Hydrogen Atom Properties - Radius, Speed and Energy

Hydrogen Atom Properties - Radius, Speed and Energy

What is the key difference between nuclear and chemical energy sources?

**Angular momentum** L_n = n h/2π, h=6.6×10⁻³⁴ J·s, L₂=2×6.6×10⁻³⁴/2π=2.11×10⁻³⁴ J·s, speed v_n = e²/(2 ε₀ h) ×1/n ≈2.2×10⁶/n m/s, kinetic energy ½ m v² =13.6/n² eV, illustrating Bohr model predictions for hydrogen-like atoms. Nuclear energy arises from changes in nuclear binding energy (MeV scale), while chemical energy comes from electron rearrangements (eV scale), making nuclear energy millions of times greater. Using E_n = -13.6/n² eV, r_n = n² a₀, L = n h/2π, R = R₀ A^¹/³, BE = Δm c² and 1 u = 931.5 MeV, evaluation yields Energy magnitude, consistent with Bohr model and nuclear binding energy systematics.

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Hydrogen Atom Properties - Radius, Speed and Energy

The mass of a proton is \( 1.67262 \times 10^{-27} \, \text{kg} \). What is its energy equivalent in Joules? (Given \( c

**Angular momentum** L_n = n h/2π, h=6.6×10⁻³⁴ J·s, L₂=2×6.6×10⁻³⁴/2π=2.11×10⁻³⁴ J·s, speed v_n = e²/(2 ε₀ h) ×1/n ≈2.2×10⁶/n m/s, kinetic energy ½ m v² =13.6/n² eV, illustrating Bohr model predictions for hydrogen-like atoms. E = m c² . m = 1.67262 × 10⁻²⁷ kg , c² = (3 × 10⁸)² = 9 × 10¹⁶ m²/s² . E = 1.67262 × 10⁻²⁷ × 9 × 10¹⁶ ≈ 1.505 × 10⁻¹⁰ J . Using E_n = -13.6/n² eV, r_n = n² a₀, L = n h/2π, R = R₀ A^¹/³, BE = Δm c² and 1 u = 931.5 MeV, evaluation yields 1.505 × 10⁻¹⁰ J, consistent

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Hydrogen Atom Properties - Radius, Speed and Energy

What is the energy equivalent of \( 0.05 \, \text{g} \) of matter in Joules? (Given \( c = 3 \times 10^8 \, \text{m/s} \

**Energy in hydrogen** total E_n = -13.6/n² eV, kinetic K = +13.6/n² eV, potential U = -27.2/n² eV, ratio K/U = -1/2, total negative indicates bound, zero at ionization, negative total means electron bound, requires energy to free. For n=4 E=-0.85 eV, U=-1.7 eV, K=0.85 eV, potential energy twice total negative. E = m c² . m = 0.05 × 10⁻³ kg = 5 × 10⁻⁵ kg , c² = 9 × 10¹⁶ m²/s² . E = 5 × 10⁻⁵ × 9 × 10¹⁶ = 4.5 × 10¹² J . Using E_n = -13.6/n² eV, r_n = n² a₀, L = n h/2π, R =

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Hydrogen Atom Properties - Radius, Speed and Energy

Why does nuclear fusion require extremely high temperatures?

**Hydrogen atom radius** r_n = n² a₀, a₀=5.3×10⁻¹¹ m first Bohr radius, r₂=4a₀=2.12×10⁻¹⁰ m, ratio r₄/r₂ =16/4=4, r₃=9a₀, circumference 2πr_n =2π n² a₀, for n=3 circumference=2π×9×5.3×10⁻¹¹=3×10⁻⁹ m. Orbital period T =2πr/v, v_n = v₁/n, v₁=2.2×10⁶ m/s, T₂=2πr₂/v₂, v₂=1.1×10⁶ m/s, T₂≈1.21×10⁻¹⁵ s. High temperatures provide nuclei with enough kinetic energy to overcome the electrostatic repulsion (Coulomb barrier) between positively charged nuclei, enabling fusion via the nuclear force. Using E_n = -13.6/n² eV, r_n = n² a₀, L = n h/2π, R = R₀ A^¹/³, BE = Δm c² and 1 u = 931.5 MeV, evaluation yields To overcome electrostatic repulsion, consistent with Bohr model and nuclear binding energy systematics.

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Hydrogen Atom Properties - Radius, Speed and Energy

What is the energy equivalent of \( 0.15 \, \text{g} \) of matter in Joules? (Given \( c = 3 \times 10^8 \, \text{m/s} \

**Angular momentum** L_n = n h/2π, h=6.6×10⁻³⁴ J·s, L₂=2×6.6×10⁻³⁴/2π=2.11×10⁻³⁴ J·s, speed v_n = e²/(2 ε₀ h) ×1/n ≈2.2×10⁶/n m/s, kinetic energy ½ m v² =13.6/n² eV, illustrating Bohr model predictions for hydrogen-like atoms. E = m c² . m = 0.15 × 10⁻³ kg = 1.5 × 10⁻⁴ kg , c² = 9 × 10¹⁶ m²/s² . E = 1.5 × 10⁻⁴ × 9 × 10¹⁶ = 1.35 × 10¹³ J . Using E_n = -13.6/n² eV, r_n = n² a₀, L = n h/2π, R = R₀ A^¹/³, BE = Δm c² and 1 u = 931.5 MeV, evaluation yields 1.35 × 10¹³ J,

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Hydrogen Atom Properties - Radius, Speed and Energy

Which of the following is true about the nuclear force at very short distances (less than 0.8 fm)?

**Hydrogen atom radius** r_n = n² a₀, a₀=5.3×10⁻¹¹ m first Bohr radius, r₂=4a₀=2.12×10⁻¹⁰ m, ratio r₄/r₂ =16/4=4, r₃=9a₀, circumference 2πr_n =2π n² a₀, for n=3 circumference=2π×9×5.3×10⁻¹¹=3×10⁻⁹ m. Orbital period T =2πr/v, v_n = v₁/n, v₁=2.2×10⁶ m/s, T₂=2πr₂/v₂, v₂=1.1×10⁶ m/s, T₂≈1.21×10⁻¹⁵ s. The nuclear force becomes strongly repulsive at distances less than 0.8 fm, preventing nucleons from collapsing into each other, as shown in the potential energy plot. Using E_n = -13.6/n² eV, r_n = n² a₀, L = n h/2π, R = R₀ A^¹/³, BE = Δm c² and 1 u = 931.5 MeV, evaluation yields It is strongly repulsive, consistent with Bohr model and nuclear binding energy systematics.

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Hydrogen Atom Properties - Radius, Speed and Energy

What is the volume of a nucleus with radius \( 3.6 \times 10^{-15} \, \text{m} \)? (Use \( \pi = 3.14 \))

**Angular momentum** L_n = n h/2π, h=6.6×10⁻³⁴ J·s, L₂=2×6.6×10⁻³⁴/2π=2.11×10⁻³⁴ J·s, speed v_n = e²/(2 ε₀ h) ×1/n ≈2.2×10⁶/n m/s, kinetic energy ½ m v² =13.6/n² eV, illustrating Bohr model predictions for hydrogen-like atoms. Volume = (4/3) π R³ . R³ = (3.6 × 10⁻¹⁵)³ = 4.6656 × 10⁻⁴⁴ m³ . Volume = (4/3) × 3.14 × 4.6656 × 10⁻⁴⁴ ≈ 1.95 × 10⁻⁴³ m³ . Using E_n = -13.6/n² eV, r_n = n² a₀, L = n h/2π, R = R₀ A^¹/³, BE = Δm c² and 1 u = 931.5 MeV, evaluation yields 1.95 × 10⁻⁴³ m³, consistent with Bohr model and nuclear binding energy systematics.

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Hydrogen Atom Properties - Radius, Speed and Energy

What is the key requirement for nuclei to undergo fusion?

**Energy in hydrogen** total E_n = -13.6/n² eV, kinetic K = +13.6/n² eV, potential U = -27.2/n² eV, ratio K/U = -1/2, total negative indicates bound, zero at ionization, negative total means electron bound, requires energy to free. For n=4 E=-0.85 eV, U=-1.7 eV, K=0.85 eV, potential energy twice total negative. Fusion requires nuclei to overcome the Coulomb barrier (electrostatic repulsion between positively charged nuclei), which is achieved by providing high kinetic energy through elevated temperatures. Using E_n = -13.6/n² eV, r_n = n² a₀, L = n h/2π, R = R₀ A^¹/³, BE = Δm c² and 1 u = 931.5 MeV, evaluation yields

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Hydrogen Atom Properties - Radius, Speed and Energy

What is the orbital period of an electron in the \( n = 3 \) orbit if \( v_1 = 2.2 \times 10^6 \, \text{m/s} \) and \( r

**Hydrogen atom radius** r_n = n² a₀, a₀=5.3×10⁻¹¹ m first Bohr radius, r₂=4a₀=2.12×10⁻¹⁰ m, ratio r₄/r₂ =16/4=4, r₃=9a₀, circumference 2πr_n =2π n² a₀, for n=3 circumference=2π×9×5.3×10⁻¹¹=3×10⁻⁹ m. Orbital period T =2πr/v, v_n = v₁/n, v₁=2.2×10⁶ m/s, T₂=2πr₂/v₂, v₂=1.1×10⁶ m/s, T₂≈1.21×10⁻¹⁵ s. v₃ = (2.2 × 10⁶/3) ≈ 7.33 × 10⁵ m/s . r₃ = 9 × 5.3 × 10⁻¹¹ = 4.77 × 10⁻¹⁰ m . T = (2π r₃/v₃) = (2 × 3.14 × 4.77 × 10⁻¹⁰/7.33 × 10⁵) ≈ 4.09 × 10⁻¹⁵ s . Using E_n = -13.6/n² eV, r_n = n² a₀, L = n h/2π, R = R₀ A^¹/³, BE = Δm

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Hydrogen Atom Properties - Radius, Speed and Energy

What is the kinetic energy of an electron in the \( n = 4 \) state of a hydrogen atom? (Use \( E_n = -\frac{13.6}{n^2} \

**Angular momentum** L_n = n h/2π, h=6.6×10⁻³⁴ J·s, L₂=2×6.6×10⁻³⁴/2π=2.11×10⁻³⁴ J·s, speed v_n = e²/(2 ε₀ h) ×1/n ≈2.2×10⁶/n m/s, kinetic energy ½ m v² =13.6/n² eV, illustrating Bohr model predictions for hydrogen-like atoms. E₄ = -0.85 eV , K = -E₄ = 0.85 eV . Using E_n = -13.6/n² eV, r_n = n² a₀, L = n h/2π, R = R₀ A^¹/³, BE = Δm c² and 1 u = 931.5 MeV, evaluation yields 0.85 eV, consistent with Bohr model and nuclear binding energy systematics.

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Hydrogen Atom Properties - Radius, Speed and Energy

An electron in a hydrogen atom has a total energy of -3.4 eV. What is its kinetic energy?

**Energy in hydrogen** total E_n = -13.6/n² eV, kinetic K = +13.6/n² eV, potential U = -27.2/n² eV, ratio K/U = -1/2, total negative indicates bound, zero at ionization, negative total means electron bound, requires energy to free. For n=4 E=-0.85 eV, U=-1.7 eV, K=0.85 eV, potential energy twice total negative. E = -K , K = -E = -(-3.4) = 3.4 eV . Using E_n = -13.6/n² eV, r_n = n² a₀, L = n h/2π, R = R₀ A^¹/³, BE = Δm c² and 1 u = 931.5 MeV, evaluation yields 3.4 eV, consistent with Bohr model and nuclear binding energy systematics.

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Hydrogen Atom Properties - Radius, Speed and Energy

In the Bohr model, what is the ratio of the kinetic energy of an electron in the \( n = 2 \) state to that in the \( n =

**Hydrogen atom radius** r_n = n² a₀, a₀=5.3×10⁻¹¹ m first Bohr radius, r₂=4a₀=2.12×10⁻¹⁰ m, ratio r₄/r₂ =16/4=4, r₃=9a₀, circumference 2πr_n =2π n² a₀, for n=3 circumference=2π×9×5.3×10⁻¹¹=3×10⁻⁹ m. Orbital period T =2πr/v, v_n = v₁/n, v₁=2.2×10⁶ m/s, T₂=2πr₂/v₂, v₂=1.1×10⁶ m/s, T₂≈1.21×10⁻¹⁵ s. K = (e²/8πepsilon₀ r) , r_n ∝ n² . K_n ∝ (1/n²) . Ratio = (K₂/K₁) = (1/2²/1/1²) = (1/4) . Using E_n = -13.6/n² eV, r_n = n² a₀, L = n h/2π, R = R₀ A^¹/³, BE = Δm c² and 1 u = 931.5 MeV, evaluation yields 0.25, consistent with Bohr model and nuclear binding energy systematics.

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