To determine the electric charge delivered to the cathode in copper ions per second, we need to follow a few logical steps involving the relationship between mass, moles, charge, and time. Let's break it down step by step.
Understanding the Problem
We know that a current passing through a copper voltameter deposits 0.0002 kg of copper on the cathode in 100 minutes. Our goal is to find out how much charge is delivered to the cathode per second.
Step 1: Convert Mass to Moles
The first step is to convert the mass of copper deposited into moles. The molar mass of copper (Cu) is approximately 63.55 g/mol. To convert kilograms to grams, we multiply by 1000:
- 0.0002 kg = 0.0002 × 1000 g = 0.2 g
Now, we can find the number of moles of copper:
- Number of moles = mass (g) / molar mass (g/mol) = 0.2 g / 63.55 g/mol ≈ 0.00315 mol
Step 2: Calculate the Number of Atoms
Next, we need to find out how many atoms of copper correspond to the moles we just calculated. Using Avogadro's number, which is approximately 6.022 × 10²³ atoms/mol:
- Number of atoms = moles × Avogadro's number = 0.00315 mol × 6.022 × 10²³ atoms/mol ≈ 1.90 × 10²¹ atoms
Step 3: Determine the Charge per Copper Ion
Each copper ion (Cu²⁺) carries a charge of 2 elementary charges (since copper typically loses two electrons to form Cu²⁺). The elementary charge (e) is approximately 1.6 × 10⁻¹⁹ C. Therefore, the charge for one copper ion is:
- Charge per Cu²⁺ = 2 × 1.6 × 10⁻¹⁹ C = 3.2 × 10⁻¹⁹ C
Step 4: Calculate Total Charge Delivered
Now, we can calculate the total charge delivered to the cathode by multiplying the number of copper ions by the charge per ion:
- Total charge (Q) = number of ions × charge per ion = 1.90 × 10²¹ ions × 3.2 × 10⁻¹⁹ C/ion ≈ 6.08 C
Step 5: Find Charge per Second
Finally, we need to find the charge delivered per second. Since the deposition occurs over 100 minutes, we convert that to seconds:
- 100 minutes = 100 × 60 seconds = 6000 seconds
Now, we can find the charge per second:
- Charge per second = Total charge / time = 6.08 C / 6000 s ≈ 0.001013 C/s
Final Calculation
To match the options provided, we can convert this to a more manageable number:
- 0.001013 C/s = 1.013 mC/s
However, it seems there might be a misunderstanding in the options provided, as none of them match our calculated value closely. If we consider rounding or potential errors in the options, the closest answer would be 0.71 C, but based on our calculations, it appears that the charge delivered is significantly lower than the options given.
In summary, the charge delivered to the cathode in Cu ions per second is approximately 0.001013 C/s, which does not align with the options provided. It’s essential to double-check the values and calculations to ensure accuracy.
