For the reaction , `FeCO_(3)(s)rarrFeO(s)+CO_(2)(g),DeltaH=82.8kJ` at `25^(@)C`, what is `(DeltaE " or "DeltaU)` at `25^(@)C`?

Options

A. 82.8 kJ
B. 80.32 kJ
C. `-2394.77 kJ`
D. `-3274.75 kJ`

Explanation

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

\[ \text{FeCO}_3(s) \rightarrow \text{FeO}(s) + \text{CO}_2(g) \]

with a given enthalpy change (ΔH) of 82.8 kJ at 25°C, we can use the relationship between enthalpy change and internal energy change:

\[ \Delta H = \Delta U + \Delta N \cdot R \cdot T \]

Where:
- ΔH = change in enthalpy
- ΔU = change in internal energy
- ΔN = change in the number of moles of gas
- R = universal gas constant (8.314 J/mol·K or 0.008314 kJ/mol·K)
- T = temperature in Kelvin

### Step 1: Identify ΔN
In the given reaction, we start with 1 mole of solid (FeCO3) and produce 1 mole of solid (FeO) and 1 mole of gas (CO2).

The change in the number of moles of gas (ΔN) is:
\[ \Delta N = \text{moles of products} - \text{moles of reactants} \]
\[ \Delta N = 1 - 0 = 1 \]

### Step 2: Convert Temperature to Kelvin
The temperature given is 25°C. To convert this to Kelvin:
\[ T = 25 + 273 = 298 \, \text{K} \]

### Step 3: Use the Gas Constant
We will use the gas constant R in kJ:
\[ R = 0.008314 \, \text{kJ/mol·K} \]

### Step 4: Substitute Values into the Equation
Now, we can substitute the known values into the equation:
\[ \Delta H = \Delta U + \Delta N \cdot R \cdot T \]
\[ 82.8 \, \text{kJ} = \Delta U + (1) \cdot (0.008314 \, \text{kJ/mol·K}) \cdot (298 \, \text{K}) \]

### Step 5: Calculate ΔN · R · T
Calculating ΔN · R · T:
\[ \Delta N \cdot R \cdot T = 1 \cdot 0.008314 \cdot 298 \]
\[ = 2.478 \, \text{kJ} \]

### Step 6: Solve for ΔU
Now, substitute this value back into the equation:
\[ 82.8 = \Delta U + 2.478 \]
\[ \Delta U = 82.8 - 2.478 \]
\[ \Delta U = 80.322 \, \text{kJ} \]

### Final Answer
Thus, the change in internal energy (ΔU) at 25°C is approximately:
\[ \Delta U \approx 80.32 \, \text{kJ} \]

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