$$(\Delta H - \Delta U)$$ for the formation of carbon monoxide (CO) from its elements at $$298\,K$$ is $$(R = 8.314\,J\,K^{-1}\,mol^{-1})$$

$$\Delta H = \Delta U + \Delta n_g RT$$

$$\Delta H - \Delta U = \Delta n_g RT$$

Where:

• $$\Delta n_g$$ is the change in the number of moles of gaseous products minus gaseous reactants.
• $$R$$ is the universal gas constant ($$8.314 \text{ J K}^{-1} \text{ mol}^{-1}$$).
• $$T$$ is the absolute temperature in Kelvin ($298 \text{ K}$).

• Step 1: Write the balanced standard formation equation

  The formation reaction involves producing exactly $$1 \text{ mole}$$ of gaseous carbon monoxide ($$\text{CO}$$) directly from its constituent elements in their standard reference states (graphite for carbon, diatomic gas for oxygen):

  $$\text{C}_{(s, \text{ graphite})} + \frac{1}{2}\text{O}_{2(g)} \rightarrow \text{CO}_{(g)}$$




• Step 2: Calculate $$\Delta n_g$$

  Count only the stoichiometric coefficients of species in the gaseous state ($g$):

  • Gaseous products = $$1 \text{ mol}$$ (from $$\text{CO}$$)
  • Gaseous reactants = $$\frac{1}{2} \text{ mol}$$ (from $$\text{O}_2$$; solid carbon is excluded)

  $$\Delta n_g = 1 - \frac{1}{2} = +0.5 \text{ mol}$$




• Step 3: Compute the value of $$\Delta n_g RT$$

  Substitute the known variables into our rearranged formula:

  $$\Delta H - \Delta U = (0.5 \text{ mol}) \times (8.314 \text{ J K}^{-1} \text{ mol}^{-1}) \times (298 \text{ K})$$

  $$\Delta H - \Delta U = 0.5 \times 2477.572 \text{ J mol}^{-1}$$

  $$\Delta H - \Delta U = 1238.78 \text{ J mol}^{-1}$$

Because there is a net increase in the moles of gas during formation, the work term adds a positive value of $$1238.78 \text{ J mol}^{-1}$$ to the internal energy change system.

Answer: Option B — $$1238.78 \text{ J mol}^{-1}$$