Question

10 ampere of current is passed through an aqueous solution of salt of $M^{4+}$ for 1 hour. 2.977 g of M is deposited. If current efficiency is 10(x)% find the value of x. (Atomic mass of M = 106.4 g/mol)

Answer 1

3

Answer 2

Here's a detailed solution:

1. Calculate the total charge (Q) passed through the solution: The current ($I$) is 10 A. The time ($t$) is 1 hour. We need to convert this to seconds: $t = 1 \text{ hour} \times 60 \text{ minutes/hour} \times 60 \text{ seconds/minute} = 3600 \text{ seconds}$ The total charge passed is given by: $Q = I \times t$ $Q = 10 \text{ A} \times 3600 \text{ s} = 36000 \text{ C}$

2. Calculate the theoretical mass of M deposited ($m_{theoretical}$): The ion is $M^{4+}$, which means 4 moles of electrons are required to deposit 1 mole of M. The reaction at the cathode is: $M^{4+} + 4e^- \rightarrow M$ According to Faraday's first law of electrolysis, the mass of a substance deposited is given by: $m = \frac{Q \times \text{Molar Mass}}{n \times F}$ Where:

• $Q$ is the total charge in Coulombs ($36000 \text{ C}$).
• Molar Mass of M is $106.4 \text{ g/mol}$.
• $n$ is the number of electrons involved per mole of substance (valency), which is 4 for $M^{4+}$.
• $F$ is Faraday's constant, approximately $96500 \text{ C/mol}$.

Substituting the values: $m_{theoretical} = \frac{36000 \text{ C} \times 106.4 \text{ g/mol}}{4 \text{ mol e}^- \times 96500 \text{ C/mol e}^-}$ $m_{theoretical} = \frac{3830400}{386000}$ $m_{theoretical} \approx 9.9233 \text{ g}$

3. Calculate the current efficiency and find the value of x: Current efficiency is defined as the ratio of the actual mass deposited to the theoretical mass deposited, expressed as a percentage: $\text{Current Efficiency} = \frac{\text{Actual mass deposited}}{\text{Theoretical mass deposited}} \times 100\%$ Given: Actual mass deposited ($m_{actual}$) = $2.977 \text{ g}$ Theoretical mass deposited ($m_{theoretical}$) = $9.9233 \text{ g}$

$\text{Current Efficiency} = \frac{2.977 \text{ g}}{9.9233 \text{ g}} \times 100\%$ $\text{Current Efficiency} \approx 0.29999 \times 100\%$ $\text{Current Efficiency} \approx 30\%$

The problem states that the current efficiency is $10(x)\%$. So, we can set up the equation: $10(x)\% = 30\%$ $10x = 30$ $x = \frac{30}{10}$ $x = 3$

The value of x is 3.

The final answer is $\boxed{3}$.

Explanation of the solution:

1. Calculate the total charge passed using $Q = I \times t$.
2. Calculate the theoretical mass of M that should have been deposited using Faraday's first law: $m_{theoretical} = \frac{Q \times \text{Molar Mass}}{n \times F}$.
3. Calculate the current efficiency using the formula: $\text{Current Efficiency} = \frac{\text{Actual mass}}{\text{Theoretical mass}} \times 100\%$.
4. Equate the calculated current efficiency to $10(x)\%$ to find the value of x.