Atom and Nuclei Notes PDF

# Atom and Nuclei ## Geiger and Marsden Experiment / Gold Foil Experiment/ alpha-particle scattering - A diagram shows a set up: - A radioactive source (U) emitting alpha particles - A lead chamber containing lead blocks and a cavity - Alpha particles are directed at a gold foil - Alpha...

# Atom and Nuclei ## Geiger and Marsden Experiment / Gold Foil Experiment/ alpha-particle scattering - A diagram shows a set up: - A radioactive source (U) emitting alpha particles - A lead chamber containing lead blocks and a cavity - Alpha particles are directed at a gold foil - Alpha particles are scattered at various angles. The diagram shows angles with values 10,000, 9990, 5-6, and 1-3 *rad*. The alpha particle is labelled as `rad ^ 2+ & particle (He-nucleus)`. ## Rutherford Atomic Model - Atom consist of large amount of free space (approx. 99.9% free space). - All positively charged particle are collectively present at the center of an atom called as nucleus. - Electrons revolve around a nucleus in a circular orbit. - A diagram shows a nucleus with positive charge (+), and arrow pointing outwards labelled with `↓F`. - A diagram shows an electron with negative charge (-), and arrow pointing inwards labelled with `e-`. ## Drawbacks of Rutherford Atomic Model - Rutherford stated that electrons perform circular motion which is an accelerated motion. - According to Maxwell theory, electrons must lose energy and eventually fall on nucleus, which isn't possible. - Rutherford cannot explain this. ### Basic: 1. Maxwell Theory - **rest:** `q = E` - **uniform velocity:** `q = E & B` - **acceleration:** `q = E & B & energy` ### Basic: 2. UCM is an accelerated motion - A diagram shows an electron labelled with `e-'` following circular path which is labelled with `As per Rutherford`, and an electron following a wavy path labelled with `As per Maxwell.` - As per Rutherford model, electrons lose energy in the form of continuous spectrum. - Practically, we observe that electrons lose energy in the form of a line spectrum. # Bohr Atomic Model - (H-atom and H-like atoms) - `[He+, Li^2+, Na^3+ etc]` - these ions have only one electron - All electrons revolve around the nucleus in a circular orbit. - The necessary centripetal force is provided by the electrostatic force of attraction. - The following equations are used to show the relation: - `CPF = ESF` - `mv^2 / r = 1/4πεο q1q2` - `mv^2 = 1/4πεο (e) (Ze)` - `mv^2 = 1/4πεο Ze^2` - `r` – radius of orbit - `Ze` – charge of nucleus - `e` – charge of electron - `πεο` – permittivity of free space - `mv^2 = Ze^2 / 4πεοr` - `r` – radius of orbit - `v` - velocity of electron ## Bohr I Postulate - The necessary electrostatic force of attraction is provided by the centripetal force to an electron revolving in a circular orbit. ## Bohr II Postulate - Electrons can only revolve in a stationary orbit where the orbital angular momentum of an electron is an integral multiple of `h/2π` - The following equation shows the relation: - `Angular momentum` = `nh / 2π` (where `n` is an integer) - `mvr` = `nh / 2π` ## Bohr III Postulate - Whenever electrons jump from a higher orbit to a lower orbit, they lose energy in the form of electromagnetic radiation. - The energy difference between the two levels is equal to the energy lost. - The following equation are used to show the relation: - `ΔE = Ei-Ef` - energy of final orbit - `ΔE` is also equal to the released energy. - `hc / λ = Ei- Ef` # nth excited state > n+1 orbit - Derive radius of orbit, velocity of electron, and energy of electron. - `# For radius` - Using Bohr's I postulate: - `mv^2 / r = Ze^2 / 4πεοr` - `mv^2 = Ze^2 / 4πεor` - From the previous equation - `mvr = nh / 2π` and `mv^2 = Ze^2 / 4πεor` - `r = εoh^2n^2 / πme^2 (Z)` - `r = 0.53 * (εoh^2n^2 / πme^2 (Z))` ­­­­Å - `# For velocity` - Substitute `r` in the equation `mvr = nh / 2π` - The equation becomes: - `v = e^2Z / 2εohn` - In the equation `v = e^2Z / 2εohn` - `v = 2.1 * 10^6 * Z/n` m/s - `# For Energy` - `TE = KE + PE` - `KE = 1/2 * mv^2` - `KE = me^4Z^2 / 8εoh^2n^2` - `PE = 1/4πεο * (q1q2/r)` - `PE = 1 / 4πεο * (-e)(Ze) / (εoh^2n^2 / πme^2 (Z))`­­­­ - `PE = -me^4Z^2 / 4εoh^2n^2` - `TE = (me^4Z^2 / 8εoh^2n^2) [1 / - 1]` - `E = -me^4Z^2 / 8εoh^2 n^2` - energy level for H atoms - For H atom only: - `E1 = -13.6 eV` - `E2 = -3.4 eV` - `E3 = -1.51 eV` - `Ea = -0.85 eV` - `E5 = -0.54 eV` - `# Derive formula for T, f, ω , ac` - `T = 2πr / v` - `ω = 2π/ T` - `f = 1/T` - `ac = v^2 / r` # Atomic Spectrum - The following equations are used to show the relation: - `ΔE = Ei-Ef` - `Ei = -me^4Z^2 / 8εoh^2n^2` - `Ef = - me^4Z^2 / 8εoh^2n^2` - The following equation is used to calculate the change in energy - `ΔE = me^4Z^2 / 8εoh^2n^2 [- (1/n^2 - 1/n^1 )]` - `ΔE = hc / λ = 13.6 * (Z^2/ (1/n^2 - 1/n^1))` eV - The following equation is used to calculate the wavelength: - `hc / λ = me^4Z^2 / 8εoh^2 [- (1/n^2 - 1/n^1)]` - `λ = 8εoh^2c / me^4Z^2 * [- (1/n^2 - 1/n^1)]` ## Rydberg Constant: - `λ = (8εoh^2c / me^4) * (RZ^2 / (- (1/n^2 - 1/n^1))` Å - `R = 1.097 * 10^7 m^-1` - `R = 912` Å # Hydrogen Energy Spectrum - H-atom energy spectrum is plotted. - Energy levels are labelled with: - `P` - `O` - `N` - `M` - `L` - `K` - `n= ∞` - `n=6` - `n=5` - `n=4` - `n=3` - `n=2` - `n=1` - Transitions between energy levels are labelled with: - `Pfund series` - `Brackett Series (IR)` - `Paschen Series (IR)` - `Balmer series (Visible)` - `Lymann Series` - The following table is shown: - Series limits: `∞ → nf` - first trans: `∞ → n=3` - second trans: `∞ → n=4` - third trans: `∞ → n=5` - fourth trans: `∞ →n=6`, etc.  - `# no. of spectral lines = (n2-n1)(n2-n1+1)` - `#no. of absorption lines = (n-1)` # Drawbacks of Bohr Atomic Model: - Bohr didn't prove his second postulate. - Bohr could not explain existence of fine-line spectrum. - Bohr didn’t discuss about intensity of radiation emitted in the atomic spectrum. - Bohr considered orbit to be circular in shape but it was found that orbits are elliptical in shape by Sommerfeld. - Bohr could not explain the Zeeman effect and Stark effect. # de Broglie's proof of Bohr 2nd Postulate - A diagram shows a circular orbit, and waves moving along the orbit. For `n=1`, there is half a wavelength shown. For `n=2`, there is a full wavelength shown. For `n=3`, there is one and a half wavelength shown. - `total length of wave` = `nλ` - `Circumference of orbit` = `2πr` - `nλ = 2πr` - According to de Broglie, `λ = h/p` - Therefore, `nh = 2πr` - p = mv. - `nh = mv` - `nh/2π = mvr` - `L = nh/2π` # 16. Data: [ASL] LY = 911.6Å - `To find: [ASL] Ba, [ASL] Pa, [ASL] Br` - Apply the formula: `Series limit: ∞ → nf` ## Solution: - **For Lyman Series**: `∞ -> 1` - `[ASL] LY= R[1/1^2 - 1/∞^2]` - `[ASL] LY = R` - `911.6 = R` - `R = 911.6Å` ­­­­ - **For Balmer series**: - ` ∞ -> 2` - `[ASL]Ba = R[1/2^2 - 1/∞^2]` - `[ASL] Ba = R/4` - `[ASL] Ba = 911.6 / 4` - `[ASL] Ba = 3646.4 Å` ­­­­ # 17. Data: n=2; t=10^8s - `To find: N = No. of revolutions` - Apply the formula: `T = 2πr / v` - `T = 2π * (εOh^2n^2 / πme^2 (Z)) / (e^2Z / 2εohn) ` - Simplify the previous equation: - `[2 / (e^2Z / 2εohn)] * (εOh^2n^2 / πme^2 (Z))` - `T = 4εoh^3n^3 / me^4` - `T = 1 rev` - `N = t / T` - `N = 10^8 / 4εoh^3n^3 / me^4` - `N = 10^8 * me^4 / 4εoh^3n^3` ## Nucleus - A diagram shows nucleus is made up of protons, electrons, and neutrons, which make up nucleons. - Nucleons are present inside the nucleus. ## Atomic Number (Z): - No. of protons present inside the nucleus - No. of electrons present in the ground state of an atom. - `Z = no. of protons` ## Atomic mass number (A): - Total no. of nucleons present inside the nucleus - `A = no. of nucleons` - `A = no. of protons + no. of neutrons` ## Atomic mass (M): - Actual mass of an atom - `M = actual mass` ## No. of neutrons (N): - The difference between atomic mass no. and atomic no. - `N` = `A- Z` ## Representation of atom - A diagram shows an Uranium atom labelled with the atomic mass number (236), and atomic number (92). - `A` (atomic mass number) goes on top - `Z` (atomic number) goes at the bottom - The symbol for the element goes on the right ## Mass of subatomic particles: - `1 amu = 1/12 mass of C-12 atom` - `1 amu = 1.6606 * 10^-27 kg = 931.5 MeV/c^2` ­­­­ - `Mp = 1.00727 amu` - `Mn = 1.00866 amu` - `me = 0.00055 amu` ## Types of nucleus - **Isotopes**: same atomic no. but different atomic mass no. - `^1H`, `^2H`, `^3H`, `^236U`, `^235U`, `^16O`, `^18O ` - the number on the top left is the atomic mass number - the number at the bottom left is atomic number - **Isobars**: same atomic mass no. but different atomic no. - `^40Ar`, `^40K`, `^40Ca` - the number on the top left is the atomic mass number - the number at the bottom left is atomic number - **Isotones**: same no. of neutrons - `^3H`, `^4He`, `^14C`, `^15N`, `^16O` - the number on the top left is the atomic mass number - the number at the bottom left is atomic number ## Size of nucleus - `Density of all nucleus is constant` ­­­­ - `Density of nucleus = mass of nucleus / volume of nucleus` - `mass of nucleus = avg mass of proton & neutron * total no. of proton & neutron` - ` (4/3)πR^3 = MA/p` ­­­­ - `R^3 = 3MA / 4πp` - `R = (3MA / 4πp)^1/3 A^(1/3) ` - `R = R0A^(1/3)` - **Radius of nucleus**: `⇒ R = R0A^(1/3)` - `Here Ro = 1.2 * 10^-15m` ­­­­ ## Nuclear force - A force of attraction present inside the nucleus is called as nuclear force. - **Characteristics** of this force: - It is always attractive in nature. - It is charge independent. - It is strongest. - It is short-range force (`10^-15 m`). ## Mass defect (Δm) - The loss in mass of an atom is called as a mass defect. - `Δm = Total mass - Actual mass` - `Δm` = Total mass of proton + Total mass of neutron - Actual mass - `Δm` = `[Zmp + (A-Z)mn - M]` ## Binding Energy (BE) - `BE = Δmc^2` - `BE = [Zmp + (A-Z)mn-M]c^2` ## Binding Energy per nucleon - `BE/A = [Zmp + (A-Z)mn-M]c^2 / A` - `BE α stability` ­­­­ ## 12. Binding Energy per nucleon v/s Mass number (A) - A graph is shown with a peak. - **Nuclear Fusion**: - This is the process of combining of light unstable nuclei to form stable nuclei. - **Nuclear Fission:** - This is the process of breaking of heavy unstable nuclei to form stable nuclei. - **Instantly disintegrate [Radioactivity]** - **56Fe (Iron) → most stable nucleus** ## Nuclear Reactions - The process of breaking or combining of unstable nucleus to form stable nuclei is called as a nuclear reaction. - It includes `Nuclear Fusion` and `Nuclear Fission.` ## Nuclear Fusion - When light unstable nuclei combine to form stable nuclei, it's called as Nuclear fusion. - `3H + 3H -> 4He + 2n` ## Nuclear Fission - When heavy unstable nuclei disintegrate to form stable nuclei, it's called as nuclear fission. - `^236U -> 140Ba + ^94Kr + 2n` - `Nuclear Energy` = `[mreact - Mprode ] c^2` ## 13. Data - `^244Am` - `Z = 95, A = 244, M= 244.06428` - `BE per nucleon = [Zmp + (A-Z)mn-M]c^2 / A` ## 13. `^3Li + p -> 2α` - `MLi = 7.016 u` - `Mp = 1. 00727 u` - `Mα = 4.0026 u` - `Nuclear Energy = [mreact - Mprode]c^2` - `[mLi + mp - 2 * Mα]c^2` - `[7.016 + 1.00727 - 8.0052]c^2` - `0.01807 * uc^2` - `0.01807 * 931.5 MeV * c^2` - `1.807 * 9.315` - `16.832 MeV` # Radioactivity - The process of disintegration of an unstable parent nucleus to form two or more stable daughter nuclei is called as radioactivity. ## Law of Radioactivity: - The rate of disintegration of a radioactive sample is directly proportional to the no. of samples present at that instant of time. - `Mathematically: rate of disintegration α no. of sample present` - `- dN/ dt α N` - `- dN / dt = λN` - `λ` is the decay constant. - `SI unit: s^-1` - `dN = -λdt` - `∫dN / N = ∫ -λ dt` - `[logN]No = -λ[t]0^t` - `logN - logNo = -λ[t-0]` - `logN = -λt - logNo` - `logN = -λt` - `N/No = e^-λt` - `N = Noe^-λt` - `N` decays exponentially - A labelled graph is shown to illustrate the exponential decay. It shows that the half-life is constant, and the rate of decay decreases over time. # Derive from equation 0 - `logN/No = -λt` - `λt = -logN/No` - `λt = 2.303 log10 (No/N)` - `t = 2.303 log10 (No/N) / λ` ## Half-life (t1/2) - The time taken by a sample to disintegrated 50% at any instant of time is called as half-life.  - `t = t1/2`, `N = No/2` - `t1/2 = 2.303 log10 (No / (No/2) / λ` - `t1/2 = 2.303 log10 (2) / λ` - `t1/2 = 2.303 * 0.3010 / λ` - `t1/2 = 0.693 / λ` ## Mean life (τ) - The time taken by a sample to disintegrate 63% at an instant of time is called as mean life. - The reciprocal of the decay constant is called mean life - `τ = 1 / λ` # Activity (A) - The rate of disintegration of a sample is called as activity. - `A = -dN/dt` - `A = λN` - `SI unit: disintegration per second(dps) or Becquerel (Bq)` - `Commonly used unit: Curie (Ci) : 1Ci = 3.7 * 10^10 Bq`   - `Rutherford (Rf) : 1Rf = 10^6 Bq` - `A = A0e^-λt` - ` t = 2.303 log10 (A0 / A) /λ`   - ` t1/2 = 2.303 log10 (A1 / A2) / λ ` ## 16. - `Data: A0 = 15.3 decays per gram per minute` - `A1 = 12.3 decays per gram per minute` - `λ = 3.839 * 10^-12 s^-1` - `t = 2.303 log10 (A0/ A1 ) / λ` - `t = 2.303 * log10(15.3 / 12.3 ) / 3.839 * 10^-12` - `t = 2.303 * log10 (1.24) / 3.839 * 10^-12` ## 17. - `Data: t1/2 = 28 years` - `t1/2 = 28 years = 28 * 3156 * 10^7 = 8.835 * 10^9` seconds - `^90Sr = 90g` - `M = 5mg = 5 * 10^-3 g` - `A = 2` - `A = λN = 0.693 * N/ t1/2 * M/Mo` - `A = 0.693 * N / 8.835 * 10^9 * (5* 10^-3) / 90 ` ## 19. - `Data: A1 = 10^10, t1 = 20 hrs` - `A2 = 6.3 * 10^9, t2 = 30 hrs` - `t1 = 2.303 log10 (A0/ A1) / λ` - `t2 = 2.303 log10 (A0/ A2) / λ` - `A0 = λN0` ## Nuclear Decay - The process in which nucleus emits light particles such as e+, position, α-particle is Nuclear decay. - A diagram shows: - Nuclear decay - α-decay - β-decay - γ-decay ## (i) α-decay [α particles are emitted] - `^A_Z X -> ^(A-4)_(Z-2) Y + ^4_2 He` - Pair annihilation: `[p]+[e-] -> γ` - Pair production: `γ -> [p] + [e-] + neutrino (ν)` - A diagram shows that a proton and an electron annihilate each other to form a gamma ray. - A diagram shows that a gamma ray breaks down to form proton and electron. - A diagram shows that a gamma ray breaks down to form proton and electron. ## β-decay (e- decay) - `^A_Z`X → `^(A)_(Z+1)Y + e- + v` - A diagram shows that a neutron breaks down to form proton, electron, and anti-neutrino. - A neutron decays to produce proton with positive charge, electron with negative charge, and antineutrino. ## β+ decay [e+ decay] - `^A_Z`X → `^(A)_(Z-1)Y + e+ + v` - A diagram shows that a proton breaks down to form neutron, positron, and neutrino. ## γ-decay - `^A_Z`X → `^A_Z`X + γ - A diagram shows that the nucleus releases a gamma ray