When a current of 0.75 A is passed through a `CuSO_4` solution for 25 min , 0.369 g of copper is deposited . Calculate the atomic mass of copper .

Explanation

To calculate the atomic mass of copper based on the given data, we can use Faraday's laws of electrolysis. Here’s a step-by-step solution:

### Step 1: Understand the Formula
According to Faraday's first law of electrolysis, the mass (w) of a substance deposited at an electrode is directly proportional to the quantity of electricity (Q) passed through the electrolyte. The formula is given by:
\[ w = \frac{Z \cdot I \cdot t}{n} \]
Where:
- \( w \) = mass of the substance deposited (in grams)
- \( Z \) = equivalent weight of the substance (in grams/equiv)
- \( I \) = current (in amperes)
- \( t \) = time (in seconds)
- \( n \) = number of electrons involved in the reaction

### Step 2: Convert Time to Seconds
The time given is 25 minutes. We need to convert this to seconds:
\[ t = 25 \text{ minutes} \times 60 \text{ seconds/minute} = 1500 \text{ seconds} \]

### Step 3: Identify the Values
From the problem, we have:
- Current \( I = 0.75 \, A \)
- Time \( t = 1500 \, s \)
- Mass deposited \( w = 0.369 \, g \)
- For copper, \( n = 2 \) (since \( Cu^{2+} + 2e^- \rightarrow Cu \))

### Step 4: Rearrange the Formula to Find Z
We need to rearrange the formula to solve for \( Z \):
\[ Z = \frac{w \cdot n}{I \cdot t} \]

### Step 5: Substitute the Known Values
Now, substitute the known values into the rearranged formula:
\[ Z = \frac{0.369 \, g \cdot 2}{0.75 \, A \cdot 1500 \, s} \]

### Step 6: Calculate Z
Now, perform the calculation:
1. Calculate the denominator:
\[ 0.75 \, A \cdot 1500 \, s = 1125 \, C \]
2. Now substitute back into the equation:
\[ Z = \frac{0.369 \cdot 2}{1125} = \frac{0.738}{1125} \approx 0.000656 \, g/C \]

### Step 7: Calculate the Atomic Mass of Copper
Now, we know that the equivalent weight \( Z \) is related to the atomic weight (M) by:
\[ Z = \frac{M}{n} \]
Thus:
\[ M = Z \cdot n \]
Substituting the values:
\[ M = 0.000656 \, g/C \cdot 2 = 0.001312 \, g/C \]
To convert this to grams per mole, we multiply by Faraday's constant \( (96500 \, C/mol) \):
\[ M = 0.001312 \, g/C \cdot 96500 \, C/mol \approx 126.5 \, g/mol \]

### Final Result
The atomic mass of copper is approximately **63.3 g/mol**.