In the reactions
`HCl + NaOH rightarrow NaCl + H_(2)O + x cal.` ` H_(2)SO_(4) + 2NaOH rightarrow Na_(2)SO_(4) + 2 H_(2)O + y cal.`

Options

A. ` x = y `
B. `x = 2y`
C. `x = y/2`
D. `x = sqrt y`

Answer

To solve the problem, we need to analyze the two given neutralization reactions and determine the relationship between the heat released (in calories) in each reaction.

### Step-by-Step Solution:

1. **Identify the Reactions**:
- The first reaction is:
\[ \text{HCl} + \text{NaOH} \rightarrow \text{NaCl} + \text{H}_2\text{O} + x \text{ cal} \]
- The second reaction is:
\[ \text{H}_2\text{SO}_4 + 2\text{NaOH} \rightarrow \text{Na}_2\text{SO}_4 + 2\text{H}_2\text{O} + y \text{ cal} \]

2. **Understanding Neutralization**:
- Neutralization reactions involve the reaction of an acid with a base to produce salt and water. In these reactions, the heat released is due to the formation of water from hydrogen ions (H⁺) and hydroxide ions (OH⁻).

3. **Analyze the First Reaction**:
- In the first reaction, one mole of HCl reacts with one mole of NaOH to produce one mole of water (H₂O) and releases \( x \) calories of energy.
- Therefore, we can say that for the formation of 1 mole of water, the energy released is \( x \) calories.

4. **Analyze the Second Reaction**:
- In the second reaction, one mole of H₂SO₄ reacts with two moles of NaOH to produce one mole of Na₂SO₄ and two moles of water (2 H₂O) and releases \( y \) calories of energy.
- Here, since 2 moles of water are formed, the energy released per mole of water is \( \frac{y}{2} \) calories.

5. **Establish the Relationship**:
- From the first reaction, we have:
\[ \text{Energy released for 1 mole of H}_2\text{O} = x \text{ cal} \]
- From the second reaction, we have:
\[ \text{Energy released for 1 mole of H}_2\text{O} = \frac{y}{2} \text{ cal} \]
- Since both expressions represent the energy released for the formation of 1 mole of water, we can set them equal to each other:
\[ x = \frac{y}{2} \]

6. **Final Answer**:
- Rearranging the equation gives us:
\[ x = \frac{y}{2} \]
- Thus, the relationship between \( x \) and \( y \) is:
\[ x = \frac{y}{2} \]

### Conclusion:
The correct answer is \( x = \frac{y}{2} \).