Assume that sodium atoms are spheres of radius 0.2 nm and that they are lined up side by side. How many miles, in length, is the line of atoms present in a 1.15 mg sample of sodium? `(N_(A) = 6 xx 10^(23))`

Options

A. `1.2 xx 10^(10)`
B. `1.2 xx 10^(8)`
C. `7.5 xx 10^(8)`
D. `7.5 xx 10^(6)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Convert the mass of sodium to grams
We have the mass of sodium given as 1.15 mg. We need to convert this to grams.

\[ 1.15 \text{ mg} = 1.15 \times 10^{-3} \text{ g} \]

### Step 2: Calculate the number of moles of sodium
The molar mass of sodium (Na) is approximately 23 g/mol. We can calculate the number of moles (n) using the formula:

\[ n = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \]

Substituting the values:

\[ n = \frac{1.15 \times 10^{-3} \text{ g}}{23 \text{ g/mol}} \approx 5 \times 10^{-5} \text{ moles} \]

### Step 3: Calculate the number of sodium atoms
Using Avogadro's number (\(N_A = 6 \times 10^{23}\) atoms/mol), we can find the total number of sodium atoms:

\[ \text{Number of atoms} = n \times N_A \]

Substituting the values:

\[ \text{Number of atoms} = 5 \times 10^{-5} \text{ moles} \times 6 \times 10^{23} \text{ atoms/mol} \approx 3 \times 10^{19} \text{ atoms} \]

### Step 4: Calculate the length of the line of sodium atoms
The radius of a sodium atom is given as 0.2 nm, which is \(0.2 \text{ nm} = 2 \times 10^{-10} \text{ m}\). The diameter (length occupied by one atom) is:

\[ \text{Diameter} = 2 \times \text{radius} = 2 \times (2 \times 10^{-10} \text{ m}) = 4 \times 10^{-10} \text{ m} \]

Now, the total length of the line of atoms is:

\[ \text{Total length} = \text{Number of atoms} \times \text{Diameter} \]

Substituting the values:

\[ \text{Total length} = 3 \times 10^{19} \text{ atoms} \times 4 \times 10^{-10} \text{ m} = 1.2 \times 10^{10} \text{ m} \]

### Step 5: Convert the length from meters to miles
To convert meters to miles, we use the conversion factor \(1 \text{ mile} = 1609.34 \text{ m}\):

\[ \text{Length in miles} = \frac{1.2 \times 10^{10} \text{ m}}{1609.34 \text{ m/mile}} \approx 7.5 \times 10^{6} \text{ miles} \]

### Final Answer
The length of the line of sodium atoms in a 1.15 mg sample of sodium is approximately \(7.5 \times 10^{6}\) miles.

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