Animated Solution for Chemistry - Electrochemistry: An acidic solution of dichromate is electrolysed for 8 minutes using 2A current. As per the following equation ${\text{Cr}}_2{\text{O}}_7^{2-}+14{\text{H}}^{+}+6{\text{e}}^{-}⟶2{\text{Cr}}^{3+}+7{\text{H}}_2\text{O}$ The amount of ${\text{Cr}}^{3+}$ obtained was 0.104 g. The efficiency of the process(in \%) is (Take : $F=96000\text{ C}$, atomic mass of chromium $=52$) ........... .

The Electrolytic Cell

Imagine you are standing in a laboratory, observing an electrolytic cell filled with a vibrant orange acidic solution of dichromate. You flip a switch, and a steady current of $2\text{ A}$ begins to flow through the circuit. You let it run for exactly $8\text{ minutes}$.

During this time, a fascinating chemical transformation is taking place at the microscopic level. The dichromate ions (${\text{Cr}}_2{\text{O}}_7^{2-}$) are being reduced at the cathode, depositing solid chromium. But in the real world, no process is perfectly efficient. Some energy is lost to heat, and side reactions might occur. Our mission is to find out exactly how efficient this specific process was, given that we only obtained $0.104\text{ g}$ of chromium.

Faraday's First Law

To understand what should have happened, we first need to quantify the electrical effort we put in. According to Faraday's First Law of Electrolysis, the amount of substance deposited is directly proportional to the total charge passed through the solution.

The formula for charge is beautifully simple:

However, there is a classic trap here! The unit of current, the Ampere, is defined as Coulombs per second. Therefore, our time must be in seconds, not minutes.

Let's calculate the total charge:

Now, we need to translate this raw electrical charge into the language of chemistry: moles of electrons. We use Faraday's constant ($F$), which represents the charge of one mole of electrons. The problem kindly provides $F=96000\text{ C/mol}$.

We have exactly $0.01\text{ moles}$ of electrons doing the heavy lifting in our cell.

Stoichiometry

Now, let's look at the blueprint of the reaction:

This balanced equation tells us a crucial story: it takes $6\text{ moles}$ of electrons to produce $2\text{ moles}$ of chromium ions (${\text{Cr}}^{3+}$).

This means the ratio of chromium produced to electrons used is $2:6$, or simplified, $1:3$. For every mole of electrons, we get one-third of a mole of chromium.

Let's apply this to our actual electrons:

To find the theoretical mass we expected to get, we multiply these moles by the atomic mass of chromium ($52\text{ g/mol}$):

Pro Tip: Notice how we kept the mass as a fraction ($\frac{52}{300}$)? In competitive exams like JEE, avoiding early decimal approximations saves time and prevents rounding errors from compounding!

Calculating Efficiency

We expected to get $\frac{52}{300}\text{ g}$ of chromium, but the problem states we only obtained $0.104\text{ g}$.

Efficiency ($\eta$) is simply the ratio of what we actually got to what we theoretically expected, expressed as a percentage:

Let's plug in our numbers and watch the math elegantly unfold:

Since $31.2$ is exactly $0.6$ times $52$, the calculation simplifies beautifully:

The final efficiency of the electrolytic process is 60%. This means 40% of our electrical energy was consumed by other processes, such as the electrolysis of water or overcoming the internal resistance of the cell. Understanding these inefficiencies is what separates a good chemist from a great one!