Predict whether it is possible or not to reduce magnesium oxide using carbon at `298K` according to the reaction.
`MgO(s) +C(s) rarr Mg(s) +CO(g)`
`Delta_(r)H^(Theta) = +491.18 kJ mol^(-1)` and `Delta_(r)S^(Theta) = 197.67 J K^(-1) mol^(-1)`
If not at what temperature, the reaction becomes spontaneous.

Explanation

We know
`Delta_(r)G^(@) = Delta_(r)H^(@) -Tdelta_(r)S^(@)`
`Delta_(r)H^(@) = + 491.18 kJ mol^(-1)`
`Delta_(r)S^(@) = 197.67 xx 10^(-3)kJ mol^(-1)K^(-1), T = 298 K `
`therefore Delta_(r)G^(@) = 491.18 kJ mol^(-1) - 298 K xx (197.67 xx 10^(-3) kJ mol^(-1)K^(-1))`
` = 491.18 kJ mol^(-1) - 58.9 kJ mol^(-1)`
` = 432.28 kJ mol^(-1)`
Since Gibbs energy change is positive, therefore at 298 K, the reaction is not possible.
Let us calculate T at which `Delta_(r)G^(@)` becomes zero
`Delta_(r)G^(@) = Delta_(r)H^(@) -T Delta_(r)S^(@) = 0`
`therefore T = (Delta_(r)H)/(Delta_(r)S) = (491.18 kJ mol^(-1))/(197.67 xx 10^(-3) kJ mol^(-1)K^(-1))`
= 2484.8 K
Therefore, the reaction will be spontaneous above `2484.8 K (or 2211.8^(@)C)`.