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Chapter 11: Problem 50

Methanol, \(\mathrm{CH}_{3} \mathrm{OH}\), a colorless, volatile liquid, was formerly known as wood alcohol. It boils at \(65.0^{\circ} \mathrm{C}\) and has a heat of vaporization of \(37.4 \mathrm{~kJ} / \mathrm{mol}\). What is its vapor pressure at \(22.0^{\circ} \mathrm{C} ?\)

Short Answer

Expert verified
The vapor pressure of methanol at 22.0掳C is approximately 0.0087 atm.

Step by step solution

01

Understand the Clausius-Clapeyron Equation

The Clausius-Clapeyron equation relates the vapor pressure and temperature of a liquid. It is given as: \[ \ln \left( \frac{P_2}{P_1} \right) = \frac{-\Delta H_{vap}}{R} \left(\frac{1}{T_2} - \frac{1}{T_1} \right) \] where \(P_1\) and \(P_2\) are the vapor pressures at temperatures \(T_1\) and \(T_2\) respectively, \(\Delta H_{vap}\) is the heat of vaporization, and \(R\) is the ideal gas constant, \(8.314\) J/mol K.
02

Convert Temperature to Kelvin

Convert the temperatures from Celsius to Kelvin by adding 273.15. For instance: \[ T_1 = 65.0^{\circ} \text{C} + 273.15 = 338.15 \text{ K} \] \[ T_2 = 22.0^{\circ} \text{C} + 273.15 = 295.15 \text{ K} \]
03

Rearrange the Clausius-Clapeyron Equation

Rearrange the equation to solve for \(P_2\), the unknown vapor pressure at \(22.0^{\circ} \text{C}\). We can assume \(P_1\) is 1 atm at the boiling point for simplification: \[ \ln(P_2) = \ln(P_1) - \frac{-\Delta H_{vap}}{R} \left(\frac{1}{T_2} - \frac{1}{T_1} \right) \] Since \( \ln(1 \text{ atm}) = 0 \), the equation simplifies to: \[ \ln(P_2) = \frac{-\Delta H_{vap}}{R} \left(\frac{1}{T_2} - \frac{1}{T_1} \right) \]
04

Solve for \(P_2\)

Insert the values into the equation. Ensure the heat of vaporization is in the correct units (J/mol) by converting: \( 37.4 \text{ kJ/mol} = 37400 \text{ J/mol} \). Then compute: \[ \ln(P_2) = \frac{-37400}{8.314} \left(\frac{1}{295.15} - \frac{1}{338.15} \right) \] Calculate the right side to find \(\ln(P_2)\), then exponentiate to solve for \(P_2\).
05

Calculate and Interpret Result

Perform the calculations: \( \ln(P_2) \approx -4.746 \). \( P_2 = e^{-4.746} \approx 0.0087 \text{ atm} \). Thus, the vapor pressure of methanol at \(22.0^{\circ} \text{C}\) is 0.0087 atm.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Clausius-Clapeyron Equation
The Clausius-Clapeyron Equation is a fundamental relation in thermodynamics used to predict the change in vapor pressure with temperature for a substance. It establishes a linkage between the vapor pressures at two different temperatures:
  • \( \ln\left( \frac{P_2}{P_1} \right) = \frac{-\Delta H_{vap}}{R} \left(\frac{1}{T_2} - \frac{1}{T_1} \right) \).
  • Here, \( P_1 \) and \( P_2 \) are the vapor pressures at temperatures \( T_1 \) and \( T_2 \) respectively, \( \Delta H_{vap} \) is the heat of vaporization, and \( R \) is the ideal gas constant, 8.314 J/mol K.
To apply this equation, understanding how each variable interacts to affect vapor pressure is essential. For instance, a greater \( \Delta H_{vap} \), which indicates stronger intermolecular forces, would typically suggest a greater resistance to change in pressure as temperature changes.
Heat of Vaporization
The heat of vaporization, denoted as \( \Delta H_{vap} \), represents the amount of energy required to convert a mole of a liquid into vapor without a temperature change.
  • For methanol, the heat of vaporization is 37.4 kJ/mol. This measure is vital, as it provides insight into the substance's volatility.
  • Substances with higher heats of vaporization have stronger intermolecular forces and require more energy to transition from liquid to gas.
In practical terms, knowing \( \Delta H_{vap} \) helps predict how the vapor pressure will respond to temperature changes. Understanding this concept deeply helps in calculating the vapor pressures at different temperatures using the Clausius-Clapeyron equation.
Temperature Conversion
Accurate temperature conversion is a crucial step in thermodynamic calculations. The Celsius and Kelvin scales are commonly used, and converting between these is straightforward:
  • To convert from Celsius to Kelvin, add 273.15 to the Celsius temperature.
  • For example, converting methanol's boiling point: 65.0掳C becomes 338.15 K.
  • Similarly, 22.0掳C converts to 295.15 K.
When working with thermodynamic equations like Clausius-Clapeyron, always ensure that temperatures are in Kelvin to maintain consistency and accuracy in calculations.
Vapor Pressure
Vapor pressure signifies the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases at a given temperature in a closed system.
  • For methanol, calculating vapor pressure at lower temperatures like 22.0掳C helps understand how its gas phase behavior changes with cooling.
  • Knowing the vapor pressure informs decisions in chemical engineering and safety procedures regarding storage and handling.
  • A higher vapor pressure indicates a substance's readiness to evaporate, reflecting its volatility.
Thus, vapor pressure is a key factor not only in theoretical calculations but also in applying principles in real-world scenarios such as methanol's utilization in various industrial applications.
Methanol Properties
Methanol, or \( \text{CH}_3\text{OH} \), is a simple alcohol with several notable properties:
  • It's a colorless, volatile liquid, often referred to as wood alcohol.
  • One important physical property is its boiling point, 65.0掳C, indicative of its relatively low boiling compared to other alcohols.
  • Methanol's heat of vaporization is 37.4 kJ/mol, which affects how readily it vaporizes.
  • These properties make methanol useful but also demand careful consideration in storage and handling due to its volatility.
Understanding methanol's properties helps in contextualizing the mathematical results of exercises like vapor pressure calculations, making the information more applicable to practical uses.

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Most popular questions from this chapter

Why does the vapor pressure of a liquid depend on the intermolecular forces?

Associate each of the solids \(\mathrm{Co}, \mathrm{LiCl}, \mathrm{SiC}\), and \(\mathrm{CH}_{3}\) with one of the following sets of properties. a. A white solid melting at \(613^{\circ} \mathrm{C}\); the liquid is electrically conducting, although the solid is not. b. A very hard, blackish solid subliming at \(2700^{\circ} \mathrm{C}\). C. A yellow solid with a characteristic odor having a melting point of \(120^{\circ} \mathrm{C}\). d. A gray, lustrous solid melting at \(1495^{\circ} \mathrm{C}\); both the solid and liquid are electrical conductors.

Associate each of the solids \(\mathrm{BN}, \mathrm{P}_{4} \mathrm{~S}_{3}, \mathrm{~Pb}\), and \(\mathrm{CaCl}_{2}\) with one of the following sets of properties. a. A bluish white, lustrous solid melting at \(327^{\circ} \mathrm{C}\); the solid is soft and malleable. b. A white solid melting at \(772^{\circ} \mathrm{C}\); the solid is an electrical nonconductor but dissolves in water to give a conducting solution. C. A yellowish green solid melting at \(172^{\circ} \mathrm{C}\). d. A very hard, colorless substance melting at about \(3000^{\circ} \mathrm{C}\).

Water at \(0^{\circ} \mathrm{C}\) was placed in a dish inside a vessel maintained at low pressure by a vacuum pump. After a quantity of water had evaporated, the remainder froze. If \(9.31 \mathrm{~g}\) of ice at \(0^{\circ} \mathrm{C}\) was obtained, how much liquid water must have evaporated? The heat of fusion of water is \(6.01 \mathrm{~kJ} / \mathrm{mol}\) and its heat of vaporization is \(44.9 \mathrm{~kJ} / \mathrm{mol}\) at \(0^{\circ} \mathrm{C}\).

a. Draw Lewis structures of each of the following compounds: \(\mathrm{LiH}, \mathrm{NH}_{3}, \mathrm{CH}_{4}, \mathrm{CO}_{2}\). b. Which of these has the highest boiling point? Why? C. Which of these has the lowest boiling point? Why? d. Which of these has the next-to-highest boiling point? Why?
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