Question

Observe the following reaction: \[ ABO_3 (s) \xrightarrow{1000 K} AO (s) + BO_2 (g) \] The enthalpy change \( \Delta H \) for this reaction is \( x \) kJ mol\(^{-1} \). What is its \( \Delta U \) (in kJ mol\(^{-1}\)) at the same temperature? (R = 8.3 \text{ J mol}^{-1} \text{ K}^{-1})

• A. \( x - 8300 \)
• B. \( x + 8.3 \)
• C. \( x + 8300 \)
• D. \( x - 8.3 \)

Hint:

For reactions involving gases, use \( \Delta H = \Delta U + \Delta n_g RT \) to account for work done by gas expansion or compression.

Answer 1

The Correct Option is D

Step 1: Understanding the Relationship Between \( \Delta H \) and \( \Delta U \)
The relationship between enthalpy change (\( \Delta H \)) and internal energy change (\( \Delta U \)) for a gaseous reaction is given by: \[ \Delta H = \Delta U + \Delta n_g RT \] where: - \( \Delta H \) = enthalpy change
- \( \Delta U \) = internal energy change
- \( \Delta n_g \) = change in the number of moles of gas
- \( R = 8.3 \text{ J mol}^{-1} \text{ K}^{-1} \)
- \( T = 1000 \) K
Step 2: Calculating \( \Delta n_g \)
The given reaction: \[ ABO_3 (s) \xrightarrow{1000 K} AO (s) + BO_2 (g) \] - Solids do not contribute to \( \Delta n_g \).
- Only gaseous species are considered.
- \( BO_2 \) is the only gaseous product.
- There are no gaseous reactants.
Thus: \[ \Delta n_g = 1 - 0 = 1 \] Step 3: Substituting Values \[ \Delta H = \Delta U + (1 \times 8.3 \times 1000) \] \[ \Delta H = \Delta U + 8.3 \text{ kJ mol}^{-1} \] \[ \Delta U = \Delta H - 8.3 \] Final Answer: The correct relation is: \[ \Delta U = x - 8.3 \] which matches Option (4): \( x - 8.3 \).