The difference between `Delta H` and `Delta U` for the combustion of methane at `27^(@)C` will be (in `J mol^(-1)`)

Options

A. `8.314 xx 27 xx - 3`
B. `8.314 xx 300 xx (-3)`
C. `8.314 xx 300 xx (-2)`
D. `8.314 xx 300 xx1`

Explanation

To find the difference between \(\Delta H\) and \(\Delta U\) for the combustion of methane at \(27^\circ C\), we can follow these steps:

### Step 1: Write the combustion reaction of methane
The balanced chemical equation for the combustion of methane (\(CH_4\)) is:
\[ CH_4(g) + 2 O_2(g) \rightarrow CO_2(g) + 2 H_2O(l) \]

### Step 2: Identify the number of gaseous molecules
Next, we need to determine the number of gaseous molecules on both the reactant and product sides:
- **Reactants**: \(1\) (from \(CH_4\)) + \(2\) (from \(O_2\)) = \(3\) gaseous molecules
- **Products**: \(1\) (from \(CO_2\)) + \(0\) (from \(H_2O\), which is liquid) = \(1\) gaseous molecule

### Step 3: Calculate \(\Delta N_G\)
\(\Delta N_G\) is the change in the number of gaseous molecules:
\[ \Delta N_G = \text{(Number of gaseous products)} - \text{(Number of gaseous reactants)} = 1 - 3 = -2 \]

### Step 4: Use the relationship between \(\Delta H\) and \(\Delta U\)
The relationship between \(\Delta H\) and \(\Delta U\) is given by the equation:
\[ \Delta H = \Delta U + \Delta N_G RT \]
Substituting \(\Delta N_G\):
\[ \Delta H = \Delta U - 2RT \]

### Step 5: Find the difference \(\Delta H - \Delta U\)
Rearranging the equation gives:
\[ \Delta H - \Delta U = -2RT \]

### Step 6: Substitute the values for \(R\) and \(T\)
- \(R = 8.314 \, J \, mol^{-1} \, K^{-1}\)
- Temperature \(T = 27^\circ C = 300 \, K\)

Now substitute these values into the equation:
\[ \Delta H - \Delta U = -2 \times 8.314 \, J \, mol^{-1} \, K^{-1} \times 300 \, K \]

### Step 7: Calculate the final value
Calculating this gives:
\[ \Delta H - \Delta U = -2 \times 8.314 \times 300 = -4988.4 \, J \, mol^{-1} \]

### Final Answer
Thus, the difference between \(\Delta H\) and \(\Delta U\) for the combustion of methane at \(27^\circ C\) is approximately:
\[ \Delta H - \Delta U \approx -4988.4 \, J \, mol^{-1} \]