Standard Gibb's energy of reaction `(Delta_(r )G^(@))` at a certain temperature can be computed `Delta_(r )G^(@)=Delta_(r)H^(@)-T.Delta_(r )S^(@)` and the change in the value of `Delta_(r)H^(@)` and `Delta_(r)S^(@)` for a reaction with temperature can be computed as follows :
`Delta_(r )H_(T_(2))^(@)-Delta_(r )H_(T_(1))^(@)=Delta_(r )C_(p)^(@)(T_(2)-T_(1))`
`Delta_(r )S_(T_(2))^(@)-Delta_(r )S_(T_(1))^(@)=Delta_(r )C_(p)^(@)ln.(T_(2)/T_(1))`
`" "Delta_(r )G^(@)=Delta_(r)H^(@)-T.Delta_(r)S^(@)`
and `" by "Delta_(r )G^(@)=-"RT " ln K_(eq)`.
Consider the following reaction : `CO(g)+2H_(2)(g)iffCH_(3)OH(g)`
Given : `Delta_(f)H^(@)(CH_(3)OH,g)=-201 " kJ"//"mol", " "Delta_(f)H^(@)(CO,g)=-114" kJ"//"mol"`
`S^(@)(CH_(3)OH,g)=240" J"//"K-mol, "S^(@)(H_(2),g)=29" JK"^(-1)" mol"^(-1)`
`S^(@)(CO,g)=198 " J"//"mol-K, "C_(p,m)^(@)(H_(2))=28.8 " J"//"mol-K"`
`C_(p,m)^(@)(CO)=29.4 " J"//"mol-K, "C_(p,m)^(@)(CH_(3)OH)=44 " J"//"mol-K"`
and `" "ln ((320)/(300))=0.06`, all data at 300 K
`Delta_(r )H^(@)` at 300 K for the reaction is :

Options

A. `-87` kJ/mol
B. 87 kJ/mol
C. `-315` kJ/mol
D. `-288` kJ/mol

Explanation

To calculate the standard enthalpy change of the reaction \( \Delta_r H^\circ \) at 300 K for the reaction

\[ \text{CO(g)} + 2\text{H}_2(g) \iff \text{CH}_3\text{OH}(g) \]

we will use the standard enthalpy of formation values provided in the question.

### Step-by-Step Solution:

1. **Identify the Standard Enthalpy of Formation Values**:
- \( \Delta_f H^\circ (\text{CH}_3\text{OH}, g) = -201 \, \text{kJ/mol} \)
- \( \Delta_f H^\circ (\text{CO}, g) = -114 \, \text{kJ/mol} \)
- \( \Delta_f H^\circ (\text{H}_2, g) = 0 \, \text{kJ/mol} \) (since it is in its standard state)

2. **Write the Reaction**:
The balanced reaction is:
\[ \text{CO(g)} + 2\text{H}_2(g) \rightarrow \text{CH}_3\text{OH}(g) \]

3. **Apply the Formula for Enthalpy Change**:
The formula for the standard enthalpy change of the reaction is:
\[ \Delta_r H^\circ = \Delta_f H^\circ (\text{products}) - \Delta_f H^\circ (\text{reactants}) \]

4. **Calculate the Enthalpy Change**:
Substitute the values into the formula:
\[ \Delta_r H^\circ = \Delta_f H^\circ (\text{CH}_3\text{OH}) - [\Delta_f H^\circ (\text{CO}) + 2 \cdot \Delta_f H^\circ (\text{H}_2)] \]
\[ \Delta_r H^\circ = -201 \, \text{kJ/mol} - [-114 \, \text{kJ/mol} + 2 \cdot 0] \]
\[ \Delta_r H^\circ = -201 \, \text{kJ/mol} + 114 \, \text{kJ/mol} \]
\[ \Delta_r H^\circ = -201 + 114 = -87 \, \text{kJ/mol} \]

5. **Final Result**:
The standard enthalpy change of the reaction \( \Delta_r H^\circ \) at 300 K is:
\[ \Delta_r H^\circ = -87 \, \text{kJ/mol} \]