Ch 19: The First Law of Thermodynamics

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 19: The First Law of Thermodynamics

Problem 25c

Chapter 19, Problem 25c

The temperature of $0.150$ mol of an ideal gas is held constant at $77.0$ °C while its volume is reduced to $25.0 percent sign$ of its initial volume. The initial pressure of the gas is $1.25$ atm. Does the gas exchange heat with its surroundings? If so, how much? Does the gas absorb or liberate heat?

Verified step by step guidance

First, understand that since the temperature of the gas is held constant, this is an isothermal process. For an ideal gas undergoing an isothermal process, the internal energy change is zero, meaning any heat exchanged is equal to the work done on or by the gas.

Use the ideal gas law, $ PV = nRT $, to relate the initial and final states of the gas. Since the temperature $ T $ and the number of moles $ n $ are constant, the product of pressure $ P $ and volume $ V $ remains constant: $ P_1V_1 = P_2V_2 $.

Given that the final volume $ V_2 $ is 25.0% of the initial volume $ V_1 $, express $ V_2 $ as $ V_2 = 0.25V_1 $. Substitute this into the equation from step 2 to find the final pressure $ P_2 $.

Calculate the work done on the gas using the formula for work in an isothermal process: $ W = nRT \ln\left(\frac{V_2}{V_1}\right) $. Since $ V_2 = 0.25V_1 $, substitute this into the equation to find $ W $.

Since the process is isothermal, the heat exchanged $ Q $ is equal to the work done $ W $. Determine whether the gas absorbs or liberates heat by considering the sign of $ W $. If $ W $ is positive, the gas absorbs heat; if negative, it liberates heat.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics, expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. It describes the relationship between these variables for an ideal gas, allowing us to predict how a gas will behave under different conditions. In this problem, it helps us understand how pressure and volume changes affect the gas when temperature is constant.

Isothermal Process

An isothermal process is a thermodynamic process in which the temperature of the system remains constant. For an ideal gas undergoing an isothermal process, the internal energy remains unchanged, and any work done by or on the gas is compensated by heat exchange with the surroundings. This concept is crucial for determining whether the gas exchanges heat with its surroundings when its volume changes at constant temperature.

First Law of Thermodynamics

The First Law of Thermodynamics, also known as the law of energy conservation, states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. In an isothermal process for an ideal gas, the internal energy change is zero, so the heat exchanged is equal to the work done. This principle helps us calculate the amount of heat exchanged when the gas volume changes at constant temperature.

Recommended video:

Guided course

08:04

The First Law of Thermodynamics

Related Practice

Textbook Question

A cylinder contains $0.0100$ mol of helium at $T = 27.0$ °C. If the gas is ideal, what is the change in its internal energy in part (a)? In part (b)? How do the two answers compare? Why?

(a) How much heat is needed to raise the temperature to $67.0$ °C while keeping the volume constant? Draw a $pV$ -diagram for this process.

(b) If instead the pressure of the helium is kept constant, how much heat is needed to raise the temperature from $27.0$ °C to $67.0$ °C? Draw a $pV$ -diagram for this process.

views

Textbook Question

A monatomic ideal gas that is initially at $1.50 \times 10^{5}$ Pa and has a volume of $0.0800$ m³ is compressed adiabatically to a volume of $0.0400$ m³. What is the final pressure?

views

Textbook Question

An experimenter adds $970$ J of heat to $1.75$ mol of an ideal gas to heat it from $10.0$ °C to $25.0$ °C at constant pressure. The gas does $positive 223$ J of work during the expansion. Calculate $γ$ for the gas.

views

Textbook Question

A monatomic ideal gas that is initially at $1.50$\times$10^5$ Pa and has a volume of $0.0800$ m³ is compressed adiabatically to a volume of $0.0400$ m³. What is the ratio of the final temperature of the gas to its initial temperature? Is the gas heated or cooled by this compression?

views

Textbook Question

Five moles of monatomic ideal gas have initial pressure $2.50 times 10 cubed$ Pa and initial volume $2.10$ m³. While undergoing an adiabatic expansion, the gas does $1480$ J of work. What is the final pressure of the gas after the expansion?

Textbook Question

Heat $Q$ flows into a monatomic ideal gas, and the volume increases while the pressure is kept constant. What fraction of the heat energy is used to do the expansion work of the gas?

views