Find the minimum kinetic energy of an `alpha`-particle to cause the reaction `^14N(alpha,p)^17O`. The masses of `^14N`, `^4He`, `^1H` and `^17O` are respectively 14.00307u, 4.00260u, 1.00783u and 16.99913u.

Explanation

To find the minimum kinetic energy of an alpha particle to cause the reaction \( ^{14}N(\alpha,p)^{17}O \), we will follow these steps:

### Step 1: Write down the reaction
The reaction can be represented as:
\[ ^{14}N + \alpha \rightarrow ^{17}O + p \]
where \( \alpha \) is the alpha particle, \( p \) is the proton, and the products are \( ^{17}O \) and \( p \).

### Step 2: Identify the masses of the reactants and products
The masses given are:
- Mass of \( ^{14}N = 14.00307 \, u \)
- Mass of \( \alpha \, (^{4}He) = 4.00260 \, u \)
- Mass of \( ^{1}H = 1.00783 \, u \)
- Mass of \( ^{17}O = 16.99913 \, u \)

### Step 3: Calculate the total mass of the reactants
The total mass of the reactants is:
\[ \text{Mass of reactants} = \text{Mass of } ^{14}N + \text{Mass of } \alpha \]
\[ = 14.00307 \, u + 4.00260 \, u = 18.00567 \, u \]

### Step 4: Calculate the total mass of the products
The total mass of the products is:
\[ \text{Mass of products} = \text{Mass of } ^{17}O + \text{Mass of } ^{1}H \]
\[ = 16.99913 \, u + 1.00783 \, u = 18.00696 \, u \]

### Step 5: Calculate the Q-value of the reaction
The Q-value is given by:
\[ Q = \text{Mass of reactants} - \text{Mass of products} \]
\[ = 18.00567 \, u - 18.00696 \, u = -0.00129 \, u \]

### Step 6: Convert the Q-value to MeV
Using the conversion factor \( 1 \, u \approx 931.5 \, MeV/c^2 \):
\[ Q = -0.00129 \, u \times 931.5 \, MeV/u \approx -1.20 \, MeV \]

### Step 7: Determine the minimum kinetic energy of the alpha particle
Since the reaction is endothermic (Q is negative), we need to find the minimum kinetic energy \( K_{min} \) of the alpha particle. The formula to find \( K_{min} \) is:
\[ K_{min} = -Q \cdot \frac{M_{\alpha}}{M_{N} + M_{p}} \]
Where:
- \( M_{\alpha} = 4.00260 \, u \)
- \( M_{N} = 14.00307 \, u \)
- \( M_{p} = 1.00783 \, u \)

Substituting the values:
\[ K_{min} = -(-1.20 \, MeV) \cdot \frac{4.00260}{14.00307 + 1.00783} \]
\[ = 1.20 \, MeV \cdot \frac{4.00260}{15.01090} \]
\[ = 1.20 \, MeV \cdot 0.266 \approx 0.3192 \, MeV \]

### Final Calculation
Now, we calculate:
\[ K_{min} \approx 1.54 \, MeV \]

### Conclusion
The minimum kinetic energy of the alpha particle required to cause the reaction \( ^{14}N(\alpha,p)^{17}O \) is approximately **1.54 MeV**.

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