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To solve this problem, let's go through the details provided and apply Raoult's Law, which will allow us to find the temperature \( T \), and the mole fractions of ethanol (\( x_E \)) and acetic acid (\( x_A \)) in the liquid phase.

### Given Data:
- Mole fraction of ethanol in the vapor phase, \( y_E = 0.40 \)
- Mole fraction of acetic acid in the vapor phase, \( y_A = 0.60 \)
- Total pressure \( P = 120.0 \ \text{mmHg} \)

### Step-by-Step Solution:

1. **Raoult's Law and Ideal Solutions:**
   For an ideal solution, Raoult’s law states:
   \[
   P_E = x_E P_E^*
   \]
   \[
   P_A = x_A P_A^*
   \]
   where:
   - \( P_E \) and \( P_A \) are the partial pressures of ethanol and acetic acid in the vapor phase, respectively.
   - \( P_E^* \) and \( P_A^* \) are the vapor pressures of pure ethanol and acetic acid at temperature \( T \).
   - \( x_E \) and \( x_A \) are the mole fractions of ethanol and acetic acid in the liquid phase.

2. **Dalton’s Law for the Vapor Phase:**
   The partial pressures in the vapor phase are given by:
   \[
   P_E = y_E \times P
   \]
   \[
   P_A = y_A \times P
   \]
   So,
   \[
   P_E = 0.40 \times 120.0 \ \text{mmHg} = 48.0 \ \text{mmHg}
   \]
   \[
   P_A = 0.60 \times 120.0 \ \text{mmHg} = 72.0 \ \text{mmHg}
   \]

3. **Using Raoult's Law to Find Mole Fractions in the Liquid Phase:**
   To find \( x_E \) and \( x_A \), we would need the vapor pressures \( P_E^* \) and \( P_A^* \) of pure ethanol and acetic acid at the unknown temperature \( T \).

   If we have or can find these values, we can solve for \( x_E \) and \( x_A \) as follows:
   \[
   x_E = \frac{P_E}{P_E^*}
   \]
   \[
   x_A = \frac{P_A}{P_A^*}
   \]

4. **Temperature Determination:**
   The temperature \( T \) is usually determined by finding a value that satisfies the relationship of the mole fractions in both the vapor and liquid phases, in conjunction with the known vapor pressures of each component at that temperature.

Without the vapor pressures \( P_E^* \) and \( P_A^* \) for ethanol and acetic acid at \( T \), we can’t directly calculate \( T \), \( x_E \), and \( x_A \) here. If you have these values or an additional source to find them, I can continue the calculations accordingly.

Would you like me to proceed with further guidance on estimating the temperature or help with a similar calculation approach if more data is provided?

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### Related Questions for Further Understanding:
1. How does Raoult’s Law apply to non-ideal solutions, and what modifications are made in such cases?
2. What methods can be used to determine the vapor pressures \( P_E^* \) and \( P_A^* \) of pure substances at a given temperature?
3. How would the presence of a non-volatile solute affect the mole fractions in the liquid and vapor phases?
4. What assumptions are made in applying Dalton’s Law to calculate partial pressures in this context?
5. Can Raoult’s Law be applied to azeotropic mixtures, and why or why not?

### Tip:
For ideal solution problems, always verify the availability of vapor pressures for each component at various temperatures, as this data is crucial for precise temperature and mole fraction calculations.

A vapor composed of 0.40 mole fraction ethanol and 0.60 mole fraction acetic acid at 120.0 mmHg (absolute) is in equilibrium with a liquid phase also composed of ethanol and acetic acid. Assume that the liquid is an ideal solution. Apply Raoult's law to find the temperature, T, of the liquid and the mole fractions, x_E and x_A, of the liquid phase.

Learn how to determine the temperature and mole fractions of ethanol and acetic acid in a liquid-vapor equilibrium system using Raoult's Law and Dalton's Law. This college-level chemistry problem involves concepts of ideal solutions and partial pressures with a detailed step-by-step solution.

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