Consider the reaction:
`2NO(g)+O_(2)(g) rarr 2NO_(2)(g)`
Calculated the standard Gibbs energy change at `298K` and predict whether the reaction is spontaneous or not. `Delta_(f)G^(Theta) (NO) = 86.69 kJ mol^(-1), Delta_(f)G^(Theta) (NO_(2)) = 51.84 kJ mol^(-1)`.

Options

A. Yes, spontaneous
B. No, the reaction is Non-spontaneous
C. Equilibrium
D. cannot predict

Answer

To calculate the standard Gibbs energy change (ΔG) for the reaction:

\[ 2NO(g) + O_2(g) \rightarrow 2NO_2(g) \]

we will use the formula:

\[ \Delta G = \Delta G_{f}^{\Theta} \text{(products)} - \Delta G_{f}^{\Theta} \text{(reactants)} \]

### Step 1: Identify the Gibbs free energy of formation values
From the question, we have the following values:
- \( \Delta_{f}G^{\Theta} (NO) = 86.69 \, \text{kJ mol}^{-1} \)
- \( \Delta_{f}G^{\Theta} (NO_2) = 51.84 \, \text{kJ mol}^{-1} \)

### Step 2: Calculate the Gibbs free energy for the products
Since there are 2 moles of \( NO_2 \) produced, we calculate:

\[ \Delta G_{f}^{\Theta} \text{(products)} = 2 \times \Delta_{f}G^{\Theta} (NO_2) \]
\[ = 2 \times 51.84 \, \text{kJ mol}^{-1} \]
\[ = 103.68 \, \text{kJ} \]

### Step 3: Calculate the Gibbs free energy for the reactants
Since there are 2 moles of \( NO \) consumed, we calculate:

\[ \Delta G_{f}^{\Theta} \text{(reactants)} = 2 \times \Delta_{f}G^{\Theta} (NO) \]
\[ = 2 \times 86.69 \, \text{kJ mol}^{-1} \]
\[ = 173.38 \, \text{kJ} \]

### Step 4: Calculate the standard Gibbs energy change (ΔG)
Now we can substitute the values into the Gibbs energy change formula:

\[ \Delta G = \Delta G_{f}^{\Theta} \text{(products)} - \Delta G_{f}^{\Theta} \text{(reactants)} \]
\[ = 103.68 \, \text{kJ} - 173.38 \, \text{kJ} \]
\[ = -69.70 \, \text{kJ} \]

### Step 5: Determine spontaneity
A reaction is spontaneous if \( \Delta G < 0 \). Since we calculated \( \Delta G = -69.70 \, \text{kJ} \), which is negative, we conclude that the reaction is spontaneous.

### Final Answer
The standard Gibbs energy change at 298 K is \( -69.70 \, \text{kJ} \), and the reaction is spontaneous.

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