Suppose that gold is being plated onto another metal in a electrolytic cell. The half`-` cell reaction producing the `Au(s)` is `AuCl_(4)^(c-) rarr Au(s)+4Cl^(c-)+3e^(-)`
If a `0.30- A` current runs for `15 mi n` , what mass of `Au(s)` will be plated, assuming all the electrons are used in the reduction of `AuCl_(4)?`

Options

A. `0.184g`
B. `0.551g`
C. `1.84g`
D. `0.613g`

Explanation

To solve the problem of how much gold will be plated when a current of 0.30 A runs for 15 minutes, we will use Faraday's laws of electrolysis. Here’s a step-by-step breakdown:

### Step 1: Calculate the total charge (Q) passed through the electrolytic cell.
The formula to calculate charge is:
\[ Q = I \times t \]
Where:
- \( I \) is the current in amperes (A)
- \( t \) is the time in seconds (s)

Given:
- \( I = 0.30 \, \text{A} \)
- \( t = 15 \, \text{minutes} = 15 \times 60 \, \text{s} = 900 \, \text{s} \)

Now substituting the values:
\[ Q = 0.30 \, \text{A} \times 900 \, \text{s} = 270 \, \text{C} \]

### Step 2: Determine the number of moles of electrons (n) transferred.
Using Faraday's constant, \( F \approx 96500 \, \text{C/mol} \), we can find the number of moles of electrons:
\[ n = \frac{Q}{F} \]
Substituting the values:
\[ n = \frac{270 \, \text{C}}{96500 \, \text{C/mol}} \approx 0.00280 \, \text{mol} \]

### Step 3: Identify the number of electrons involved in the half-cell reaction.
From the half-cell reaction:
\[ \text{AuCl}_4^{-} + 3e^{-} \rightarrow \text{Au}(s) + 4\text{Cl}^{-} \]
We see that 3 moles of electrons are required to produce 1 mole of gold (Au).

### Step 4: Calculate the moles of gold plated.
Using the stoichiometry from the half-cell reaction:
\[ \text{Moles of Au} = \frac{n}{3} \]
Substituting the value of \( n \):
\[ \text{Moles of Au} = \frac{0.00280 \, \text{mol}}{3} \approx 0.000933 \, \text{mol} \]

### Step 5: Calculate the mass of gold plated.
The molar mass of gold (Au) is approximately 197 g/mol. The mass of gold can be calculated using:
\[ \text{Mass of Au} = \text{Moles of Au} \times \text{Molar mass of Au} \]
Substituting the values:
\[ \text{Mass of Au} = 0.000933 \, \text{mol} \times 197 \, \text{g/mol} \approx 0.184 \, \text{g} \]

### Final Answer:
The mass of gold plated is approximately **0.184 g**.

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