Question 9 of 13intermediate🔍 AnalyzeNumerical3 marks

Suppose, we think of fission of a 2656Fe^{56}_{26}\text{Fe} nucleus into two equal fragments, 1328Al^{28}_{13}\text{Al}. Is the fission energetically possible? Argue by working out QQ of the process. Given m(2656Fe)=55.93494um(^{56}_{26}\text{Fe}) = 55.93494 \text{u} and m(1328Al)=27.98191um(^{28}_{13}\text{Al}) = 27.98191 \text{u}.

Correct Answer

No, the fission is not energetically possible.

Step 1: Write the nuclear reaction

2656Fe→2×1328Al^{56}_{26}\text{Fe} \rightarrow 2 \times ^{28}_{13}\text{Al}

Step 2: Calculate total mass of products Mass of two 1328Al^{28}_{13}\text{Al} fragments = 2×27.98191 u=55.96382 u2 \times 27.98191 \text{ u} = 55.96382 \text{ u}

Step 3: Calculate Q value

Q=[mreactant−mproducts]c2Q = [m_{\text{reactant}} - m_{\text{products}}]c^2 Q=[55.93494−55.96382] u×c2Q = [55.93494 - 55.96382] \text{ u} \times c^2 Q=−0.02888 u×c2Q = -0.02888 \text{ u} \times c^2

Step 4: Convert to energy units Using 1 u=931.5 MeV/c21 \text{ u} = 931.5 \text{ MeV}/c^2

Q=−0.02888×931.5 MeV=−26.90 MeVQ = -0.02888 \times 931.5 \text{ MeV} = -26.90 \text{ MeV}

Conclusion: Since the Q value is negative (-26.90 MeV), energy must be supplied for this fission to occur. The reaction is endothermic. Therefore, the fission of 2656Fe^{56}_{26}\text{Fe} into two 1328Al^{28}_{13}\text{Al} fragments is not energetically possible as a spontaneous process.

Exercise: EXERCISES | Q: 13.6 | (Page 16)
For More Understanding

Explanation

The Q value determines whether a nuclear reaction is energetically favorable. A positive Q indicates energy release (exothermic, spontaneous), while a negative Q indicates energy absorption is required (endothermic, not spontaneous). The calculation shows that the combined mass of two Al-28 fragments exceeds the mass of Fe-56, resulting in a negative Q value. This is consistent with the textbook's discussion that fission energy release depends on binding energy per nucleon - lighter nuclei like Fe have higher binding energy per nucleon and are more stable, so splitting them requires energy input.

Solution Steps

  1. Step 1: Write the fission reaction equation

  2. Step 2: Calculate total mass of product fragments (2×27.981912 \times 27.98191 u)

  3. Step 3: Apply Q value formula: Q = [mass of reactant - mass of products]c²

  4. Step 4: Convert mass difference to MeV using 1 u = 931.5 MeV/c²

  5. Step 5: Interpret the sign of Q to determine if fission is energetically possible

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