Calculating Vapor Pressure Of Heptane And Octane Ideal Solutions At 373K

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Calculate the vapor pressure of a mixture containing 25g of heptane and 35g of octane, given that at 373K, the vapor pressures of heptane and octane are 105.2 kPa and 46.8 kPa, respectively. The molar mass of heptane is approximately 100 g/mol.

When dealing with solutions, especially in chemistry, understanding vapor pressure is crucial. Vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (solid or liquid) at a given temperature in a closed system. This article delves into calculating the vapor pressure of an ideal solution formed by heptane and octane, two common hydrocarbons.

Problem Statement: Heptane and Octane Ideal Solution

We are given that heptane and octane form an ideal solution. At a temperature of 373K, the vapor pressures of the two pure liquids are provided: heptane has a vapor pressure of 105.2 kPa, and octane has a vapor pressure of 46.8 kPa. The task is to determine the vapor pressure, in bar, of a mixture containing 25g of heptane and 35g of octane.

Key Concepts: Raoult's Law and Ideal Solutions

To tackle this problem, we rely on Raoult's Law, which is fundamental in understanding the behavior of ideal solutions. An ideal solution is one where the interactions between the molecules of the different components are of the same magnitude as the interactions between the molecules of each individual component. In simpler terms, the components mix ideally without any excess heat being evolved or absorbed.

Raoult's Law states that the partial vapor pressure of each component of an ideal mixture of liquids is equal to the vapor pressure of the pure component multiplied by its mole fraction in the mixture. Mathematically, it is expressed as:

  • Pᵢ = xᵢ * Pᵢ⁰

Where:

  • Pᵢ is the partial vapor pressure of component i in the solution.
  • xᵢ is the mole fraction of component i in the solution.
  • Pᵢ⁰ is the vapor pressure of pure component i.

Steps to Calculate Vapor Pressure

To calculate the vapor pressure of the mixture, we need to follow these steps:

  1. Calculate the number of moles of each component:

    • This involves using the given masses of heptane and octane, along with their respective molar masses.

  2. Determine the mole fraction of each component in the solution:

    • The mole fraction of a component is the ratio of its number of moles to the total number of moles of all components in the mixture.

  3. Apply Raoult's Law to find the partial vapor pressure of each component:

    • Multiply the mole fraction of each component by its pure vapor pressure.

  4. Calculate the total vapor pressure of the solution:

    • According to Dalton's Law of Partial Pressures, the total vapor pressure of a mixture is the sum of the partial vapor pressures of the individual components.

  5. Convert the total vapor pressure from kPa to bar:

    • Use the conversion factor 1 bar = 100 kPa.

Detailed Calculation

Step 1: Calculate the Number of Moles of Each Component

First, we need to find the molar masses of heptane (C₇H₁₆) and octane (C₈H₁₈).

  • Molar mass of heptane (C₇H₁₆) = (7 * 12.01) + (16 * 1.01) = 84.07 + 16.16 = 100.23 g/mol (approximately 100 g/mol)
  • Molar mass of octane (C₈H₁₈) = (8 * 12.01) + (18 * 1.01) = 96.08 + 18.18 = 114.26 g/mol (approximately 114 g/mol)

Now, calculate the number of moles for each component:

  • Moles of heptane = mass of heptane / molar mass of heptane = 25 g / 100 g/mol = 0.25 mol
  • Moles of octane = mass of octane / molar mass of octane = 35 g / 114 g/mol = 0.307 mol

Step 2: Determine the Mole Fraction of Each Component

The mole fraction of a component is calculated as:

  • xᵢ = moles of component i / total moles

Total moles in the mixture = moles of heptane + moles of octane = 0.25 mol + 0.307 mol = 0.557 mol

Now, calculate the mole fractions:

  • Mole fraction of heptane (x_heptane) = 0.25 mol / 0.557 mol = 0.449
  • Mole fraction of octane (x_octane) = 0.307 mol / 0.557 mol = 0.551

Step 3: Apply Raoult's Law to Find Partial Vapor Pressures

Using Raoult's Law, we calculate the partial vapor pressure of each component:

  • Partial vapor pressure of heptane (P_heptane) = x_heptane * P_heptane⁰ = 0.449 * 105.2 kPa = 47.23 kPa
  • Partial vapor pressure of octane (P_octane) = x_octane * P_octane⁰ = 0.551 * 46.8 kPa = 25.78 kPa

Step 4: Calculate the Total Vapor Pressure of the Solution

According to Dalton's Law, the total vapor pressure is the sum of the partial pressures:

  • Total vapor pressure (P_total) = P_heptane + P_octane = 47.23 kPa + 25.78 kPa = 73.01 kPa

Step 5: Convert Total Vapor Pressure from kPa to Bar

To convert kPa to bar, we use the conversion factor 1 bar = 100 kPa:

  • Total vapor pressure in bar = 73.01 kPa / 100 kPa/bar = 0.7301 bar

Result and Conclusion

The vapor pressure of the mixture containing 25g of heptane and 35g of octane at 373K is approximately 0.7301 bar. This calculation demonstrates the application of Raoult's Law in determining the vapor pressure of ideal solutions. Understanding these concepts is crucial in various chemical and engineering applications, especially in processes involving distillation and separation of liquids.

By following the steps outlined above, we can accurately determine the vapor pressure of ideal solutions, providing valuable insights into their behavior under different conditions. The principles of ideal solutions and vapor pressure are fundamental in physical chemistry and are widely used in industrial processes and research.

Understanding these concepts is vital for students and professionals in chemistry and related fields. This example showcases how theoretical laws like Raoult's Law can be applied to practical problems, thereby enhancing our understanding of the physical properties of solutions.

Additional Insights and Considerations

Deviations from Ideal Behavior

While Raoult's Law provides a good approximation for ideal solutions, it's important to note that many real solutions exhibit deviations from ideal behavior. These deviations can be either positive or negative, depending on the intermolecular forces between the components. Positive deviations occur when the intermolecular forces between the different components are weaker than those between the molecules of the same component, leading to a higher vapor pressure than predicted by Raoult's Law. Conversely, negative deviations occur when the intermolecular forces between the different components are stronger, resulting in a lower vapor pressure.

Temperature Dependence of Vapor Pressure

The vapor pressure of a liquid is highly dependent on temperature. As temperature increases, the kinetic energy of the molecules increases, allowing more molecules to escape into the vapor phase. This relationship is described by the Clausius-Clapeyron equation, which relates the vapor pressure of a liquid to its temperature and enthalpy of vaporization.

Applications in Distillation

The principles of vapor pressure and Raoult's Law are fundamental to understanding distillation, a widely used separation technique in the chemical industry. Distillation involves separating components of a liquid mixture based on their boiling points. The component with the higher vapor pressure (lower boiling point) will vaporize more readily, allowing for its separation from the mixture. Fractional distillation, a more advanced technique, allows for the separation of multiple components with close boiling points.

Importance of Accurate Molar Mass Values

In the calculations above, we used approximate molar mass values for heptane and octane. While this provides a reasonable estimate, using more precise molar mass values will yield a more accurate result. The molar mass values can be obtained from the periodic table and should be calculated using the atomic weights of the elements to several decimal places.

Practical Implications and Further Learning

The calculation of vapor pressure is not just an academic exercise; it has significant practical implications in various fields, including chemical engineering, pharmaceuticals, and environmental science. For instance, understanding vapor pressure is crucial in designing distillation columns, predicting the evaporation rates of solvents, and assessing the environmental impact of volatile organic compounds (VOCs).

To further enhance your understanding of this topic, consider exploring the following:

  • Non-ideal solutions: Learn about the different types of deviations from Raoult's Law and the factors that cause them.
  • Clausius-Clapeyron equation: Study the relationship between vapor pressure, temperature, and enthalpy of vaporization.
  • Distillation techniques: Investigate simple, fractional, and azeotropic distillation methods and their applications.
  • Thermodynamics of solutions: Delve deeper into the thermodynamic principles governing the behavior of solutions.

By understanding the vapor pressure of solutions, we gain valuable insights into their physical properties and behavior, which is essential for various scientific and industrial applications.

Conclusion

In summary, this article has provided a comprehensive guide to calculating the vapor pressure of an ideal solution, using a mixture of heptane and octane as an example. By understanding and applying Raoult's Law, we can accurately predict the behavior of such solutions. Further exploration of non-ideal solutions, temperature dependence of vapor pressure, and distillation techniques will provide a more complete understanding of the topic. This knowledge is crucial for students, researchers, and professionals in the fields of chemistry, chemical engineering, and related disciplines.