To solve the problem of how much weight a 75 kg person would gain if all \( ^{1}H \) (hydrogen-1) atoms are replaced by \( ^{2}H \) (deuterium) atoms, we can follow these steps:
### Step 1: Determine the mass of hydrogen in the body
The percentage of hydrogen in the body is given as \( 10.0\% \). To find the mass of hydrogen in a 75 kg person, we can calculate:
\[
\text{Mass of } H = \left( \frac{10.0}{100} \right) \times 75 \, \text{kg} = 7.5 \, \text{kg}
\]
### Step 2: Understand the replacement of hydrogen atoms
When \( ^{1}H \) atoms are replaced by \( ^{2}H \) atoms, the number of hydrogen atoms in the body remains the same, but the mass of each hydrogen atom changes.
### Step 3: Calculate the mass difference
- The mass of \( ^{1}H \) (hydrogen) is approximately \( 1 \, \text{g/mol} \).
- The mass of \( ^{2}H \) (deuterium) is approximately \( 2 \, \text{g/mol} \).
### Step 4: Calculate the weight gain
Since the mass of hydrogen in the body is \( 7.5 \, \text{kg} \) (which is equivalent to \( 7500 \, \text{g} \)), we can find the number of moles of \( ^{1}H \):
\[
\text{Moles of } ^{1}H = \frac{7500 \, \text{g}}{1 \, \text{g/mol}} = 7500 \, \text{mol}
\]
When these \( 7500 \, \text{mol} \) of \( ^{1}H \) are replaced by \( ^{2}H \):
\[
\text{Mass of } ^{2}H = 7500 \, \text{mol} \times 2 \, \text{g/mol} = 15000 \, \text{g} = 15 \, \text{kg}
\]
### Step 5: Calculate the weight gain
The weight gain from this replacement is:
\[
\text{Weight gain} = \text{Mass of } ^{2}H - \text{Mass of } ^{1}H = 15 \, \text{kg} - 7.5 \, \text{kg} = 7.5 \, \text{kg}
\]
### Final Answer
Thus, the weight that a 75 kg person would gain if all \( ^{1}H \) atoms are replaced by \( ^{2}H \) atoms is \( 7.5 \, \text{kg} \).
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