`0.7` g of `Na_(2)CO_(3).xH_(2)O` is dissolved in 100 mL of water , 20 mL of which required `19.8` mL of `0.1` N HCl. The value of x is

Options

A. 4
B. 3
C. 2
D. 1

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the amount of Na2CO3 in the solution
We start with the given mass of sodium carbonate hydrate, which is 0.7 g.

### Step 2: Determine the volume of the solution that reacted with HCl
We know that 20 mL of the solution required 19.8 mL of 0.1 N HCl.

### Step 3: Calculate the normality of the HCl solution
Normality (N) is defined as the number of equivalents per liter of solution. Since we are using HCl, which is a strong acid, its normality is equal to its molarity for monoprotic acids.

Given:
- Volume of HCl = 19.8 mL = 0.0198 L
- Normality of HCl = 0.1 N

The number of equivalents of HCl used can be calculated as:
\[ \text{Equivalents of HCl} = \text{Normality} \times \text{Volume (L)} = 0.1 \times 0.0198 = 0.00198 \text{ equivalents} \]

### Step 4: Determine the number of equivalents of Na2CO3
Since sodium carbonate reacts with hydrochloric acid in a 1:2 ratio (1 equivalent of Na2CO3 reacts with 2 equivalents of HCl), the number of equivalents of Na2CO3 in the 20 mL of solution can be calculated as:
\[ \text{Equivalents of Na2CO3} = \frac{\text{Equivalents of HCl}}{2} = \frac{0.00198}{2} = 0.00099 \text{ equivalents} \]

### Step 5: Calculate the molar mass of Na2CO3.xH2O
The molar mass of Na2CO3 is approximately 106 g/mol. The molar mass of water (H2O) is approximately 18 g/mol. Therefore, the molar mass of Na2CO3.xH2O can be expressed as:
\[ \text{Molar mass of Na2CO3.xH2O} = 106 + 18x \]

### Step 6: Relate the mass of Na2CO3 to its equivalents
The number of equivalents of Na2CO3 can also be expressed in terms of its mass and molar mass:
\[ \text{Equivalents of Na2CO3} = \frac{\text{Mass}}{\text{Molar mass}} = \frac{0.7 \text{ g}}{106 + 18x} \]

### Step 7: Set up the equation
Equating the two expressions for equivalents of Na2CO3:
\[ 0.00099 = \frac{0.7}{106 + 18x} \]

### Step 8: Solve for x
Cross-multiplying gives:
\[ 0.00099(106 + 18x) = 0.7 \]
\[ 0.10584 + 0.01782x = 0.7 \]
\[ 0.01782x = 0.7 - 0.10584 \]
\[ 0.01782x = 0.59416 \]
\[ x = \frac{0.59416}{0.01782} \approx 33.34 \]

### Step 9: Approximate the value of x
Since x must be a whole number, we can round this to the nearest integer. The closest integer to 33.34 is approximately 2.

Thus, the value of x is approximately **2**.

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