Consider the reaction `2SO_(2)(g)+O_(2)(g)hArr2SO_(3)(g)` for which `K_(c)=278M^(-1)`.0.001 mole ofeach of the reagents `SO_(2)(g),O_(2)(g)and SO_(3)(g)` are mixed in a 1.0 L flask . Dterminr=e the reaction quotient of the system and the spontaneus direction of the system:

Options

A. `Q_(c)=1000,`the equilibrium shifts to the right
B. `Q_(c)=1000,`the equilibrium shifts to the left
C. `Q_(c)=0.001,` the equilibrium shifts to the left
D. `Q_(c)=0.001, ` theequilibrium shifts to the right

Answer

To solve the problem, we need to determine the reaction quotient (Q) for the reaction:

\[ 2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g) \]

Given that the equilibrium constant \( K_c = 278 \, M^{-1} \) and that we have 0.001 moles of each of the reactants and products in a 1.0 L flask, we can follow these steps:

### Step 1: Calculate the Concentrations
Since we have 0.001 moles of each gas in a 1.0 L flask, the concentration of each gas can be calculated as follows:

\[ \text{Concentration} = \frac{\text{Number of moles}}{\text{Volume in L}} \]

For \( SO_2 \):
\[ [\text{SO}_2] = \frac{0.001 \, \text{mol}}{1.0 \, \text{L}} = 0.001 \, M \]

For \( O_2 \):
\[ [\text{O}_2] = \frac{0.001 \, \text{mol}}{1.0 \, \text{L}} = 0.001 \, M \]

For \( SO_3 \):
\[ [\text{SO}_3] = \frac{0.001 \, \text{mol}}{1.0 \, \text{L}} = 0.001 \, M \]

### Step 2: Write the Expression for the Reaction Quotient (Q)
The reaction quotient \( Q \) is calculated using the formula:

\[ Q = \frac{[\text{Products}]^{\text{coefficients}}}{[\text{Reactants}]^{\text{coefficients}}} \]

For our reaction:
\[ Q = \frac{[\text{SO}_3]^2}{[\text{SO}_2]^2 \cdot [\text{O}_2]} \]

### Step 3: Substitute the Concentrations into the Q Expression
Substituting the concentrations we calculated:

\[ Q = \frac{(0.001)^2}{(0.001)^2 \cdot (0.001)} = \frac{0.000001}{0.000001 \cdot 0.001} = \frac{0.000001}{0.000000001} = 1000 \]

### Step 4: Compare Q with Kc
Now we compare \( Q \) with \( K_c \):

- \( Q = 1000 \)
- \( K_c = 278 \)

Since \( Q > K_c \), this indicates that the reaction will shift to the left (towards the reactants) to reach equilibrium.

### Conclusion
The reaction quotient \( Q \) is \( 1000 \, M^{-1} \) and the spontaneous direction of the system is towards the left (favoring the formation of reactants).

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