The heats of combustion of `C_(x)H_(y)`, carbon and hydrogen are a, b and c cal respectively. The heat of formation of `C_(x)H_(y)`, will be:

Options

A. `-(xb+(yc)/(2)-(a)/(2))" cal"`
B. `-(xb+(yc)/(2)-a)" cal"`
C. `(xb-(yc)/(2)+(a)/(2))" cal"`
D. `(xb-(yc)/(2)-(a)/(2))" cal"`

Answer

To find the heat of formation of the compound \( C_xH_y \), we can use the heats of combustion of the compound, carbon, and hydrogen. Here’s a step-by-step solution:

### Step 1: Understand the Concept of Heat of Formation
The heat of formation (\( \Delta H_f \)) of a compound is the heat change that results when one mole of the compound is formed from its elements in their standard states.

### Step 2: Write the Combustion Reactions
1. The combustion of \( C_xH_y \):
\[ C_xH_y + \left( x + \frac{y}{4} \right) O_2 \rightarrow xCO_2 + \frac{y}{2} H_2O \]
The heat of combustion for this reaction is given as \( A \).

2. The combustion of carbon:
\[ C + O_2 \rightarrow CO_2 \]
The heat of combustion for carbon is given as \( B \).

3. The combustion of hydrogen:
\[ H_2 + \frac{1}{2} O_2 \rightarrow H_2O \]
The heat of combustion for hydrogen is given as \( C \).

### Step 3: Write the Heat of Combustion in Terms of Formation
Using Hess's law, we can relate the heat of formation to the heats of combustion:
\[ \Delta H_f (C_xH_y) = \Delta H_{combustion}(C_xH_y) - \left[ x \Delta H_{combustion}(C) + \frac{y}{2} \Delta H_{combustion}(H_2) \right] \]

### Step 4: Substitute the Known Values
Substituting the values of heats of combustion:
\[ \Delta H_f (C_xH_y) = A - \left[ xB + \frac{y}{2}C \right] \]

### Step 5: Rearranging the Equation
Rearranging gives us:
\[ \Delta H_f (C_xH_y) = A - xB - \frac{y}{2}C \]

### Final Answer
Thus, the heat of formation of \( C_xH_y \) is:
\[ \Delta H_f (C_xH_y) = A - xB - \frac{y}{2}C \]

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