The value of enthalpy change (`DeltaH`) for the reaction
`C_2H_5OH(l) + 3O_2(g) to 2CO_2(g) +3H_2O(l)`
at `27^@C` is `-1366.5 kJ mol^(-1)`. The value of internal energy change for the above reaction at this temperature will be

Options

A. `-1371.5 kJ`
B. `-1369.0 kJ`
C. `-1364.0 kJ`
D. `-1361.5 kJ`

Explanation

To find the internal energy change (ΔU) for the reaction:

\[ C_2H_5OH(l) + 3O_2(g) \rightarrow 2CO_2(g) + 3H_2O(l) \]

at a temperature of \(27^\circ C\) (which is \(300 K\)), we can use the relationship between enthalpy change (ΔH) and internal energy change (ΔU):

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

Where:
- ΔH = Enthalpy change
- ΔU = Internal energy change
- Δn_g = Change in the number of moles of gas (products - reactants)
- R = Universal gas constant (8.314 J/mol·K)
- T = Temperature in Kelvin

### Step 1: Calculate Δn_g
Identify the number of moles of gaseous products and reactants.

- **Products**:
- \(2CO_2(g)\): 2 moles of gas
- **Reactants**:
- \(3O_2(g)\): 3 moles of gas

Now, calculate Δn_g:

\[ \Delta n_g = \text{(moles of products)} - \text{(moles of reactants)} = 2 - 3 = -1 \]

### Step 2: Convert ΔH to the same units
Given ΔH = -1366.5 kJ/mol, we convert this to Joules for consistency:

\[ \Delta H = -1366.5 \text{ kJ/mol} \times 1000 \text{ J/kJ} = -1366500 \text{ J/mol} \]

### Step 3: Substitute values into the equation
Now, substitute ΔH, Δn_g, R, and T into the equation:

\[ -1366500 \text{ J/mol} = \Delta U + (-1)(8.314 \text{ J/mol·K})(300 \text{ K}) \]

Calculate the term involving R and T:

\[ -1 \times 8.314 \times 300 = -2494.2 \text{ J/mol} \]

### Step 4: Solve for ΔU
Now substitute this value back into the equation:

\[ -1366500 \text{ J/mol} = \Delta U - 2494.2 \text{ J/mol} \]

Rearranging gives:

\[ \Delta U = -1366500 \text{ J/mol} + 2494.2 \text{ J/mol} \]

Calculating the right-hand side:

\[ \Delta U = -1364005.8 \text{ J/mol} \]

### Step 5: Convert ΔU back to kJ
Convert ΔU back to kJ:

\[ \Delta U = -1364005.8 \text{ J/mol} \div 1000 = -1364.0 \text{ kJ/mol} \]

### Final Answer
The value of internal energy change (ΔU) for the reaction at 27°C is:

\[ \Delta U = -1364.0 \text{ kJ/mol} \]

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