The electrochemical equivalents of two substances are E1 and E2. The current that must pass to depos — Electrochemistry Chemistry Question

💡 Solution & Explanation

Step 1 - State Faraday's First Law of Electrolysis Faraday's First Law of Electrolysis states that the mass ($W$) of a substance deposited or liberated at any electrode during electrolysis is directly proportional to the quantity of electricity ($Q$) passed through the electrolyte: $$W \propto Q$$ Since the quantity of electricity is the product of the electric current ($I$) and the time of passage ($t$), we can write: $$W = Z \cdot I \cdot t$$ Where: * $W$ is the mass of the substance deposited. * $I$ is the electric current passed. * $t$ is the time of electrolysis. * $Z$ is the proportionality constant known as the **electrochemical equivalent** of the substance. In this question, the electrochemical equivalents of the two substances are explicitly denoted as $E_1$ and $E_2$ (instead of the standard notation $Z$). Thus, Faraday's first law for these two substances can be written as: $$W = E \cdot I \cdot t$$ Step 2 - Apply the Given Conditions We are given the following conditions for the electrolysis of the two substances: 1. The same mass (amount) of both substances must be deposited at the cathodes: $$W_1 = W_2 = W$$ 2. The electrolysis is carried out for the same duration of time: $$t_1 = t_2 = t$$ Let $I_1$ and $I_2$ be the currents passed through the respective cells to deposit these equal masses. Step 3 - Set up the Equations and Determine the Current Ratio Using Faraday's First Law, we write the deposition equations for both substances: $$\text{For Substance 1: } W_1 = E_1 \cdot I_1 \cdot t_1$$ $$\text{For Substance 2: } W_2 = E_2 \cdot I_2 \cdot t_2$$ Substituting the given conditions ($W_1 = W_2 = W$ and $t_1 = t_2 = t$) into these equations: $$W = E_1 \cdot I_1 \cdot t$$ $$W = E_2 \cdot I_2 \cdot t$$ Equating the two expressions for $W$: $$E_1 \cdot I_1 \cdot t = E_2 \cdot I_2 \cdot t$$ Since the time of passage ($t$) is non-zero, we can divide both sides of the equation by $t$: $$E_1 \cdot I_1 = E_2 \cdot I_2$$ Rearranging the equation to solve for the ratio of the currents ($\frac{I_1}{I_2}$): $$\frac{I_1}{I_2} = \frac{E_2}{E_1}$$ Therefore, the ratio of the currents must be: $$I_1 : I_2 = \mathbf{E_2 : E_1}$$ Step 4 - Evaluate the Options * **Option (A) is incorrect:** This is the direct ratio $E_1 : E_2$. This would only be correct if the deposited mass were inversely proportional to the current, which contradicts Faraday's first law. * **Option (B) is correct:** As mathematically derived, the currents must be in the inverse ratio of their electrochemical equivalents, which is $E_2 : E_1$. * **Option (C) is incorrect:** This is a mathematically incorrect expression that does not represent the physical relationship of the system. * **Option (D) is incorrect:** This is a mathematically incorrect expression that does not represent the physical relationship of the system. $$\text{Correct Option: } \boxed{\text{B}}$$