Ch. 19 - Heat and the First Law of Thermodynamics

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

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Giancoli Douglas 5th edition

Ch. 19 - Heat and the First Law of Thermodynamics

Problem 38b

Chapter 19, Problem 38b

A 1.0-L volume of air initially at 3.5 atm of (gauge)pressure is allowed to expand isothermally until the (gauge) pressure is 1.0 atm. It is then compressed at constant pressure to its initial volume, and lastly is brought back to its original pressure by heating at constant volume. How much work does the 1.0 L of air do in this process?

Verified step by step guidance

Step 1: Understand the problem and identify the process. The problem involves three thermodynamic processes: (1) isothermal expansion, (2) isobaric compression, and (3) isochoric heating. The goal is to calculate the total work done by the air during these processes.

Step 2: Write the formula for work done during isothermal expansion. For an isothermal process, the work done is given by:

W = n R T ln (\frac{P_{i}}{P_{f}})

, where

n

is the number of moles,

R

is the gas constant,

T

is the temperature, and

P_{i}

and

P_{f}

are the initial and final pressures, respectively. Use the given pressures and assume the temperature remains constant.

Step 3: Calculate the work done during the isobaric compression. For an isobaric process, the work done is given by:

W = P_{f} \cdot (V_{f} - V_{i})

, where

P_{f}

is the constant pressure, and

V_{i}

and

V_{f}

are the initial and final volumes. Use the given data to calculate the change in volume during this step.

Step 4: Note that no work is done during the isochoric heating process. In an isochoric process, the volume remains constant, so the work done is zero:

W = 0

. This simplifies the calculation for this step.

Step 5: Add the work contributions from all three processes. The total work done by the air is the sum of the work done during the isothermal expansion, the isobaric compression, and the isochoric heating. Use the results from Steps 2, 3, and 4 to find the total work done.

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Key Concepts

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

Isothermal Process

An isothermal process occurs when a system undergoes a change at a constant temperature. In the context of gases, this means that the internal energy remains constant, and any heat added to the system is used to do work. For an ideal gas, the work done during an isothermal expansion can be calculated using the formula W = nRT ln(Vf/Vi), where Vf and Vi are the final and initial volumes, respectively.

Work Done by a Gas

The work done by a gas during expansion or compression is defined as the product of pressure and the change in volume. For a constant pressure process, the work can be calculated using W = PΔV, where ΔV is the change in volume. In the case of isothermal expansion and compression, the work done can vary significantly depending on the initial and final pressures and volumes.

First Law of Thermodynamics

The First Law of Thermodynamics states that energy cannot be created or destroyed, only transformed from one form to another. In thermodynamic processes, this law can be 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. Understanding this principle is crucial for analyzing energy transfers in the described process.

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Related Practice

Textbook Question

Show, using Eqs. 19–7 and 19–16, that the work done by a gas that slowly expands adiabatically from pressure P₁ and volume V₁ , to P₂ and V₂, is given by W = (P₁V₁ - P₂V₂) / (γ - 1).

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Textbook Question

At a crime scene, the forensic investigator notes that the 6.2-g lead bullet that was stopped in a doorframe apparently melted completely on impact. Assuming the bullet was shot at room temperature (20°C), what does the investigator calculate as the minimum muzzle velocity of the gun?

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Textbook Question

If a heater supplies 1.8 x 10⁶ J/h to a room 3.5 m x 4.6 m x 3.0 m containing air at 20°C and 1.0 atm, by how much will the temperature rise in one hour, assuming no losses of heat or air mass to the outside? Assume air is an ideal diatomic gas with molecular mass 29.

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Textbook Question

A ceramic teapot (e = 0.70) and a shiny metal one (e = 0.10) each hold 0.55 L of tea at 85°C. (a) Estimate the rate of heat loss from each, and (b) estimate the temperature drop after 30 min for each. Consider only radiation, and assume the surroundings are at 20°C.

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Textbook Question

Determine the latent heat of fusion of mercury using the following calorimeter data: 1.00 kg of solid Hg at its melting point of −39.0°C is placed in a 0.620-kg aluminum calorimeter with 0.400 kg of water at 12.80°C; the resulting equilibrium temperature is 5.06°C.