The mass of a nucleus _Z^AX is less than the sum of the masses of (A-Z) number of neutrons and Z number of protons in the nucleus. The energy equivalent to the corresponding mass difference is known as the binding energy of the nucleus. A heavy nucleus of mass M can break into two light nuclei of masses m 1 and m 2 only if (m 1 + m 2 ) 3 and m 4 can undergo complete fusion and form a heavy nucleus of mass M' only if (m 3 + m 4 ) > M'. The masses of some neutral atoms are given in the table below : .tg border-collapse:collapse;border-spacing:0; .tg tdborder-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal; .tg thborder-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal; .tg .tg-0laxtext-align:left;vertical-align:top _1^1H 1.007825 u _1^2H 2.014102 u _3^6Li 6.015123 u _3^7Li 7.016004 u _64^152Gd 151.919803 u _82^206Pb 205.974455 u _1^3H 3.016050 u _2^4He 4.002603 u _30^70Zn 69.925325 u _34^82Se 81.916709 u _83^209Bi 208.980388 u _84^210Po 209.982876 u (1 u = 932 MeV/c 2 ) The correct statement is

deuteron and alpha particle can undergo complete fusion.

the nucleus _3^6Li can emit an alpha particle.

the nucleus _84^210Po can emit a proton.

the nuclei _30^70Zn and _34^82Se can undergo complete fusion.

Atoms and Nuclei PYQ — JEE Advanced 2013 · Shift 2 · Q56 with Solution

The mass of a nucleus $_Z^AX$ is less than the sum of the masses of (A-Z) number of neutrons and Z number of protons in the nucleus. The energy equivalent to the corresponding mass difference is known as the binding energy of the nucleus. A heavy nucleus of mass M can break into two light nuclei of masses m₁ and m₂ only if (m₁ + m₂) < M. Also two light nuclei of masses m₃ and m₄ can undergo complete fusion and form a heavy nucleus of mass M' only if (m₃ + m₄) > M'. The masses of some neutral atoms are given in the table below :

$_1^1H$	1.007825 u	$_1^2H$	2.014102 u
$_3^6Li$	6.015123 u	$_3^7Li$	7.016004 u
$_{64}^{152}Gd$	151.919803 u	$_{82}^{206}Pb$	205.974455 u
$_1^3H$	3.016050 u	$_2^4He$	4.002603 u
$_{30}^{70}Zn$	69.925325 u	$_{34}^{82}Se$	81.916709 u
$_{83}^{209}Bi$	208.980388 u	$_{84}^{210}Po$	209.982876 u

(1 u = 932 MeV/c²)

The correct statement is

A
the nucleus $_3^6Li$ can emit an alpha particle.
B
the nucleus $_{84}^{210}Po$ can emit a proton.
C
deuteron and alpha particle can undergo complete fusion.
D
the nuclei $_{30}^{70}Zn$ and $_{34}^{82}Se$ can undergo complete fusion.

View written solutionFree

Correct answer: C

Criterion to check spontaneity

For a decay/fusion process to be energetically possible, the total mass of initial neutral atoms must be greater than the total mass of final neutral atoms.

For emission/decay: possible if $M_{\text{initial}} > M_{\text{final products}}$
For complete fusion: possible if $m_1+m_2 > M'$

Since atomic masses are given, we can directly compare atomic masses for these nuclear reactions when electrons balance appropriately.

Check option A: ${}_3^6\text{Li}$ can emit an alpha particle

If ${}_3^6\text{Li}$ emits an alpha particle, the daughter nucleus is hydrogen-2: ${}_3^6\text{Li} \rightarrow {}_2^4\text{He} + {}_1^2\text{H}$

Now compare masses: $m({}_3^6\text{Li}) = 6.015123\ \text{u}$ $m({}_2^4\text{He}) + m({}_1^2\text{H}) = 4.002603 + 2.014102 = 6.016705\ \text{u}$

Since $6.016705 > 6.015123$ we have $m_{\text{products}} > m_{\text{initial}}$ So this decay is not possible energetically.

Therefore, A is false.

Check option B: ${}_{84}^{210}\text{Po}$ can emit a proton

Proton emission would be: ${}_{84}^{210}\text{Po} \rightarrow {}_{83}^{209}\text{Bi} + {}_1^1\text{H}$

Compare masses: $m({}_{84}^{210}\text{Po}) = 209.982876\ \text{u}$ $m({}_{83}^{209}\text{Bi}) + m({}_1^1\text{H}) = 208.980388 + 1.007825 = 209.988213\ \text{u}$

Since $209.988213 > 209.982876$ again products are heavier.

So proton emission is not possible energetically.

Therefore, B is false.

Check option C: deuteron and alpha particle can undergo complete fusion

Reaction: ${}_1^2\text{H} + {}_2^4\text{He} \rightarrow {}_3^6\text{Li}$

Compare masses: $m({}_1^2\text{H}) + m({}_2^4\text{He}) = 2.014102 + 4.002603 = 6.016705\ \text{u}$ $m({}_3^6\text{Li}) = 6.015123\ \text{u}$

Since $6.016705 > 6.015123$ complete fusion is energetically possible.

Mass defect: $\Delta m = 6.016705 - 6.015123 = 0.001582\ \text{u}$ Energy released: $Q = 0.001582 \times 932 \approx 1.47\ \text{MeV}$

Therefore, C is true.

Check option D: ${}_{30}^{70}\text{Zn}$ and ${}_{34}^{82}\text{Se}$ can undergo complete fusion

Fusion product would be: ${}_{30}^{70}\text{Zn} + {}_{34}^{82}\text{Se} \rightarrow {}_{64}^{152}\text{Gd}$

Compare masses: $m({}_{30}^{70}\text{Zn}) + m({}_{34}^{82}\text{Se}) = 69.925325 + 81.916709 = 151.842034\ \text{u}$ $m({}_{64}^{152}\text{Gd}) = 151.919803\ \text{u}$

Since $151.842034 < 151.919803$ initial mass is smaller than final mass, so complete fusion is not possible energetically.

Therefore, D is false.

Final conclusion

Only option C is correct.

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$_1^1H$	1.007825 u	$_1^2H$	2.014102 u
$_3^6Li$	6.015123 u	$_3^7Li$	7.016004 u
$_{64}^{152}Gd$	151.919803 u	$_{82}^{206}Pb$	205.974455 u
$_1^3H$	3.016050 u	$_2^4He$	4.002603 u
$_{30}^{70}Zn$	69.925325 u	$_{34}^{82}Se$	81.916709 u
$_{83}^{209}Bi$	208.980388 u	$_{84}^{210}Po$	209.982876 u

(1 u = 932 MeV/c²)

The correct statement is

A
the nucleus $_3^6Li$ can emit an alpha particle.
B
the nucleus $_{84}^{210}Po$ can emit a proton.
C
deuteron and alpha particle can undergo complete fusion.
D
the nuclei $_{30}^{70}Zn$ and $_{34}^{82}Se$ can undergo complete fusion.

View written solutionFree

Correct answer: C

Criterion to check spontaneity

For a decay/fusion process to be energetically possible, the total mass of initial neutral atoms must be greater than the total mass of final neutral atoms.

For emission/decay: possible if $M_{\text{initial}} > M_{\text{final products}}$
For complete fusion: possible if $m_1+m_2 > M'$

Since atomic masses are given, we can directly compare atomic masses for these nuclear reactions when electrons balance appropriately.

Check option A: ${}_3^6\text{Li}$ can emit an alpha particle

If ${}_3^6\text{Li}$ emits an alpha particle, the daughter nucleus is hydrogen-2: ${}_3^6\text{Li} \rightarrow {}_2^4\text{He} + {}_1^2\text{H}$

Now compare masses: $m({}_3^6\text{Li}) = 6.015123\ \text{u}$ $m({}_2^4\text{He}) + m({}_1^2\text{H}) = 4.002603 + 2.014102 = 6.016705\ \text{u}$

Since $6.016705 > 6.015123$ we have $m_{\text{products}} > m_{\text{initial}}$ So this decay is not possible energetically.

Therefore, A is false.

Check option B: ${}_{84}^{210}\text{Po}$ can emit a proton

Proton emission would be: ${}_{84}^{210}\text{Po} \rightarrow {}_{83}^{209}\text{Bi} + {}_1^1\text{H}$

Compare masses: $m({}_{84}^{210}\text{Po}) = 209.982876\ \text{u}$ $m({}_{83}^{209}\text{Bi}) + m({}_1^1\text{H}) = 208.980388 + 1.007825 = 209.988213\ \text{u}$

Since $209.988213 > 209.982876$ again products are heavier.

So proton emission is not possible energetically.

Therefore, B is false.

Check option C: deuteron and alpha particle can undergo complete fusion

Reaction: ${}_1^2\text{H} + {}_2^4\text{He} \rightarrow {}_3^6\text{Li}$

Compare masses: $m({}_1^2\text{H}) + m({}_2^4\text{He}) = 2.014102 + 4.002603 = 6.016705\ \text{u}$ $m({}_3^6\text{Li}) = 6.015123\ \text{u}$

Since $6.016705 > 6.015123$ complete fusion is energetically possible.

Mass defect: $\Delta m = 6.016705 - 6.015123 = 0.001582\ \text{u}$ Energy released: $Q = 0.001582 \times 932 \approx 1.47\ \text{MeV}$

Therefore, C is true.

Check option D: ${}_{30}^{70}\text{Zn}$ and ${}_{34}^{82}\text{Se}$ can undergo complete fusion

Fusion product would be: ${}_{30}^{70}\text{Zn} + {}_{34}^{82}\text{Se} \rightarrow {}_{64}^{152}\text{Gd}$

Compare masses: $m({}_{30}^{70}\text{Zn}) + m({}_{34}^{82}\text{Se}) = 69.925325 + 81.916709 = 151.842034\ \text{u}$ $m({}_{64}^{152}\text{Gd}) = 151.919803\ \text{u}$

Since $151.842034 < 151.919803$ initial mass is smaller than final mass, so complete fusion is not possible energetically.

Therefore, D is false.

Final conclusion

Only option C is correct.

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