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Atoms and Nuclei question

2020 · 6 Sep · Shift 1 · Q51
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Atoms and Nuclei question

2020 · 6 Sep · Shift 1 · Q51

JEE MainPhysicsAtoms and NucleiMCQ+4 / −1
You are given that Mass of 37Li{}_3^7Li37​Li= 7.0160u, Mass of 24He{}_2^4He24​He= 4.0026u and Mass of 11H{}_1^1H11​H= 1.0079u. When 20 g of 37Li{}_3^7Li37​Li is converted into 24He{}_2^4He24​He by proton capture, the energy liberated, (in kWh), is : [Mass of nucleon = 1 GeV/c2]
  1. A
    6.82 ×\times× 105
  2. B
    4.5 ×\times× 105
  3. C
    8 ×\times× 106
  4. D
    1.33 ×\times× 106
View written solutionFree

Correct answer: D

  1. Write the nuclear reaction

    Proton capture by lithium gives: 37Li+11H→2 24He+Q{}_3^7\text{Li} + {}_1^1\text{H} \rightarrow 2\,{}_2^4\text{He} + Q37​Li+11​H→224​He+Q

  2. Calculate mass defect per reaction

    Initial mass: mi=m(37Li)+m(11H)=7.0160+1.0079=8.0239 um_i = m({}_3^7\text{Li}) + m({}_1^1\text{H}) = 7.0160 + 1.0079 = 8.0239\,umi​=m(37​Li)+m(11​H)=7.0160+1.0079=8.0239u

    Final mass: mf=2×m(24He)=2×4.0026=8.0052 um_f = 2\times m({}_2^4\text{He}) = 2\times 4.0026 = 8.0052\,umf​=2×m(24​He)=2×4.0026=8.0052u

    Mass defect: Δm=mi−mf=8.0239−8.0052=0.0187 u\Delta m = m_i - m_f = 8.0239 - 8.0052 = 0.0187\,uΔm=mi​−mf​=8.0239−8.0052=0.0187u

  3. Convert mass defect into energy per reaction

    Given: 1 nucleon mass≈1u≈1 GeV/c21\text{ nucleon mass} \approx 1u \approx 1\,\text{GeV}/c^21 nucleon mass≈1u≈1GeV/c2

    Hence, Q=0.0187 GeV=18.7 MeVQ = 0.0187\,\text{GeV} = 18.7\,\text{MeV}Q=0.0187GeV=18.7MeV

    In joules: 18.7 MeV=18.7×106×1.6×10−1918.7\,\text{MeV} = 18.7\times 10^6 \times 1.6\times 10^{-19}18.7MeV=18.7×106×1.6×10−19 Q≈2.992×10−12 JQ \approx 2.992\times 10^{-12}\,\text{J}Q≈2.992×10−12J

  4. Find number of lithium nuclei in 20 g of 7Li{}^7\text{Li}7Li

    Molar mass of 7Li{}^7\text{Li}7Li is approximately 7 g/mol7\,\text{g/mol}7g/mol.

    Number of moles in 20 g20\,\text{g}20g: n=207 moln = \frac{20}{7}\,\text{mol}n=720​mol

    Number of nuclei: N=207NA=207×6.02×1023N = \frac{20}{7}N_A = \frac{20}{7}\times 6.02\times 10^{23}N=720​NA​=720​×6.02×1023 N≈1.72×1024N \approx 1.72\times 10^{24}N≈1.72×1024

  5. Total energy released

    E=NQ=1.72×1024×2.992×10−12E = NQ = 1.72\times 10^{24}\times 2.992\times 10^{-12}E=NQ=1.72×1024×2.992×10−12 E≈5.15×1012 JE \approx 5.15\times 10^{12}\,\text{J}E≈5.15×1012J

  6. Convert joules to kWh

    1 kWh=3.6×106 J1\,\text{kWh} = 3.6\times 10^6\,\text{J}1kWh=3.6×106J

    Therefore, E=5.15×10123.6×106 kWhE = \frac{5.15\times 10^{12}}{3.6\times 10^6} \,\text{kWh}E=3.6×1065.15×1012​kWh E≈1.43×106 kWhE \approx 1.43\times 10^6\,\text{kWh}E≈1.43×106kWh

  7. Match with the nearest option

    The computed value is closest to: 1.33×106 kWh1.33\times 10^6\,\text{kWh}1.33×106kWh

    So the correct option is D.


Note: Using more accurate constants gives about 1.39×106 kWh1.39\times 10^6\,\text{kWh}1.39×106kWh, but among the given choices, D is the best match.

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