Variation of equilibrium constan K with temperature is given by van't Hoff equation
`InK=(Delta_(r)S^(@))/R-(Delta_(r)H^(@))/(RT)`
for this equation, `(Delta_(r)H^(@))` can be evaluated if equilibrium constans `K_(1)` and `K_(2)` at two temperature `T_(1)` and `T_(2)` are known.
`log(K_(2)/K_(1))=(Delta_(r)H^(@))/(2.303R)[1/T_(1)-1/T_(2)]`
Variation of `log_(10)` K with `1/T` is shown by the following graph in which straight line is at `45^(@)` hence `DeltaH^(@)` is :

Answer

(a) `9.574J mol^(-1);` (b)`A=10^(10);` (c) `9.96xx10^(9);` (d)`9.98xx10^(9)`

Explanation

(a) `log_(10)K=log_(10)A-(DeltaH_(@))/(2.303RT)`
It is an equation of a straight line of the type `y=c+mx`
Slope 'm'=`tan0=(DeltaH^(@))/(2.303R)`
`0.5=(DeltaH^(@))/2.303xx8.314`
`DeltaH^(@)=9.574J mol^(-1)`
(b) `"Intercept",C'=log_(10)A=10` `A=10^(10)`
(c) `log K=10-(9.574)/(2.303xx8.314xx298)`
`K=9.96xx10^(9)`
(d) `log((K_(2))/(K_(1)))=(DeltaH)/2.303R {(1)/(T_(1))-(1)/(T_(2))}`
`log (K_(2))/(9.96xx10_(9))=(9.574)/(2.303xx8.314) {(1)/(298)-(1)/(798)}`
On solving `K_(2)=9.98xx10_(9)`