A current of 10.0 A flows for 2.00 h through an electrolytic cell containing a molten salt of metal X. This results in the decomposition of 0.250 mol of metal X at the cathode. The oxidation state of X in the molten salt is: (F = 96,500 C)

To find the oxidation state of metal X in the molten salt, we start by recalling Faraday's law of electrolysis. The law states that the amount of substance deposited at an electrode is proportional to the charge passed through the cell. The formula relating the moles of metal deposited ($$n$$) to the charge ($$Q$$) is:

$$ n = \frac{Q}{z \times F} $$

where $$z$$ is the number of electrons required to reduce one mole of metal ions (which is the oxidation state of the metal), and $$F$$ is the Faraday constant (96,500 C/mol). We can rearrange this formula to solve for $$z$$:

$$ z = \frac{Q}{n \times F} $$

We are given:

• Current, $$I = 10.0$$ A (amperes)
• Time, $$t = 2.00$$ h (hours)
• Moles of metal deposited, $$n = 0.250$$ mol
• Faraday constant, $$F = 96,500$$ C/mol

First, we need to calculate the total charge passed, $$Q$$. Charge is given by the product of current and time:

$$ Q = I \times t $$

Since time is in hours but current is in amperes (coulombs per second), we must convert time to seconds. There are 3600 seconds in one hour, so:

$$ t = 2.00 \times 3600 = 7200 \text{ s} $$

Now, substitute the values to find $$Q$$:

$$ Q = 10.0 \times 7200 = 72,000 \text{ C} $$

So, the total charge passed is 72,000 coulombs.

Next, substitute the known values into the formula for $$z$$:

$$ z = \frac{Q}{n \times F} = \frac{72,000}{0.250 \times 96,500} $$

First, calculate the denominator:

$$ 0.250 \times 96,500 = \frac{1}{4} \times 96,500 = 24,125 $$

So,

$$ z = \frac{72,000}{24,125} $$

Now, perform the division:

$$ z = \frac{72,000 \div 125}{24,125 \div 125} = \frac{576}{193} $$

Dividing 576 by 193:

$$ 193 \times 2 = 386 $$

$$ 576 - 386 = 190 $$

So, $$ z = 2 + \frac{190}{193} \approx 2.984 $$.

This value is very close to 3. Considering that oxidation states are integers and the given data (current, time, moles) have three significant figures, the slight discrepancy is due to rounding. The calculated value 2.984 rounds to 3, which matches one of the options.

Therefore, the oxidation state of metal X is 3+.

Hence, the correct answer is Option C.