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 `1.50 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. `(0.551)`
C. `(1.84g)`
D. `(0.613g)`

Explanation

To solve the problem of how much gold (Au) will be plated in an electrolytic cell given the current and time, we can follow these steps:

### Step 1: Convert Time to Seconds
First, we need to convert the time from minutes to seconds since the current is given in amperes (A), which is equivalent to coulombs per second (C/s).

\[ \text{Time in seconds} = 1.50 \text{ minutes} \times 60 \text{ seconds/minute} = 90 \text{ seconds} \]

**Hint:** Remember that 1 minute = 60 seconds.

### Step 2: Calculate Total Charge (Q)
Next, we calculate the total charge (Q) using the formula:

\[ Q = I \times t \]

Where:
- \(I\) = current in amperes (0.30 A)
- \(t\) = time in seconds (90 s)

\[ Q = 0.30 \, \text{A} \times 90 \, \text{s} = 27 \, \text{C} \]

**Hint:** The total charge is the product of current and time.

### Step 3: Calculate the Number of Moles of Electrons
Using Faraday's constant (\(F = 96500 \, \text{C/mol}\)), we can find the number of moles of electrons (\(n_e\)):

\[ n_e = \frac{Q}{F} = \frac{27 \, \text{C}}{96500 \, \text{C/mol}} \approx 0.00028 \, \text{mol} \]

**Hint:** Faraday's constant tells us how much charge is needed to produce one mole of electrons.

### Step 4: Relate Moles of Electrons to Moles of Gold
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). Therefore, the moles of gold produced (\(n_{Au}\)) can be calculated as:

\[ n_{Au} = \frac{n_e}{3} = \frac{0.00028 \, \text{mol}}{3} \approx 0.0000933 \, \text{mol} \]

**Hint:** Use the stoichiometry of the reaction to relate moles of electrons to moles of gold.

### Step 5: Calculate the Mass of Gold Plated
Now, we can calculate the mass of gold plated using the molar mass of gold (approximately 197 g/mol):

\[ \text{Mass of Au} = n_{Au} \times \text{Molar mass of Au} = 0.0000933 \, \text{mol} \times 197 \, \text{g/mol} \approx 0.0184 \, \text{g} \]

**Hint:** The mass can be found by multiplying the number of moles by the molar mass.

### Final Answer
The mass of gold (Au) that will be plated is approximately **0.0184 grams**.