The `Delta H^(0)` for the following reaction at 298 K is -36.4 kJ
`1//2 H_(2) (g) + 1//2Br_(2) (I) rarr HBr(g)`
Calculate `DeltaU^(0)` at 298 K. The universal gas constant, R, is 8.314 J/mol K.

Options

A. `-35.2kJ`
B. `+35.2kJ`
C. `-36.4 kJ`
D. `-37.6 kJ`

Answer

To calculate the change in internal energy (ΔU°) for the given reaction at 298 K, we can use the relationship between the change in enthalpy (ΔH°) and the change in internal energy (ΔU°). The formula we will use is:

\[ \Delta H = \Delta U + \Delta N_g R T \]

Where:
- ΔH = change in enthalpy
- ΔU = change in internal energy
- ΔN_g = change in the number of moles of gas
- R = universal gas constant
- T = temperature in Kelvin

### Step 1: Identify the values given in the problem
- ΔH° = -36.4 kJ
- R = 8.314 J/mol·K = 0.008314 kJ/mol·K (after converting from Joules to kilojoules)
- T = 298 K

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

From the reaction:
\[ \frac{1}{2} H_2 (g) + \frac{1}{2} Br_2 (l) \rightarrow HBr (g) \]

- Moles of gaseous products = 1 (from HBr)
- Moles of gaseous reactants = 0.5 (from H2)

Thus,
\[ \Delta N_g = \text{Moles of products} - \text{Moles of reactants} = 1 - 0.5 = 0.5 \]

### Step 3: Substitute the values into the equation
Now we can substitute the values into the equation:

\[ \Delta H = \Delta U + \Delta N_g R T \]

Rearranging gives us:

\[ \Delta U = \Delta H - \Delta N_g R T \]

Substituting the known values:

\[ \Delta U = -36.4 \text{ kJ} - (0.5)(0.008314 \text{ kJ/mol·K})(298 \text{ K}) \]

### Step 4: Calculate the term involving R and T
Calculating the term:

\[ 0.5 \times 0.008314 \text{ kJ/mol·K} \times 298 \text{ K} = 1.239 \text{ kJ} \]

### Step 5: Final calculation of ΔU
Now substitute this value back into the equation:

\[ \Delta U = -36.4 \text{ kJ} - 1.239 \text{ kJ} \]
\[ \Delta U = -37.639 \text{ kJ} \approx -37.6 \text{ kJ} \]

### Final Answer
\[ \Delta U^{0} \text{ at } 298 \text{ K} = -37.6 \text{ kJ} \]

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