If the cost of electricity required to discharge 10g of an ion X ^ 3+ is N20.00, how much would it cost to…

1. Analyze the Question

We need to find the cost of discharging 6g of ion $Y^{2+}$, given the cost of discharging 10g of ion $X^{3+}$. This is a Faraday’s Law of Electrolysis problem where the cost is directly proportional to the quantity of electricity (charge) required.

2. Explain the Concept

According to Faraday’s First Law of Electrolysis, the mass ($m$) of a substance deposited is given by:

Where:

• $M$ = Molar mass
• $Q$ = Quantity of electricity (Charge)
• $n$ = Valency (number of electrons transferred)
• $F$ = Faraday's constant

Since the cost is proportional to the quantity of electricity ($Q$), we can rearrange the formula to find $Q$:

The cost is directly proportional to $Q$. Therefore, the ratio of costs equals the ratio of charges:

3. Show the Solution

Step 1: Calculate the charge ($Q_x$) for ion $X^{3+}$

• Mass ($m_x$) = 10g
• Valency ($n_x$) = 3
• Molar mass ($M_x$) = 27

Step 2: Calculate the charge ($Q_y$) for ion $Y^{2+}$

• Mass ($m_y$) = 6g
• Valency ($n_y$) = 2
• Molar mass ($M_y$) = 24

Step 3: Set up the proportion to find the cost Let the cost for $Y^{2+}$ be $C_y$.

Cancel $F$ from both sides:

The cost to discharge 6g of $Y^{2+}$ is N9.00.

4. Identify the Answer

Comparing our result to the options provided, the correct choice is D.

5. Tip

When dealing with electrolysis cost problems, always remember that the charge required is proportional to the number of moles multiplied by the valency ($n$). You can simplify your work by using the formula: $\text{Cost} \propto \frac{m \times n}{M}$. This saves you from having to write out the Faraday constant every time!