How may moles of electrons weigh one kilogram?

Options

A. `6.023 xx ^(23)`
B. `(1)/(9.108) xx 10^(31)`
C. `(6.023)/(9.108) xx 10^(54)`
D. `(1)/(9.108 xx 6.023) xx 10^(8)`

Explanation

To determine how many moles of electrons weigh one kilogram, we can follow these steps:

### Step 1: Find the mass of a single electron
The mass of an electron is given as:
\[ m_e = 9.108 \times 10^{-31} \text{ kg} \]

### Step 2: Calculate the mass of one mole of electrons
To find the mass of one mole of electrons, we need to multiply the mass of a single electron by Avogadro's number (\(N_A\)), which is approximately \(6.022 \times 10^{23}\) particles/mol:
\[ \text{Mass of 1 mole of electrons} = m_e \times N_A \]
\[ = 9.108 \times 10^{-31} \text{ kg} \times 6.022 \times 10^{23} \text{ mol}^{-1} \]

### Step 3: Perform the calculation
Calculating the above expression:
\[ \text{Mass of 1 mole of electrons} = 9.108 \times 6.022 \times 10^{-31 + 23} \]
\[ = 9.108 \times 6.022 \times 10^{-8} \text{ kg} \]

### Step 4: Find the number of moles of electrons that weigh 1 kg
To find the number of moles of electrons that weigh 1 kg, we set up the equation:
\[ \text{Number of moles} = \frac{\text{Total mass}}{\text{Mass of 1 mole}} \]
Substituting the values:
\[ \text{Number of moles} = \frac{1 \text{ kg}}{9.108 \times 6.022 \times 10^{-8} \text{ kg}} \]

### Step 5: Simplify the expression
This simplifies to:
\[ \text{Number of moles} = \frac{1}{9.108 \times 6.022 \times 10^{-8}} \]
Rearranging gives:
\[ \text{Number of moles} = \frac{10^8}{9.108 \times 6.022} \]

### Step 6: Final calculation
Now, we can calculate the final value. The result will give us the number of moles of electrons that weigh 1 kg.

### Final Result
After performing the calculations, we find that the number of moles of electrons that weigh 1 kg is approximately:
\[ \text{Number of moles} \approx 1.82 \times 10^{8} \text{ moles} \]