For the hypothetical reaction
`A_(2)(g)+B_(2)(g) to 2AB_(g)`
`DeltaG_(r)^(@)" and " DeltaS_(r)^(@)" are " 20KJ//moland-20JK^(-1)mol^(-1) "respectively at" 200K`
`Delta_(r)C_(p)" is "20 JK^(-1) " then " DeltaH_(r)^(@) "at" 400K is `

Options

A. 20 kJ/mol
B. `7.98` kJ/mos
C. 28 kJ/mol
D. None of these

Explanation

To find the standard enthalpy change (ΔH_r^(@)) for the reaction at 400 K, we will follow these steps:

### Step 1: Calculate ΔH_r^(@) at 200 K
We can use the Gibbs free energy equation:
\[ \Delta G_r^(@) = \Delta H_r^(@) - T \Delta S_r^(@) \]
Rearranging gives us:
\[ \Delta H_r^(@) = \Delta G_r^(@) + T \Delta S_r^(@) \]

### Step 2: Substitute the known values at 200 K
Given:
- ΔG_r^(@) = 20 kJ/mol
- ΔS_r^(@) = -20 J/K·mol (which is -0.020 kJ/K·mol when converted to kJ)
- T = 200 K

Substituting these values into the equation:
\[ \Delta H_r^(@) = 20 \text{ kJ/mol} + (200 \text{ K})(-0.020 \text{ kJ/K·mol}) \]
\[ \Delta H_r^(@) = 20 \text{ kJ/mol} - 4 \text{ kJ/mol} \]
\[ \Delta H_r^(@) = 16 \text{ kJ/mol} \]

### Step 3: Calculate ΔH_r^(@) at 400 K using ΔC_p
We can use the formula:
\[ \Delta H_r^(@) (T_2) - \Delta H_r^(@) (T_1) = \Delta C_p (T_2 - T_1) \]
Where:
- ΔC_p = 20 J/K (which is 0.020 kJ/K when converted to kJ)
- T_1 = 200 K
- T_2 = 400 K

Substituting the values:
\[ \Delta H_r^(@) (400) - 16 \text{ kJ/mol} = (0.020 \text{ kJ/K})(400 \text{ K} - 200 \text{ K}) \]
\[ \Delta H_r^(@) (400) - 16 \text{ kJ/mol} = (0.020 \text{ kJ/K})(200 \text{ K}) \]
\[ \Delta H_r^(@) (400) - 16 \text{ kJ/mol} = 4 \text{ kJ/mol} \]

### Step 4: Solve for ΔH_r^(@) at 400 K
\[ \Delta H_r^(@) (400) = 16 \text{ kJ/mol} + 4 \text{ kJ/mol} \]
\[ \Delta H_r^(@) (400) = 20 \text{ kJ/mol} \]

### Final Answer
Thus, the standard enthalpy change (ΔH_r^(@)) at 400 K is **20 kJ/mol**.

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