Six moles of an ideal gas are in a cylinder fitted at one end with a movable piston. The initial temperature of the gas is 27.027.0°C and the pressure is constant. As part of a machine design project, calculate the final temperature of the gas after it has done 2.40×1032.40\$$times10^3 J of $$work.

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

All textbooks

Young & Freedman Calc 15th Edition

Ch 19: The First Law of Thermodynamics

Problem 2

Chapter 19, Problem 2

Six moles of an ideal gas are in a cylinder fitted at one end with a movable piston. The initial temperature of the gas is $27.0$ °C and the pressure is constant. As part of a machine design project, calculate the final temperature of the gas after it has done $2.40 times 10 cubed$ J of work.

Verified step by step guidance

Convert the initial temperature from Celsius to Kelvin by adding 273.15 to the Celsius temperature. This is necessary because the ideal gas law calculations require temperature in Kelvin.

Use the first law of thermodynamics, which states that the change in internal energy (ΔU) of a system is equal to the heat added to the system (Q) minus the work done by the system (W). Since the problem does not mention heat exchange, assume it is an adiabatic process where Q = 0, thus ΔU = -W.

Recall that for an ideal gas, the change in internal energy (ΔU) can also be expressed as ΔU = n * C_v * ΔT, where n is the number of moles, C_v is the molar specific heat at constant volume, and ΔT is the change in temperature.

Rearrange the equation ΔU = n * C_v * ΔT to solve for the final temperature (T_f). Substitute ΔU = -W, n = 6 moles, and the given work done by the gas (W = 2.40 * 10^3 J).

Calculate the final temperature (T_f) in Kelvin using the rearranged equation. Finally, if needed, convert the final temperature back to Celsius by subtracting 273.15 from the Kelvin temperature.

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 gas constant, and T is temperature. It describes the relationship between these variables for an ideal gas, assuming no interactions between molecules and that the gas occupies no volume.

First Law of Thermodynamics

The First Law of Thermodynamics states that energy cannot be created or destroyed, only transferred or converted. In the context of gases, it is often expressed as ΔU = Q - W, where ΔU is the change in internal energy, Q is the heat added to the system, and W is the work done by the system. This principle helps in calculating changes in temperature when work is done.

Work Done by a Gas

Work done by a gas during expansion or compression is given by W = PΔV, where P is the constant pressure and ΔV is the change in volume. In thermodynamic processes, work is a form of energy transfer, and understanding how it affects the system's internal energy and temperature is crucial for solving problems involving gas expansion.

Recommended video:

Guided course

03:47

Calculating Work Done on Monoatomic Gas

Related Practice

Textbook Question

Two moles of an ideal gas are heated at constant pressure from $T equals 27$ °C to $T equals 107$ °C. Calculate the work done by the gas.

views

Textbook Question

The graph in Fig. E $19.4$ shows a $pV$ -diagram of the air in a human lung when a person is inhaling and then exhaling a deep breath. Such graphs, obtained in clinical practice, are normally somewhat curved, but we have modeled one as a set of straight lines of the same general shape. (Important: The pressure shown is the gauge pressure, not the absolute pressure.) The process illustrated here is somewhat different from those we have been studying, because the pressure change is due to changes in the amount of gas in the lung, not to temperature changes. (Think of your own breathing. Your lungs do not expand because they've gotten hot.) If the temperature of the air in the lung remains a reasonable $20$ °C, what is the maximum number of moles in this person's lung during a breath?

views

Textbook Question

Two moles of an ideal gas are heated at constant pressure from $T = 27$ °C to $T = 107$ °C. Draw a $p V$ -diagram for this process.

views

Textbook Question

A gas undergoes two processes. In the first, the volume remains constant at $0.200$ m³ and the pressure increases from $2.00$\times$10^5$ Pa to $5.00$\times$10^5$ Pa. The second process is a compression to a volume of $0.120$ m³ at a constant pressure of $5.00$\times$10^5$ Pa. In a $p V$ -diagram, show both processes.

views

Textbook Question