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**Hint:**Calculate the charge required for the electrolysis $2M$ solution of $Ni{\left( {N{O_3}} \right)_2}$ by using $1F = 96500C$. Then, use Faraday's first law of electrolysis to calculate the quantity of charge passed.

By using the combination of Faraday’s law of electrolysis, we will calculate the number of moles of the nickel. Then, by using the formula of molarity we will calculate the molarity of solution at the end of electrolysis.

**Complete step by step answer:**

From the question, the electrode can be represented by –

$N{i^{2 + }} + 2{e^ - } \to Ni$

Now, we know that –

$1F = 96500C$

where, $C$ is the unit of charge, Coulomb.

Therefore, now to calculate the charge required for the electrolysis $2M$ solution of $Ni{\left( {N{O_3}} \right)_2}$ -

$\therefore 2 \times 96500 = 193000C$

Now, we will use Faraday's first law of electrolysis which states that “During electrolysis, the substance which undergoes the chemical reaction at any electrode is proportional to the quantity of electricity passed through the electrolyte.”

$

W\propto Q \\

W\propto It \\

\Rightarrow W = kIt \\

$

where, $W$ is the weight of substance

$Q$ is the charge

$t$ is the time at which current flows

$k$ is the proportionality constant

$I$ is the amount of flowing of current

According to the question it is given that –

Time taken for electrolysis, $t = 6h = 6 \times 60 \times 60s = 21600s$

Therefore, to calculate the quantity of charge passed in the electrolysis of nickel we use Faraday’s first law of electrolysis –

$ \Rightarrow 3.7 \times 21600 = 79920C$

Now, we will use the combination of Faraday’s Law of Electrolysis which states that, “by passing the 1 mole of electrons $1F = 96500C$, then, the equivalent weight of the substance is deposited.”

$W = \dfrac{E}{F}It$

where, $E$ is the equivalent weight.

$W$ is the weight deposited.

Therefore, the number of moles of nickel in the electrolysis can be given by –

$ \Rightarrow \dfrac{{79920}}{{2 \times 96500}} = 0.414mol$

Thus, the number of moles of nickel which are deposited –

$ \Rightarrow 2 \times 0.5 - 0.414 = 0.586mol$

Hence, the number of moles of nickel which are present in $0.5l$ is $0.586mol$.

Now, the molarity can be given by formula –

$M = \dfrac{n}{V} \cdots \left( 1 \right)$

where, $n$ is the number of moles of nickel

$V$ is the volume

Putting the values of $n$ and $V$ in the equation $\left( 1 \right)$, so, the molarity of the left $2M$ solution –

$M = \dfrac{{0.586}}{{0.5}}M = 1.172M$

**So, the correct answer is option B.**

**Note:**The process of decomposition of the substance by passing the electric current is called Electrolysis. Now, the combination of Faraday’s law of electrolysis can be set up by both Faraday’s first law and second law of electrolysis. So, the Faraday’s second law of electrolysis states that, “the deposited mass of substance on passing a certain amount of charge is directly proportional to its chemical equivalent weight. Mathematically, it can be represented by –

$\dfrac{{{W_1}}}{{{W_2}}} = \dfrac{{{E_1}}}{{{E_2}}}$

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