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Step-by-Step Explanation
1. Understanding the Question
The question provides the standard enthalpy of formation of methane (
$ \Delta_fH^\circ $
for
$ CH_4(g) $
) at 298 K and asks which additional information is needed to determine the average energy of a C–H bond in methane. The correct answer given is the enthalpy of sublimation of carbon and the dissociation energy of
$ H_2 $
.
2. Standard Enthalpy of Formation of Methane
The standard enthalpy of formation of methane is defined by the reaction:
$ C(s,\text{graphite}) + 2\,H_2(g) \;\longrightarrow\; CH_4(g) $
and the value at 298 K is
$ \Delta_fH^\circ = -74.8\;\text{kJ mol}^{-1}.$
3. From Elements in Standard State to Atoms
To find the average C–H bond enthalpy, we conceptually need to consider the process that goes all the way from individual atoms
($ C(g) $
and
$ H(g) $
) to the molecule
($ CH_4(g) $
). This reasoning helps us connect the tabulated values (such as the standard enthalpy of formation) with the bond enthalpies.
The necessary steps are:
Step for Carbon: Sublimate the carbon from solid graphite to gaseous atoms:
$ C(s,\text{graphite}) \;\longrightarrow\; C(g). $
The enthalpy change here is the enthalpy of sublimation of carbon.
Step for Hydrogen: Dissociate hydrogen molecules into atoms:
$ H_2(g) \;\longrightarrow\; 2\,H(g). $
The enthalpy change here is the bond dissociation energy of
$ H_2 $
.
Once we have gas-phase carbon and hydrogen atoms, these combine to form methane:
$ C(g) + 4\,H(g) \;\longrightarrow\; CH_4(g). $
The enthalpy change involved in this step is related to the C–H bond formation energies.
4. Why These Two Pieces of Data Are Needed
Enthalpy of sublimation of carbon is necessary to evaluate the energy change when converting
$ C(s,\text{graphite}) $
to
$ C(g) $
.
Bond dissociation energy of
$ H_2 $
is required to break
$ H_2 $
gas into hydrogen atoms,
which will later bond to carbon atoms to form C–H bonds.
Without these two quantities, we cannot correctly account for the energy changes from the elemental forms of carbon and hydrogen to the separate atoms needed to build the
$ CH_4 $
molecule. When we know how much energy it takes to create atoms from elements in their standard states (solid C and
$ H_2 $
gas), and we also know the standard enthalpy of formation of methane, we can calculate the total energy involved per C–H bond.
5. Conclusion
Therefore, to determine the average energy of a C–H bond in methane from its standard enthalpy of formation, we must know:
the enthalpy of sublimation of carbon (to get from solid C to gaseous C atoms), and
the bond dissociation energy of
$ H_2 $
(to get from the hydrogen molecule to separate hydrogen atoms).
These data, combined with the given
$ \Delta_fH^\circ(CH_4) $
, allow us to compute the average C–H bond enthalpy.
