Total mass of neutrons in `7 mg` of `.^(14)C` is:

Options

A. `3xx10^(20) kg`
B. `4xx10^(-6) kg`
C. `5xx10^(-7) kg`
D. `4xx10^(-7)kg`

Explanation

To find the total mass of neutrons in 7 mg of carbon-14 (C-14), we can follow these steps:

### Step 1: Determine the number of neutrons in C-14
The mass number of carbon-14 is 14, and the atomic number of carbon is 6. The number of neutrons can be calculated using the formula:

\[ \text{Number of neutrons} = \text{Mass number} - \text{Atomic number} \]

For C-14:

\[ \text{Number of neutrons} = 14 - 6 = 8 \]

### Step 2: Convert the mass of carbon-14 from mg to grams
Given mass of carbon-14 is 7 mg. We convert this to grams:

\[ 7 \text{ mg} = 7 \times 10^{-3} \text{ g} \]

### Step 3: Calculate the number of moles of carbon-14
To find the number of moles, we use the formula:

\[ \text{Number of moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \]

The molar mass of carbon-14 is approximately 14 g/mol. Therefore:

\[ \text{Number of moles} = \frac{7 \times 10^{-3} \text{ g}}{14 \text{ g/mol}} = 5 \times 10^{-4} \text{ moles} \]

### Step 4: Calculate the total number of neutrons
Since each mole of C-14 contains 8 neutrons, the total number of neutrons in the given mass is:

\[ \text{Total neutrons} = \text{Number of moles} \times \text{Number of neutrons per mole} \]

\[ \text{Total neutrons} = 5 \times 10^{-4} \text{ moles} \times 8 = 4 \times 10^{-3} \text{ neutrons} \]

### Step 5: Calculate the total mass of neutrons
The mass of one neutron is approximately \(1.675 \times 10^{-27} \text{ kg}\). To find the total mass of neutrons, we use:

\[ \text{Total mass of neutrons} = \text{Total neutrons} \times \text{mass of one neutron} \]

First, we need to convert the total number of neutrons to kg:

\[ \text{Total mass of neutrons} = 4 \times 10^{-3} \times 1.675 \times 10^{-27} \text{ kg} \]

Calculating this gives:

\[ \text{Total mass of neutrons} = 6.7 \times 10^{-30} \text{ kg} \]

### Step 6: Convert the mass to mg
To express the total mass of neutrons in mg, we convert kg to mg:

\[ \text{Total mass of neutrons in mg} = 6.7 \times 10^{-30} \text{ kg} \times 10^6 \text{ mg/kg} = 6.7 \times 10^{-24} \text{ mg} \]

### Final Answer
The total mass of neutrons in 7 mg of carbon-14 is approximately \(6.7 \times 10^{-24} \text{ mg}\).

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