To calculate the enthalpy of the overall reaction
[tex]\[
C(s) + H_2O(g) \rightarrow CO(g) + H_2(g),
\][/tex]
we need to manipulate the given intermediate equations in such a way that they sum up to the overall reaction.
Intermediate Equations and Steps:
1. First Equation:
[tex]\[
C(s) + O_2(g) \rightarrow CO_2(g), \quad \Delta H_1 = -393.5 \text{ kJ}
\][/tex]
To use this equation in the overall reaction, we need to reverse it to produce [tex]\( C(s) \)[/tex] and [tex]\( O_2(g) \)[/tex] from [tex]\( CO_2(g) \)[/tex]:
[tex]\[
CO_2(g) \rightarrow C(s) + O_2(g)
\][/tex]
Reversing the equation changes the sign of [tex]\(\Delta H_1\)[/tex]:
[tex]\[
\Delta H_1 = +393.5 \text{ kJ}
\][/tex]
2. Second Equation:
[tex]\[
2 CO(g) + O_2(g) \rightarrow 2 CO_2(g), \quad \Delta H_2 = -566.0 \text{ kJ}
\][/tex]
We need to produce [tex]\( CO(g) \)[/tex] on the product side, so let's halve the equation first:
[tex]\[
CO(g) + \frac{1}{2} O_2(g) \rightarrow CO_2(g)
\][/tex]
Halving the equation also halves the enthalpy change:
[tex]\[
\Delta H_2 = \frac{-566.0}{2} = -283.0 \text{ kJ}
\][/tex]
3. Third Equation:
[tex]\[
2 H_2O(g) \rightarrow 2 H_2(g) + O_2(g), \quad \Delta H_3 = 483.6 \text{ kJ}
\][/tex]
To balance the overall reaction, we also need to halve this equation:
[tex]\[
H_2O(g) \rightarrow H_2(g) + \frac{1}{2} O_2(g)
\][/tex]
Halving this equation halves the enthalpy change:
[tex]\[
\Delta H_3 = \frac{483.6}{2} = 241.8 \text{ kJ}
\][/tex]
Next, Sum up the modified enthalpies to get the overall reaction enthalpy.
Modified enthalpies:
[tex]\[
\Delta H_1 = +393.5 \text{ kJ}
\][/tex]
[tex]\[
\Delta H_2 = -283.0 \text{ kJ}
\][/tex]
[tex]\[
\Delta H_3 = 241.8 \text{ kJ}
\][/tex]
Now, summing these values:
[tex]\[
\Delta H_{\text{overall}} = \Delta H_1 + \Delta H_2 + \Delta H_3 = 393.5 \text{ kJ} - 283.0 \text{ kJ} + 241.8 \text{ kJ} = 352.3 \text{ kJ}
\][/tex]
Therefore, the enthalpy change for the overall reaction is:
[tex]\[
\Delta H_{\text{xn}} = 352.3 \text{ kJ}
\][/tex]
Necessary Steps to Calculate:
- The first equation must be reversed.
- The second equation must be halved.
- The third equation must be halved.
These are the steps required to determine the overall enthalpy change.
