no volume and experience no intermolecular forces, which is an approximation that holds best at low pressures and high temperatures. > **Example:** To calculate the moles of nitrogen gas needed to pressurize a 60-liter air bag to 2.37 atm at 25°C, you would use the ideal gas law: > $V = 60 \text{ L}$ > $P = 2.37 \text{ atm}$ > $T = 25^\circ\text{C} = 298.15 \text{ K}$ > $R = 0.0821 \text{ L} \cdot \text{atm} / (\text{mol} \cdot \text{K})$ > > $$ n = \frac{PV}{RT} = \frac{(2.37 \text{ atm})(60 \text{ L})}{(0.0821 \text{ L} \cdot \text{atm} / (\text{mol} \cdot \text{K}))(298.15 \text{ K})} \approx 5.79 \text{ mol} $$ ### 3.4 Diffusion and Effusion Diffusion and effusion are phenomena related to the movement of gas particles. #### 3.4.1 Diffusion Diffusion is the process by which particles of one gas spread out and mix with particles of another gas, driven by the random motion of the particles. #### 3.4.2 Graham's Law of Diffusion and Effusion Graham's Law describes the rates of diffusion and effusion of gases. It states that the rate of diffusion or effusion of a gas is inversely proportional to the square root of its molar mass. Mathematically, for two gases A and B: $$ \frac{\text{Rate of diffusion of A}}{\text{Rate of diffusion of B}} = \sqrt{\frac{\text{Molar mass of B}}{\text{Molar mass of A}}} $$ > **Example:** Hydrogen ($H_2$, molar mass $\approx 2.0 \text{ g/mol}$) diffuses much faster than oxygen ($O_2$, molar mass $\approx 32.0 \text{ g/mol}$). The rate of diffusion of hydrogen to oxygen is approximately: > $$ \sqrt{\frac{32.0 \text{ g/mol}}{2.0 \text{ g/mol}}} = \sqrt{16} = 4 $$ > This means hydrogen diffuses about 4 times faster than oxygen. #### 3.4.3 Effusion Effusion is the process where gas molecules escape from a container through a small opening. Graham's Law also applies to effusion. > **Tip:** If gas A and gas B effuse in the same amount of time, and gas A contains 2 moles while gas B contains 1 mole, and assuming they have the same molar mass, then the rate of effusion for gas A is not necessarily twice that of gas B. The rate of effusion is dependent on molar mass and temperature, not just the number of moles. If the molar masses and temperatures are the same, then the rates would be equal. However, if their molar masses differ, then the comparison must be made using Graham's Law. ### 3.5 Deviations from ideal gas behavior Real gases deviate from ideal gas behavior, particularly at high pressures and low temperatures. These deviations occur because: * **Finite molecular volume:** Real gas molecules do occupy a finite volume, which becomes significant when the molecules are close together (high pressure). * **Intermolecular forces:** Real gas molecules do experience attractive and repulsive forces, which become more important at low temperatures when the molecules move slower and are closer. The Van der Waals equation is a modification of the ideal gas law that accounts for these deviations: $$ \left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT $$ where $a$ and $b$ are constants specific to each gas that correct for intermolecular attractions and molecular volume, respectively. --- # Thermodynamics and energy transformations This section explores the fundamental principles of thermodynamics, focusing on energy interconversion, heat, work, enthalpy, and the classification of thermodynamic systems. ### 4.1 Introduction to thermodynamics Thermodynamics is the branch of physics that deals with the interconversion of various kinds of energy, studying processes in which energy is transferred as heat and as work, along with the changes in physical properties involved. ### 4.2 Laws of thermodynamics The first law of thermodynamics states that energy can neither be created nor destroyed but can only be transformed from one form to another. ### 4.3 Energy transfer: heat and work * **Heat** is a transfer of energy due to a difference in temperature. At a molecular level, heat is a transfer of energy that achieves chaotic or random motion in the surroundings. * **Work** is a transfer of energy that is not due to a temperature difference. Specifically, work arises when an object moves a distance $\Delta x$ against an opposing force $f$, described by the formula $w = f \Delta x$. ### 4.4 Work done by expanding gases The work done when a system expands through a given volume is determined by the opposing pressure. The greater the opposing pressure, the greater the work the system does. * **Free expansion**: Expansion against zero external pressure, where there is no external pressure, results in zero work done. * **Electrical work**: Electrical work is done when a body having a charge $q$ moves through a potential difference $\Delta V$. Work can be completely converted into heat by electrical resistance. ### 4.5 Exothermic and endothermic processes * A **process** where energy is released in the form of heat by the system into the surroundings is an **exothermic process**. The sign of $q$ for an exothermic process is negative, as is its enthalpy change. * An **endothermic process** involves energy absorbed by the system from the surroundings. ### 4.6 Thermodynamic systems and boundaries * A **system** is a part of the universe where we have a special interest. * A **boundary** is a real or imaginary surface that separates the system from its surroundings. ### 4.7 Types of thermodynamic systems * **Open systems** allow the free exchange of energy and matter through the boundary between the system and its surroundings. A cell can be thermodynamically regarded as an open system. * **Closed systems** allow the exchange of energy but not matter. * **Isolated systems** do not allow the exchange of either energy or matter. A thermos flask is an example of an isolated system, designed to keep its contents interacting minimally with the surroundings. ### 4.8 Enthalpy The enthalpy of a thermodynamic system is given by the equation $H = U + PV$, where $U$ is the internal energy, $P$ is the pressure, and $V$ is the volume. ### 4.9 Internal energy Internal energy, denoted by $U$, is the sum of the kinetic and potential energies of the particles that form the system. ### 4.10 Physical properties: intensive and extensive Physical properties of materials and systems can be categorized as either intensive or extensive. * An **intensive property** is a bulk property, meaning it is a local physical property of a system that does not depend on the system size or the amount of material. Examples include friction, temperature ($T$), refractive index ($n$), and density ($\rho$). Density is considered an intensive property because it remains constant regardless of the amount of substance. * An **extensive property** is a quantity that depends on the system size or the amount of material. Examples include heat capacity and mass. ### 4.11 Heat capacity Heat capacity is defined as the heat required to raise the temperature of a system or a unit mass of substance by one degree rise. ### 4.12 State functions and path functions * A **state function** is a quantity that describes the state of a system and is independent of the path by which the system arrived at that state. Examples include internal energy and enthalpy. * A **path function** is a quantity that depends on the history of a system and is dependent on the route by which the system arrives at its present location or state. Examples include heat ($q$) and work ($w$). ### 4.13 Isochoric processes An isochoric process is a thermodynamic process during which the volume of the closed system undergoing such a process remains constant. --- # Radioactivity and nuclear processes This topic explores the phenomena of radioactivity, including the different types of radioactive decay, the concept of half-life, and the processes of nuclear fusion and fission, along with their practical applications. ### 5.1 Radioactivity Radioactivity is the spontaneous emission of radiation from unstable atomic nuclei. The stability of a nucleus is dependent on the balance of forces within it. #### 5.1.1 Types of radioactive decay Radioactive decay involves the emission of particles or energy from the nucleus to achieve a more stable configuration. The rate of reaction during radioactivity is not influenced by external factors. ##### 5.1.1.1 Alpha ($\alpha$) decay Alpha decay involves the emission of an alpha particle, which is a helium nucleus ($_{2}^{4}\text{He}$). This results in a decrease in both the atomic number and the mass number of the parent nucleus. In radioactivity, the emission of an $\alpha$-particle is represented by the following general equation: $$_{Z}^{A}\text{X} \rightarrow _{Z-2}^{A-4}\text{Y} + _{2}^{4}\alpha$$ ##### 5.1.1.2 Beta ($\beta$) decay Beta decay involves the emission of a beta particle. There are two main types: * **Beta-minus ($\beta^-$) decay:** This involves the conversion of a neutron into a proton and an electron, with the electron (beta particle) being emitted. The atomic number increases by one, while the mass number remains the same. $$_{Z}^{A}\text{X} \rightarrow _{Z+1}^{A}\text{Y} + e^{-} + \bar{\nu}_e$$ * **Beta-plus ($\beta^+$) decay:** This involves the conversion of a proton into a neutron and a positron (the antiparticle of an electron), with the positron being emitted. The atomic number decreases by one, while the mass number remains the same. The other name for a $\beta^+$ particle is a positron. $$_{Z}^{A}\text{X} \rightarrow _{Z-1}^{A}\text{Y} + e^{+} + \nu_e$$ ##### 5.1.1.3 Gamma ($\gamma$) emission Gamma emission often accompanies alpha or beta decay. It involves the release of high-energy photons from an excited nucleus as it transitions to a lower energy state. Gamma rays are electromagnetic radiation and do not change the atomic or mass number of the nucleus. ##### 5.1.1.4 Other decay processes * **Electron capture:** A nucleus captures an inner-shell electron, which then combines with a proton to form a neutron. This results in a decrease in atomic number by one, while the mass number remains unchanged. $$_{Z}^{A}\text{X} + e^{-} \rightarrow _{Z-1}^{A}\text{Y} + \nu_e$$ * **Spontaneous fission:** A heavy nucleus splits into two or more lighter nuclei, along with the release of neutrons and energy. ##### 5.1.1.5 Alpha, Beta, and Gamma emissions During radioactive decay, particles like alpha, beta, and gamma rays are produced. Radioactivity is defined as the process by which unstable atomic nuclei lose energy by emitting radiation. ### 5.2 Half-life Half-life ($t_{1/2}$) is the time required for half of the radioactive nuclei in a sample to decay. This is a fundamental concept in understanding the rate of radioactive decay. The half-life refers to the time required for the mass of nuclides to reduce to half its initial value. The rate constant ($k$) for radioactive decay is related to the half-life by the equation: $$t_{1/2} = \frac{\ln(2)}{k}$$ > **Tip:** Half-life is a constant for a given isotope and is independent of the initial amount of the substance. **Example:** Carbon-14 dating is a method used for dating ancient artefacts made from wood or cloth. It relies on the predictable decay rate of carbon-14. ### 5.3 Nuclear processes Nuclear processes involve transformations within the nucleus of an atom, releasing significant amounts of energy. #### 5.3.1 Nuclear fission Nuclear fission involves the splitting of a heavy nucleus into two nuclei with smaller mass numbers. This process is typically initiated by the absorption of a neutron. The neutron bombardment of Uranium-235 often results in the release of energy and additional neutrons, which can sustain a chain reaction. #### 5.3.2 Nuclear fusion Nuclear fusion is the process where atomic nuclei with low atomic numbers fuse to form a heavier nucleus. This process releases a tremendous amount of energy and is the primary energy source for stars. In the Earth’s natural environment, the sun produces energy via nuclear fusion processes. #### 5.3.3 Mass defect and binding energy * **Mass defect ($\Delta m$)**: This refers to the change in mass that occurs when a nucleus is formed from its constituent nucleons (protons and neutrons). The mass of a nucleus is typically less than the sum of the masses of its individual components. * **Binding energy**: This is the energy required to decompose a nucleus into its constituent nucleons. It is also the energy released when a nucleus is formed from its nucleons. The relationship between the mass defect and energy released is given by Einstein's famous equation: $$E = \Delta m c^2$$ where $E$ is the energy released and $c$ is the speed of light. ### 5.4 Applications of radioactivity Radioactivity has numerous applications in various fields: * **Medical imaging and treatment:** Radioactive isotopes are used in diagnostic imaging techniques and in cancer therapy. * **Dating:** Radiometric dating techniques, such as carbon-14 dating, are used to determine the age of archaeological artifacts and geological formations. * **Industrial applications:** Radioactivity is used in gauging thickness, detecting leaks, and sterilizing materials. * **Nuclear power:** Nuclear fission is the basis for generating electricity in nuclear power plants. --- ## Common mistakes to avoid - Review all topics thoroughly before exams - Pay attention to formulas and key definitions - Practice with examples provided in each section - Don't memorize without understanding the underlying concepts

| Term | Definition | |------|------------| | Atomic orbital | A region of space around the nucleus of an atom where there is a high probability of finding an electron. | | Mass number | The total number of protons and neutrons in an atom's nucleus. | | Atomic number | The number of protons in the nucleus of an atom, which determines the element. | | Electron configuration | The arrangement of electrons in the orbitals of an atom. | | Isotopes | Atoms of the same element that have the same number of protons but different numbers of neutrons. | | Neutrons | Subatomic particles with no electrical charge, found in the nucleus of an atom. | | Protons | Positively charged subatomic particles found in the nucleus of an atom. | | Electrons | Negatively charged subatomic particles that orbit the nucleus of an atom. | | Sublevel | A subdivision of an electron shell, characterized by a specific value of the azimuthal quantum number. | | Energy level | A region around the nucleus where electrons can be found, each with a specific amount of energy. | | Mass spectrometry | A technique used to measure the mass-to-charge ratio of ions, often used to determine the isotopic composition of an element. | | Atomic mass unit (amu) | A unit of mass used to express the mass of atoms and molecules, defined as 1/12th the mass of a carbon-12 atom. | | Relative atomic mass | The ratio of the average mass of atoms of an element to 1/12th the mass of a carbon-12 atom. | | Molecular mass | The sum of the atomic masses of all atoms in a molecule. | | Empirical formula | The simplest whole-number ratio of atoms of each element present in a compound. | | Molecular formula | A formula that shows the number of atoms of each element in a molecule. | | Ionic bonding | A chemical bond formed by the electrostatic attraction between oppositely charged ions. | | Covalent bonding | A chemical bond formed by the sharing of electrons between atoms. | | Electronegativity | A measure of the tendency of an atom to attract electrons towards itself in a chemical bond. | | Lewis dot structure | A representation of a molecule or ion that shows valence electrons as dots around the chemical symbols of the atoms. | | Aufbau principle | A rule stating that electrons fill atomic orbitals in order of increasing energy. | | Pauli exclusion principle | A principle stating that no two electrons in an atom can have the same set of four quantum numbers. | | Hund's rule | A rule stating that electrons occupy orbitals within a sublevel singly before pairing up. | | Quantum numbers | A set of numbers that describe the properties of an electron in an atom, including its energy level, shape of orbital, orientation of orbital, and spin. | | Principal quantum number ($n$) | Denotes the main energy level or shell of an electron. | | Azimuthal quantum number ($l$) | Denotes the shape of an atomic orbital and the subshell it belongs to. | | Magnetic quantum number ($m_l$) | Denotes the orientation of an atomic orbital in space. | | Spin quantum number ($m_s$) | Denotes the intrinsic angular momentum of an electron, often referred to as its spin. | | Radioactivity | The spontaneous emission of radiation from the nucleus of an unstable atom. | | Alpha particle ($\alpha$) | A helium nucleus consisting of two protons and two neutrons, emitted during alpha decay. | | Beta particle ($\beta$) | An electron or positron emitted during beta decay. | | Half-life | The time required for half of the radioactive atoms in a sample to decay. | | Nuclear fusion | A nuclear reaction in which two or more atomic nuclei collide at very high speed and join to form a new type of atomic nucleus. | | Nuclear fission | A nuclear reaction in which a heavy nucleus splits into two or more lighter nuclei, releasing a large amount of energy. | | Thermodynamics | The branch of physics that deals with heat, work, temperature, and energy, and their interconversions. | | Exothermic process | A process that releases heat into its surroundings. | | Endothermic process | A process that absorbs heat from its surroundings. | | Enthalpy ($H$) | A thermodynamic quantity representing the total heat content of a system. | | Internal energy ($U$) | The sum of the kinetic and potential energies of the particles that form a system. | | Work ($w$) | The transfer of energy that is not due to a temperature difference. | | Heat ($q$) | The transfer of energy due to a difference in temperature. | | Ideal gas law | An equation of state that describes the behavior of a hypothetical ideal gas: $PV = nRT$. | | Pressure ($P$) | The force exerted per unit area. | | Volume ($V$) | The amount of space occupied by a substance. | | Amount of substance ($n$) | The quantity of a substance, measured in moles. | | Gas constant ($R$) | A physical constant that appears in the ideal gas law and other equations of state. | | Diffusion | The process by which particles spread out from an area of high concentration to an area of low concentration. | | Effusion | The process by which gas molecules escape from a container through a small opening. | | Electrochemistry | The branch of chemistry that deals with the relationship between electrical and chemical phenomena. | | Electrolysis | The decomposition of a substance by an electric current. | | Faraday's laws of electrolysis | Laws that relate the amount of substance produced or consumed at an electrode to the quantity of electricity passed. | | Catalyst | A substance that increases the rate of a chemical reaction without itself being consumed. | | Ionic compound | A compound formed by the electrostatic attraction between oppositely charged ions. | | Metallic bonding | A type of chemical bonding that arises from the electrostatic attractive force between conduction electrons and positively charged metal ions. |