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Chapter 12: Problem 27

A cylinder contains \(3.0 \mathrm{L}\) of oxygen at \(300 \mathrm{K}\) and \(2.4 \mathrm{atm}\). The gas is heated, causing a piston in the cylinder to move outward. The heating causes the temperature to rise to \(600 \mathrm{K}\) and the volume of the cylinder to increase to \(9.0 \mathrm{L}\). What is the final gas pressure?

Short Answer

Expert verified
The final gas pressure after the heating process is \(0.8 atm\).

Step by step solution

01

Identify Known Parameters

From the question, we have the following known parameters: Initial Volume \(V_1 = 3.0 L\), Initial Temperature \(T_1 = 300 K\), Initial Pressure \(P_1 = 2.4 atm\), Final Volume \(V_2 = 9.0 L\), and Final Temperature \(T_2 = 600 K\).
02

Applying the Combined Gas Law

We can plug these values into the combined gas law formula \(P_1 V_1 / T_1 = P_2 V_2 / T_2\), with \(P_2\) unknown. By rearranging the formula for \(P_2\), we get \(P_2 = P_1 V_1 T_2 / (T_1 V_2)\).
03

Substituting the Known Values

Now, substitute the given values into the equation to find \(P_2\), which comes out to be \(P_2 = 2.4 atm * 3.0 L * 600 K / (300 K * 9.0 L) = 0.8 atm\)

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Key Concepts

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

Gas Pressure
Gas pressure is a fundamental concept in the study of gases, physics, and chemistry. It refers to the force that the gas molecules exert per unit area of the walls of their container. The incessant, random motion of gas particles collides with the container walls, leading to pressure. Understanding gas pressure is essential for various practical applications, such as in the functioning of an engine, the inflation of tires, or the human respiratory system.
Thermodynamics
Thermodynamics is the branch of physics that deals with the relationships between heat, work, temperature, and energy. The main objective of thermodynamics is to understand how energy is transformed from one form to another and how it affects matter. It includes laws that govern these transformations, and one of its primary concerns is how heat transfer corresponds to work done by or on a system. The behavior of gases under changing temperatures, as in our exercise, is well understood within the framework of thermodynamics.
Ideal Gas Equation
The ideal gas equation, \( PV=nRT \), is a cornerstone of gas laws synthesizing Boyle's, Charles's, and Avogadro's laws. It鈥檚 a mathematical model that approximates the behavior of most gases under various conditions. The equation relates the pressure (P), volume (V), and temperature (T) of a given amount of gas with its number of moles (n), and the ideal gas constant (R). Even though real gases don't adhere strictly to the predictions of the ideal gas law due to intermolecular forces and the finite volume of gas particles, this equation is highly useful for calculations and predictions in both academic and industrial settings.

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Most popular questions from this chapter

The maximum amount of water an adult in temperate climates can perspire in one hour is typically \(1.8 \mathrm{L}\). However, after several weeks in a tropical climate the body can adapt, increasing the maximum perspiration rate to \(3.5 \mathrm{L} / \mathrm{h}\). At what rate, in watts, is energy being removed when perspiring that rapidly? Assume all of the perspired water evaporates. At body temperature, the heat of vaporization of water is \(L_{\mathrm{v}}=24 \times 10^{5} \mathrm{J} / \mathrm{kg} .\)

An increase in temperature of the water may cause other changes. An increase in surface water temperature is likely to the rate of evaporation. A. Increase B. Not affect C. Decrease

For a normal car riding on tires with relatively flexible sidewalls, the weight of the car is held up, in large measure, by the pressure of the air in the tires. If you look at one of your car's tires, you'll note that the tire is flattened slightly to make a rectangle where it touches the ground. The area of the resulting "contact patch" depends on the pressure in the tires. To a good approximation, the upward normal force of the ground (which we can assume is equal to \(1 / 4\) of the car's weight) on this patch of the tire is equal to the downward pressure force on the patch. a. Suppose you inflate your \(2000 \mathrm{kg}\) car's tires to the recommended pressure, as measured by a gauge. The resulting contact patch is \(18 \mathrm{cm}\) wide and \(12 \mathrm{cm}\) long. What does the gauge read? b. If you let a bit of air out of your tire, what happens to the area of the contact patch?

When air is inhaled, it quickly becomes saturated with water vapor as it passes through the moist upper airways. When a person breathes dry air, about 25 mg of water are exhaled with each breath. At 12 breaths/min, what is the rate of energy loss due to evaporation? Express your answer in both watts and Calories per day. At body temperature, the heat of vaporization of water is \(L_{\mathrm{v}}=24 \times 10^{5} \mathrm{J} / \mathrm{kg}\).

Homes are often insulated with fiberglass insulation in their walls and ceiling. The thermal conductivity of fiberglass is \(0.040 \mathrm{W} / \mathrm{m} \cdot \mathrm{K} .\) Suppose that the total surface area of the walls and roof of a windowless house is \(370 \mathrm{m}^{2}\) and that the thickness of the insulation is \(10 \mathrm{cm}\). At what rate does heat leave the house on a day when the outside temperature is \(30^{\circ} \mathrm{C}\) colder than the inside temperature?
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