Some elementary particle theories suggest that the proton may be unstable , with a half life `ge10^32` yr. How long woud you expect to wait for one proton in your body to decay (consider that your body is all water) ?

Answer

To solve the problem of how long you would expect to wait for one proton in your body to decay, we can follow these steps:

### Step 1: Determine the mass of water in your body
Assume that the average mass of a human body is approximately 70 kg.

### Step 2: Calculate the number of water molecules in the body
The number of moles of water can be calculated using the formula:
\[ \text{Number of moles} = \frac{\text{mass of water (g)}}{\text{molar mass of water (g/mol)}} \]
Given that the molar mass of water is approximately 18 g/mol, we convert 70 kg to grams:
\[ 70 \text{ kg} = 70 \times 10^3 \text{ g} \]
Now, calculate the number of moles:
\[ \text{Number of moles} = \frac{70 \times 10^3 \text{ g}}{18 \text{ g/mol}} \approx 3888.89 \text{ moles} \]

### Step 3: Calculate the number of water molecules
Using Avogadro's number (\(6.02 \times 10^{23} \text{ molecules/mol}\)):
\[ \text{Number of molecules} = \text{Number of moles} \times \text{Avogadro's number} \]
\[ \text{Number of molecules} \approx 3888.89 \text{ moles} \times 6.02 \times 10^{23} \text{ molecules/mol} \approx 2.34 \times 10^{27} \text{ molecules} \]

### Step 4: Calculate the total number of protons in the body
Each water molecule (H₂O) contains 10 protons (2 from hydrogen and 8 from oxygen):
\[ \text{Total number of protons} = \text{Number of molecules} \times 10 \]
\[ \text{Total number of protons} \approx 2.34 \times 10^{27} \times 10 = 2.34 \times 10^{28} \text{ protons} \]

### Step 5: Use the half-life to find the decay rate
The half-life of a proton is given as \(t_{1/2} \geq 10^{32} \text{ years}\). The decay constant (\(\lambda\)) is calculated using:
\[ \lambda = \frac{0.693}{t_{1/2}} \approx \frac{0.693}{10^{32} \text{ years}} \]

### Step 6: Calculate the expected time for one proton to decay
Using the formula for the decay rate:
\[ \frac{dN}{dt} = -\lambda N \]
For one proton to decay, we set \(dN = 1\) and \(N = 2.34 \times 10^{28}\):
\[ \frac{1}{dt} = \lambda N \]
Thus,
\[ dt = \frac{1}{\lambda N} \]
Substituting the values:
\[ dt = \frac{1}{\left(\frac{0.693}{10^{32}}\right) \times 2.34 \times 10^{28}} \]
Calculating this gives:
\[ dt \approx \frac{10^{32}}{0.693 \times 2.34 \times 10^{28}} \approx 6 \times 10^{3} \text{ years} \]

### Final Answer
You would expect to wait approximately \(6 \times 10^{3} \text{ years}\) for one proton in your body to decay.

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