Consider the reacting at `300K`
`C_(6)H_(6)(l)+(15)/(2)O_(2)(g)rarr 6CO_(2)(g)+3H_(2)O(l),DeltaH=-3271`kJ
What is `DeltaU` for the combustion of `1.5` mole of benzene at `27^(@)C` ?

Options

A. `-3267.25 kJ`
B. `-4900.88 kJ`
C. `-4906.5 kJ`
D. `-3274.75 kJ`

Explanation

To find the change in internal energy (ΔU) for the combustion of 1.5 moles of benzene (C₆H₆) at 27°C, we can use the relationship between enthalpy change (ΔH) and internal energy change (ΔU). The formula we will use is:

\[ \Delta H = \Delta U + \Delta N_g RT \]

Where:
- ΔH is the change in enthalpy,
- ΔU is the change in internal energy,
- ΔN_g is the change in the number of moles of gas,
- R is the universal gas constant (8.314 J/mol·K),
- T is the temperature in Kelvin.

### Step 1: Identify Given Values
- ΔH = -3271 kJ (for the combustion of 1 mole of benzene)
- T = 27°C = 27 + 273 = 300 K
- R = 8.314 J/mol·K = 0.008314 kJ/mol·K (to convert to kJ)

### Step 2: Calculate ΔNg
ΔN_g is calculated as the difference between the moles of gaseous products and the moles of gaseous reactants.

**Products:**
- 6 moles of CO₂ (g)
- 3 moles of H₂O (l) - not considered since it's not a gas

**Reactants:**
- (15/2) moles of O₂ (g)

Thus,
\[ \Delta N_g = \text{moles of products} - \text{moles of reactants} = 6 - \frac{15}{2} = 6 - 7.5 = -1.5 \]

### Step 3: Calculate ΔNgRT
Now, we can calculate ΔN_g RT:
\[ \Delta N_g RT = (-1.5) \times (0.008314 \text{ kJ/mol·K}) \times (300 \text{ K}) \]
\[ \Delta N_g RT = -1.5 \times 0.008314 \times 300 = -3.747 kJ \]

### Step 4: Calculate ΔU
Now we can rearrange the formula to find ΔU:
\[ \Delta U = \Delta H - \Delta N_g RT \]
Substituting the values:
\[ \Delta U = -3271 \text{ kJ} - (-3.747 \text{ kJ}) \]
\[ \Delta U = -3271 + 3.747 = -3267.253 \text{ kJ} \]

### Step 5: Calculate for 1.5 moles
Since the ΔU we calculated is for 1 mole of benzene, we need to multiply it by 1.5 to find the ΔU for 1.5 moles:
\[ \Delta U_{1.5} = -3267.253 \text{ kJ} \times 1.5 = -4900.88 \text{ kJ} \]

### Final Answer
Thus, the change in internal energy (ΔU) for the combustion of 1.5 moles of benzene at 27°C is:
\[ \Delta U = -4900.88 \text{ kJ} \]

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