A sample consisting of 22.7 g of a nongaseous, unstable compound X is placed inside a metal cyl

Name: Problem 128, Chapter 6: Chemistry
Uploaded: 2025-08-01T12:34:12-08:00
Duration: 16 min 1 s
Channel: Susan Hallstrom

step 1 I have an absolutely huge problem here. And we're, it's a marathon problem. We're given an absol

Question

A sample consisting of $22.7 \mathrm{~g}$ of a nongaseous, unstable compound $\mathrm{X}$ is placed inside a metal cylinder with a radius of $8.00 \mathrm{~cm}$, and a piston is carefully placed on the surface of the compound so that, for all practical purposes, the distance between the bottom of the cylinder and the piston is zero. (A hole in the piston allows trapped air to escape as the piston is placed on the compound; then this hole is plugged so that nothing inside the cylinder can escape.) The piston-and-cylinder apparatus is carefully placed in $10.00 \mathrm{~kg}$ water at $25.00^{\circ} \mathrm{C}$. The barometric pressure is 778 torr. When the compound spontaneously decomposes, the piston moves up, the temperature of the water reaches a maximum of $29.52^{\circ} \mathrm{C}$, and then it gradually decreases as the water loses heat to the surrounding air. The distance between the piston and the bottom of the cylinder, at the maximum temperature, is $59.8 \mathrm{~cm}$. Chemical analysis shows that the cylinder contains $0.300 \mathrm{~mol}$ carbon dioxide, $0.250$ mol liquid water, $0.025$ mol oxygen gas, and an undetermined amount of a gaseous element $\mathrm{A}$. It is known that the enthalpy change for the decomposition of $X$, according to the reaction described above, is $-1893$ $\mathrm{kJ} / \mathrm{mol} \mathrm{X}$. The standard enthalpies of formation for gaseous carbon dioxide and liquid water are $-393.5 \mathrm{~kJ} / \mathrm{mol}$ and $-286 \mathrm{~kJ} / \mathrm{mol}$, respectively. The heat capacity for water is $4.184 \mathrm{~J} /{ }^{\circ} \mathrm{C} \cdot \mathrm{g}$. The conversion factor between $\mathrm{L} \cdot \mathrm{atm}$ and $\mathrm{J}$ can be determined from the two values for the gas constant $R$, namely, $0.08206 \mathrm{~L}$. $\mathrm{atm} / \mathrm{K} \cdot \mathrm{mol}$ and $8.3145 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol}$. The vapor pressure of water at $29.5^{\circ} \mathrm{C}$ is 31 torr. Assume that the heat capacity of the pistonand-cylinder apparatus is negligible and that the piston has negligible mass. Given the preceding information, determine a. The formula for $\mathrm{X}$. b. The pressure-volume work (in $\mathrm{kJ}$ ) for the decomposition of the $22.7-\mathrm{g}$ sample of $\mathrm{X}$. c. The molar change in internal energy for the decomposition of $X$ and the approximate standard enthalpy of formation for $X$.

   A sample consisting of $22.7 \mathrm{~g}$ of a nongaseous, unstable compound $\mathrm{X}$ is placed inside a metal cylinder with a radius of $8.00 \mathrm{~cm}$, and a piston is carefully placed on the surface of the compound so that, for all practical purposes, the distance between the bottom of the cylinder and the piston is zero. (A hole in the piston allows trapped air to escape as the piston is placed on the compound; then this hole is plugged so that nothing inside the cylinder can escape.) The piston-and-cylinder apparatus is carefully placed in $10.00 \mathrm{~kg}$ water at $25.00^{\circ} \mathrm{C}$. The barometric pressure is 778 torr.

When the compound spontaneously decomposes, the piston moves up, the temperature of the water reaches a maximum of $29.52^{\circ} \mathrm{C}$, and then it gradually decreases as the water loses heat to the surrounding air. The distance between the piston and the bottom of the cylinder, at the maximum temperature, is $59.8 \mathrm{~cm}$. Chemical analysis shows that the cylinder contains $0.300 \mathrm{~mol}$ carbon dioxide, $0.250$ mol liquid water, $0.025$ mol oxygen gas, and an undetermined amount of a gaseous element $\mathrm{A}$.

It is known that the enthalpy change for the decomposition of $X$, according to the reaction described above, is $-1893$ $\mathrm{kJ} / \mathrm{mol} \mathrm{X}$. The standard enthalpies of formation for gaseous carbon dioxide and liquid water are $-393.5 \mathrm{~kJ} / \mathrm{mol}$ and $-286 \mathrm{~kJ} / \mathrm{mol}$, respectively. The heat capacity for water is $4.184 \mathrm{~J} /{ }^{\circ} \mathrm{C} \cdot \mathrm{g}$. The conversion factor between $\mathrm{L} \cdot \mathrm{atm}$ and $\mathrm{J}$ can be determined from the two values for the gas constant $R$, namely, $0.08206 \mathrm{~L}$. $\mathrm{atm} / \mathrm{K} \cdot \mathrm{mol}$ and $8.3145 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol}$. The vapor pressure of water
at $29.5^{\circ} \mathrm{C}$ is 31 torr. Assume that the heat capacity of the pistonand-cylinder apparatus is negligible and that the piston has negligible mass. Given the preceding information, determine
a. The formula for $\mathrm{X}$.
b. The pressure-volume work (in $\mathrm{kJ}$ ) for the decomposition of the $22.7-\mathrm{g}$ sample of $\mathrm{X}$.
c. The molar change in internal energy for the decomposition of $X$ and the approximate standard enthalpy of formation for $X$.

Chemistry

Steven S. Zumdahl,… 8th Edition

Chapter 6, Problem 128

Instant Answer

Solved by Expert Susan Hallstrom

08/01/2025

Step 1

To find the number of moles of $ X $, we need its molar mass. However, we will first analyze the products of the decomposition reaction to help deduce the formula for $ X $. Show more…

Show all steps

Thanks for your feedback!

A sample consisting of $22.7 \mathrm{~g}$ of a nongaseous, unstable compound $\mathrm{X}$ is placed inside a metal cylinder with a radius of $8.00 \mathrm{~cm}$, and a piston is carefully placed on the surface of the compound so that, for all practical purposes, the distance between the bottom of the cylinder and the piston is zero. (A hole in the piston allows trapped air to escape as the piston is placed on the compound; then this hole is plugged so that nothing inside the cylinder can escape.) The piston-and-cylinder apparatus is carefully placed in $10.00 \mathrm{~kg}$ water at $25.00^{\circ} \mathrm{C}$. The barometric pressure is 778 torr. When the compound spontaneously decomposes, the piston moves up, the temperature of the water reaches a maximum of $29.52^{\circ} \mathrm{C}$, and then it gradually decreases as the water loses heat to the surrounding air. The distance between the piston and the bottom of the cylinder, at the maximum temperature, is $59.8 \mathrm{~cm}$. Chemical analysis shows that the cylinder contains $0.300 \mathrm{~mol}$ carbon dioxide, $0.250$ mol liquid water, $0.025$ mol oxygen gas, and an undetermined amount of a gaseous element $\mathrm{A}$. It is known that the enthalpy change for the decomposition of $X$, according to the reaction described above, is $-1893$ $\mathrm{kJ} / \mathrm{mol} \mathrm{X}$. The standard enthalpies of formation for gaseous carbon dioxide and liquid water are $-393.5 \mathrm{~kJ} / \mathrm{mol}$ and $-286 \mathrm{~kJ} / \mathrm{mol}$, respectively. The heat capacity for water is $4.184 \mathrm{~J} /{ }^{\circ} \mathrm{C} \cdot \mathrm{g}$. The conversion factor between $\mathrm{L} \cdot \mathrm{atm}$ and $\mathrm{J}$ can be determined from the two values for the gas constant $R$, namely, $0.08206 \mathrm{~L}$. $\mathrm{atm} / \mathrm{K} \cdot \mathrm{mol}$ and $8.3145 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol}$. The vapor pressure of water at $29.5^{\circ} \mathrm{C}$ is 31 torr. Assume that the heat capacity of the pistonand-cylinder apparatus is negligible and that the piston has negligible mass. Given the preceding information, determine a. The formula for $\mathrm{X}$. b. The pressure-volume work (in $\mathrm{kJ}$ ) for the decomposition of the $22.7-\mathrm{g}$ sample of $\mathrm{X}$. c. The molar change in internal energy for the decomposition of $X$ and the approximate standard enthalpy of formation for $X$.

Play audio

Feedback

Susan Hallstrom and 72 other subject Chemistry 102 educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Stoichiometry of Chemical Reactions

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It involves using balanced chemical equations to calculate the amounts in moles, masses, or volumes of reactants and products. This concept is essential for determining how far a reaction will proceed, linking the measurable quantities to the theoretical predictions, and is indispensable for energy calculations in chemical processes.

Vapor Pressure Considerations

Vapor pressure is the pressure exerted by a vapor when it is in equilibrium with its liquid (or solid) phase at a given temperature. It plays an important role in determining the total pressure in a system, especially when volatile components such as water are involved. Understanding vapor pressure is necessary for accurately accounting for all gaseous contributions in a system, particularly in systems where temperature changes can alter the balance between phases.

Conversion Factors and Unit Conversions

Accurate conversion between units (such as between liters·atmosphere and Joules, or between Celsius and Kelvin) is fundamental in thermodynamics and chemistry. These conversion factors ensure consistency in calculations and allow one to correctly apply formulas such as the ideal gas law and energy balance equations. Mastery of unit conversions is critical for bridging experimental results with theoretical analyses.

Standard Enthalpy of Formation

The standard enthalpy of formation is defined as the heat change when one mole of a compound is formed from its elements in their standard states. It represents a key reference point in thermodynamic calculations, allowing the use of Hess’s law to determine reaction enthalpies. Understanding these values is crucial for predicting reaction energetics and comparing the stability of different compounds.

Enthalpy and Internal Energy Relationships

This concept involves understanding the connection between the heat content of a system (enthalpy, ?H) and the internal energy changes (?U) during a process. In many chemical reactions, energy is released or absorbed not only as heat but also as work done by or on the system, and the relationship is often expressed as ?H = ?U + P?V, where P?V represents the pressure-volume work. This relationship is fundamental in thermochemistry to interconvert between changes in internal energy and enthalpy particularly when gases are produced or consumed.

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, relates the pressure, volume, temperature, and number of moles of a gas. This equation is critical for calculating any one of these properties when the others are known, and it is ubiquitous in studies dealing with gas-phase reactions and work calculations. Its use allows chemists to relate macroscopic measurements, like volume and pressure, to the amount of substance present, which is essential for understanding reaction stoichiometry and energetic contributions in gaseous systems.

Pressure-Volume Work

Pressure-volume work is the work done by a system when it expands against an external pressure, and it is calculated using the formula w = P?V for processes occurring at constant pressure. In scenarios involving gaseous products, the expansion or compression of gases performs work on the surroundings, and this work can have a notable effect on the overall energy balance of the reaction. The concept is essential in determining how much energy is 'spent' or 'released' due to changes in volume during a chemical process.

Calorimetry

Calorimetry is the experimental measurement of the heat exchanged in chemical or physical processes. By measuring temperature changes and knowing the heat capacity of a medium (such as water), one can determine the heat absorbed or released. This technique is central in thermochemistry for quantifying reaction enthalpies, validating reaction stoichiometry, and linking thermal measurements to chemical energy changes.

Please give Ace some feedback