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The vapor pressure of an organic solvent is \(50 \mathrm{mm}\) Hg at \(25^{\circ} \mathrm{C}\) and \(200 \mathrm{mm} \mathrm{Hg}\) at \(45^{\circ} \mathrm{C}\). The solvent is the only species in a closed flask at \(35^{\circ} \mathrm{C}\) and is present in both liquid and vapor states. The volume of gas above the liquid is \(150 \mathrm{mL}\). (a) Estimate the amount of the solvent \((\mathrm{mol})\)contained in the gas phase. (b) What assumptions did you make? How would your answer change if the species dimerized (one molecule results from two molecules of the species combining)?

Short Answer

Expert verified
The amount of the solvent in the gas phase is calculated from the Clausius-Clapeyron and Ideal Gas Law equations. If the species was dimerized, the number of moles would be halved.

Step by step solution

01

Determine the Clausius-Clapeyron equation

The Clausius-Clapeyron equation provides a mathematical relationship between vapor pressure and temperature of a substance. It's given by: \[ \ln \left( \frac{{P_2}}{{P_1}} \right) = -\frac{{\Delta H_{vap}}}{{R}} \left( \frac{{1}}{{T_2}} - \frac{{1}}{{T_1}} \right) \] Here, \(P_1\) and \(P_2\) are the vapor pressures at temperatures \(T_1\) and \(T_2\) (all temperatures must be in Kelvin). \( \Delta H_{vap} \) is the enthalpy of vaporization, R is the gas constant. We don't have \(\Delta H_{vap}\), so we need to solve this equation with the given pressures and temperatures to find it.
02

Find the enthalpy of vaporization

Convert temperatures to Kelvin (25掳C = 298.15 K, 45掳C = 318.15 K). Then, substitute \(P_1 = 50 \, \text{mm Hg}\), \(T_1 = 298.15 \, \text{K}\), \(P_2 = 200 \, \text{mm Hg}\), and \(T_2 = 318.15 \, \text{K}\) into the Clausius-Clapeyron equation and solve to find \(\Delta H_{vap}\) .
03

Estimate gas-phase moles at 35掳C (308.15 K)

First, find the vapor pressure at 35掳C by re-writing the Clausius-Clapeyron equation to solve for the required \(P\): \[ \ln \left( \frac{{P}}{{50}} \right) = -\frac{{\Delta H_{vap}}}{{R}} \left( \frac{{1}}{{308.15}} - \frac{{1}}{{298.15}} \right) \]Solve this to find \(P\). Now that we have the pressure, calculate the number of moles (\(n\)) using the ideal gas law (\(PV = nRT\)), where \(V = 150 \, \text{mL} = 0.150 \, \text{L}\), \(P\) is the calculated pressure (converted to atmospheres), \(R = 0.0821 \, \text{L路atm/K路mol}\) and \(T = 308.15 \, \text{K}\).
04

Answer the second part of the problem

The assumptions made in this problem come from the ideal gas law and the Clausius-Clapeyron equation. We assume the gas behaves ideally (no intermolecular interactions and occupies no volume). We also assume constant enthalpy of vaporization over the temperature range and that the system is in equilibrium (liquid and vapor phases). If the species dimerized, the number of moles would be halved, because two molecules would combine to form one.

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Key Concepts

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

Vapor Pressure
Vapor pressure is a fundamental concept in thermodynamics related to phases of matter. It refers to the pressure exerted by a vapor in equilibrium with its liquid form at a given temperature. When a substance has a higher vapor pressure, it means more molecules are escaping from the liquid phase into the vapor phase. This is because the molecules are energetic enough to overcome the attractive forces within the liquid.
Understanding vapor pressure is crucial when considering how substances evaporate and boil. For example, at higher temperatures, molecules have more kinetic energy, leading to increased vapor pressure. This signifies that more molecules have the necessary energy to transition into the vapor state, leading to evaporation.
  • It is temperature-dependent: As temperature increases, vapor pressure also increases.
  • High vapor pressure implies greater volatility of the substance.
  • Equilibrium is achieved when evaporation and condensation occur at the same rate.
It's important to remember that at any given temperature, a phase equilibrium will exist when the vapor pressure matches the atmospheric pressure. This is what happens, for example, at the boiling point of a liquid.
Clausius-Clapeyron Equation
The Clausius-Clapeyron equation is a vital tool in thermodynamics and chemistry, providing a way to describe the relationship between temperature and vapor pressure. It is particularly useful when comparing the vapor pressures of a substance at two different temperatures.
The equation is represented 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, - \(\Delta H_{vap}\) represents the enthalpy of vaporization,- \(R\) is the universal gas constant, and - \(T_1\) and \(T_2\) are temperatures in Kelvin.
To use the equation effectively, temperatures should always be in Kelvin, and it may require solving for \( \Delta H_{vap} \) if it is unknown. By rearranging the equation, we can also estimate the vapor pressure at any temperature given we know the pressures and temperatures at two points.
This is particularly useful in predicting how a system will behave at temperatures that may not be directly measurable.
Ideal Gas Law
The Ideal Gas Law is a cornerstone of chemistry that helps us understand and predict gas behavior under various conditions. It is a simplified equation of state for gases, bridging pressure, volume, temperature, and the number of moles of gas. It is expressed mathematically as:
\[ PV = nRT \] Here, - \(P\) represents the pressure of the gas,- \(V\) is the volume of the gas,- \(n\) is the number of moles of the gas,- \(R\) is the ideal gas constant, which is usually 0.0821 L路atm K鈦宦 mol鈦宦,- \(T\) stands for the temperature in Kelvin.
The Ideal Gas Law assumes that gases are composed of particles that occupy no volume and experience no intermolecular attractions or repulsions. This is an approximation, as real gases exhibit these properties, but for many cases鈥攅specially at high temperatures and low pressures鈥攖his assumption allows us to predict and calculate gas properties effectively.
  • It is used for estimating gas moles from pressure, volume, and temperature.
  • Assumes ideal behavior, which means the equation is best applied when deviations due to intermolecular forces and molecular size are minimal.
  • Commonly applied to estimate moles of gas in closed systems or compare reactions where gases are involved.

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Most popular questions from this chapter

An aqueous waste stream leaving a process contains 10.0 wt\% sulfuric acid and 1 kg nitric acid per \(\mathrm{kg}\) sulfuric acid. The flow rate of sulfuric acid in the waste stream is \(1000 \mathrm{kg} / \mathrm{h}\). The acids are neutralized before being sent to a wastewater treatment facility by combining the waste stream with an aqueous slurry of solid calcium carbonate that contains 2 kg of recycled liquid per \(\mathrm{kg}\) solid calcium carbonate. (The source of the recycled liquid will be given later in the process description.) The following neutralization reactions occur in the reactor:$$\begin{array}{l} \mathrm{CaCO}_{3}+\mathrm{H}_{2} \mathrm{SO}_{4} \rightarrow \mathrm{CaSO}_{4}+\mathrm{H}_{2} \mathrm{O}+\mathrm{CO}_{2} \\ \mathrm{CaCO}_{3}+2 \mathrm{HNO}_{3} \rightarrow \mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}+\mathrm{H}_{2} \mathrm{O}+\mathrm{CO}_{2} \end{array}$$,The sulfuric and nitric acids and calcium carbonate fed to the reactor are completely consumed. The carbon dioxide leaving the reactor is compressed to 30 atm absolute and \(40^{\circ} \mathrm{C}\) and sent elsewhere in the plant. The remaining reactor effluents are sent to a crystallizer operating at \(30^{\circ} \mathrm{C},\) at which temperature the solubility of calcium sulfate is \(2.0 \mathrm{g} \mathrm{CaSO}_{4} / 1000 \mathrm{g} \mathrm{H}_{2} \mathrm{O} .\) Calcium sulfate crystals form in the crystallizer and all other species remain in solution.The slurry leaving the crystallizer is filtered to produce (i) a filter cake containing \(96 \%\) calcium sulfate crystals and the remainder entrained saturated calcium sulfate solution, and (ii) a filtrate solution saturated with \(\mathrm{CaSO}_{4}\) at \(30^{\circ} \mathrm{C}\) that also contains dissolved calcium nitrate. The filtrate is split, with a portion being recycled to mix with the solid calcium carbonate to form the slurry fed to the reactor, and the remainder being sent to the wastewater treatment facility.(a) Draw and completely label a flowchart for this process. (b) Speculate on why the acids must be neutralized before being sent to the wastewater treatment facility.(c) Calculate the mass flow rates ( \(\mathrm{kg} / \mathrm{h}\) ) of the calcium carbonate fed to the process and of the filter cake; also determine the mass flow rates and compositions of the solution sent to the wastewater facility and of the recycle stream. (Caution: If you write a water balance around the reactor or the overall system, remember that water is a reaction product and not just an inert solvent.)(d) Calculate the volumetric flow rate ( \(L / h\) ) of the carbon dioxide leaving the process at 30 atm absolute and 40^0 C. Do not assume ideal-gas behavior. (e) The solubility of \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}\) at \(30^{\circ} \mathrm{C}\) is \(152.6 \mathrm{kg} \mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}\) per \(100 \mathrm{kg} \mathrm{H}_{2} \mathrm{O}\). What is the maximum ratio of nitric acid to sulfuric acid in the feed that can be tolerated without encountering difficulties associated with contamination of the calcium sulfate by-product by \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2} ?\)

Liquid methyl ethyl ketone \((\mathrm{MEK})\) is introduced into a vessel containing air. The system temperature is increased to \(55^{\circ} \mathrm{C},\) and the vessel contents reach equilibrium with some MEK remaining in the liquid state. The equilibrium pressure is \(1200 \mathrm{mm} \mathrm{Hg}\).(a) Use the Gibbs phase rule to determine how many degrees of freedom exist for the system at equilibrium. State the meaning of your result in your own words.(b) Mixtures of MEK vapor and air that contain between 1.8 mole\% MEK and 11.5 mole\% MEK can ignite and burn explosively if exposed to a flame or spark. Determine whether or not the given vessel constitutes an explosion hazard.

A gas containing nitrogen, benzene, and toluene is in equilibrium with a liquid mixture of 40 mole \(\%\) benzene-60 mole\% toluene at 100^'C and 10 atm. Estimate the gas-phase composition (mole fractions) using Raoult's law. State your assumptions. Why would you have confidence in the accuracy of Raoult's law?

Pure chlorobenzene is contained in a flask attached to an open-end mercury manometer. When the flask contents are at \(58.3^{\circ} \mathrm{C}\), the height of the mercury in the arm of the manometer connected to the flask is \(747 \mathrm{mm}\) and that in the arm open to the atmosphere is \(52 \mathrm{mm} . \mathrm{At} 110^{\circ} \mathrm{C},\) the mercury level is \(577 \mathrm{mm}\) in the arm connected to the flask and \(222 \mathrm{mm}\) in the other arm. Atmospheric pressure is \(755 \mathrm{mm} \mathrm{Hg}\). (a) Extrapolate the data using the Clausius-Clapeyron equation to estimate the vapor pressure of chlorobenzene at \(130^{\circ} \mathrm{C}\). (b) Air saturated with chlorobenzene at \(130^{\circ} \mathrm{C}\) and \(101.3 \mathrm{kPa}\) is cooled to \(58.3^{\circ} \mathrm{C}\) at constant pressure. Estimate the percentage of the chlorobenzene originally in the vapor that condenses. (See Example 6.3-2.)(c) Summarize the assumptions you made in doing the calculation of Part (b).

An adult inhales approximately 12 times per minute, taking in about 500 mL of air with each inhalation. Oxygen and carbon dioxide are exchanged in the lungs, but there is essentially no exchange of nitrogen. The exhaled air has a mole fraction of nitrogen of 0.75 and is saturated with water vapor at body temperature, \(37^{\circ} \mathrm{C}\). If ambient conditions are \(25^{\circ} \mathrm{C}, 1\) atm, and \(50 \%\) relative humidity, what volume of liquid water (mL) would have to be consumed over a two-hour period to replace the water loss from breathing? How much would have to be consumed if the person is on an airplane where the temperature, pressure, and relative humidity are respectively \(25^{\circ} \mathrm{C}, 1 \mathrm{atm},\) and \(10 \% ?\)

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