To solve the problem of determining the mass of nickel deposited at the cathode during the electrolysis of a nickel nitrate solution, we can follow these steps:
### Step 1: Understand the Electrolysis Process
When nickel nitrate (Ni(NO₃)₂) is electrolyzed, nickel ions (Ni²⁺) are reduced at the cathode, gaining electrons to form solid nickel (Ni). The half-reaction at the cathode can be represented as:
\[ \text{Ni}^{2+} + 2e^- \rightarrow \text{Ni (s)} \]
### Step 2: Calculate the Total Charge (Q)
We need to calculate the total charge passed through the solution during the electrolysis. The formula for charge is:
\[ Q = I \times t \]
Where:
- \( I \) = current in amperes (5 A)
- \( t \) = time in seconds
Convert the time from minutes to seconds:
\[ t = 20 \text{ minutes} \times 60 \text{ seconds/minute} = 1200 \text{ seconds} \]
Now, calculate \( Q \):
\[ Q = 5 \, \text{A} \times 1200 \, \text{s} = 6000 \, \text{C} \]
### Step 3: Calculate the Number of Moles of Electrons
Using Faraday's constant, which is approximately \( 96500 \, \text{C/mol} \), we can find the number of moles of electrons (n):
\[ n = \frac{Q}{F} \]
Where \( F \) is Faraday's constant.
Substituting the values:
\[ n = \frac{6000 \, \text{C}}{96500 \, \text{C/mol}} \approx 0.0621 \, \text{mol} \]
### Step 4: Calculate the Moles of Nickel Deposited
From the half-reaction, we see that 2 moles of electrons are required to deposit 1 mole of nickel. Therefore, the moles of nickel deposited (\( n_{\text{Ni}} \)) can be calculated as:
\[ n_{\text{Ni}} = \frac{n}{2} = \frac{0.0621}{2} = 0.03105 \, \text{mol} \]
### Step 5: Calculate the Mass of Nickel Deposited
Now, we can calculate the mass of nickel deposited using its molar mass (58.7 g/mol):
\[ \text{mass} = n_{\text{Ni}} \times \text{molar mass of Ni} \]
\[ \text{mass} = 0.03105 \, \text{mol} \times 58.7 \, \text{g/mol} \approx 1.82 \, \text{g} \]
### Step 6: Round to the Nearest Whole Number
Finally, rounding the mass to the nearest whole number gives:
\[ \text{mass} \approx 2 \, \text{g} \]
### Final Answer
The mass of nickel deposited at the cathode is approximately **2 grams**.
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