In the complete oombustion of butanol `C_4H_9OH (I)", if "triangleH` is enthalpy of combustion and AE is the heat of combustion at constant volume, then

Options

A. `triangleH lt triangleE`
B. `triangleH=triangleE`
C. `triangleH gt triangleE`
D. `triangleH, triangle` relation cannot be predicted

Answer

To find the relationship between the enthalpy of combustion (ΔH) and the heat of combustion at constant volume (ΔE) for the complete combustion of butanol (C₄H₉OH), we will follow these steps:

### Step 1: Write the balanced chemical equation for the combustion of butanol.
The complete combustion of butanol can be represented as:
\[ C_4H_9OH (l) + O_2 (g) \rightarrow 4 CO_2 (g) + 5 H_2O (l) \]

### Step 2: Identify the number of moles of gaseous products and reactants.
In the balanced equation:
- Products: 4 moles of CO₂ (g) and 5 moles of H₂O (l)
- Reactants: 1 mole of C₄H₉OH (l) and x moles of O₂ (g)

To find the moles of O₂ needed, we can balance the equation:
\[ C_4H_9OH + 6 O_2 \rightarrow 4 CO_2 + 5 H_2O \]
Thus, we have:
- Total moles of products (gaseous): 4 (from CO₂) + 0 (from H₂O) = 4
- Total moles of reactants (gaseous): 6 (from O₂)

### Step 3: Calculate ΔN_G (change in the number of moles of gas).
Using the formula:
\[ \Delta N_G = \text{(moles of gaseous products)} - \text{(moles of gaseous reactants)} \]
Substituting the values:
\[ \Delta N_G = 4 - 6 = -2 \]

### Step 4: Use the relationship between ΔH and ΔE.
The relationship between the enthalpy change (ΔH) and the internal energy change (ΔE) at constant volume is given by:
\[ \Delta H = \Delta E + \Delta N_G RT \]
Substituting ΔN_G:
\[ \Delta H = \Delta E + (-2)RT \]
This simplifies to:
\[ \Delta H = \Delta E - 2RT \]

### Step 5: Analyze the relationship.
From the equation:
\[ \Delta H = \Delta E - 2RT \]
We can deduce that:
\[ \Delta E = \Delta H + 2RT \]
This implies that ΔE is greater than ΔH because 2RT is a positive term.

### Conclusion:
Thus, the relationship shows that the heat of combustion at constant volume (ΔE) is greater than the enthalpy of combustion (ΔH).