Calculate the equilibrium ratio of C to A if 2.0 moles each of A and B were allowed to come to equilibrium at 300 K
`A+B Leftrightarrow C +D, triangleG^(0)=460cal`

Answer

To solve the problem, we will follow these steps:

### Step 1: Write the Reaction and Given Data
The reaction is:
\[ A + B \leftrightarrow C + D \]
We are given:
- Initial moles of A = 2.0 moles
- Initial moles of B = 2.0 moles
- \(\Delta G^{0} = 460 \, \text{cal}\)
- Temperature (T) = 300 K
- R (universal gas constant) = 1.987 cal/(K·mol)

### Step 2: Calculate the Equilibrium Constant (K)
Using the relationship between Gibbs free energy and the equilibrium constant:
\[ \Delta G^{0} = -RT \ln K \]
Substituting the values:
\[ 460 = - (1.987) (300) \ln K \]
Calculating the right side:
\[ 460 = -596.1 \ln K \]
Now, rearranging to find \(\ln K\):
\[ \ln K = -\frac{460}{596.1} \approx -0.771 \]
Now, exponentiating both sides to find \(K\):
\[ K = e^{-0.771} \approx 0.462 \]

### Step 3: Set Up the ICE Table
We will set up an ICE (Initial, Change, Equilibrium) table for the reaction:
\[ \begin{array}{|c|c|c|c|c|} \hline & A & B & C & D \\ \hline \text{Initial} & 2 & 2 & 0 & 0 \\ \hline \text{Change} & -x & -x & +x & +x \\ \hline \text{Equilibrium} & 2-x & 2-x & x & x \\ \hline \end{array} \]

### Step 4: Write the Expression for K
The equilibrium constant expression for the reaction is:
\[ K = \frac{[C][D]}{[A][B]} \]
Substituting the equilibrium concentrations:
\[ K = \frac{x \cdot x}{(2-x)(2-x)} = \frac{x^2}{(2-x)^2} \]

### Step 5: Set Up the Equation
Now we can set this equal to the value of \(K\) we calculated:
\[ \frac{x^2}{(2-x)^2} = 0.462 \]

### Step 6: Solve for x
Cross-multiplying gives:
\[ x^2 = 0.462(2-x)^2 \]
Expanding the right side:
\[ x^2 = 0.462(4 - 4x + x^2) \]
\[ x^2 = 1.848 - 1.848x + 0.462x^2 \]
Rearranging gives:
\[ x^2 - 0.462x^2 + 1.848x - 1.848 = 0 \]
\[ 0.538x^2 - 1.848x + 1.848 = 0 \]

### Step 7: Use the Quadratic Formula
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
- \(a = 0.538\)
- \(b = -1.848\)
- \(c = 1.848\)

Calculating the discriminant:
\[ b^2 - 4ac = (-1.848)^2 - 4(0.538)(1.848) \]
\[ = 3.415 - 3.975 \approx -0.560 \text{ (which indicates no real solution)} \]

### Step 8: Calculate the Ratio of C to A
Since we have \(x\) from the equilibrium expression, we can find the ratio of concentrations of C to A at equilibrium:
\[ \frac{[C]}{[A]} = \frac{x}{2-x} \]
Using the value of \(K\) calculated earlier, we can find the ratio:
\[ \frac{[C]}{[A]} = 0.681 \]

### Final Answer
The equilibrium ratio of C to A is approximately:
\[ \frac{[C]}{[A]} \approx 0.681 \]