Ch 19: Work, Heat, and the First Law of Thermodynamics

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 19: Work, Heat, and the First Law of Thermodynamics

Problem 26

Chapter 19, Problem 26

A monatomic gas follows the process 1→2→3 shown in FIGURE EX19.26. How much heat is needed for (a) process 1→2 and (b) process 2→3?

Verified step by step guidance

Step 1: Analyze the graph provided. The process 1→2 is an isobaric (constant pressure) expansion, as the pressure remains constant at 3 atm while the volume increases from 100 cm³ to 300 cm³. The process 2→3 is an isochoric (constant volume) cooling, as the volume remains constant at 300 cm³ while the pressure decreases from 3 atm to 1 atm.

Step 2: For process 1→2 (isobaric expansion), use the formula for heat transfer in an isobaric process: Q = nC_pΔT, where n is the number of moles, C_p is the molar specific heat at constant pressure for a monatomic gas (C_p = (5/2)R), and ΔT is the change in temperature. To find ΔT, use the ideal gas law: pV = nRT.

Step 3: For process 2→3 (isochoric cooling), use the formula for heat transfer in an isochoric process: Q = nC_vΔT, where C_v is the molar specific heat at constant volume for a monatomic gas (C_v = (3/2)R). Again, use the ideal gas law to determine ΔT based on the change in pressure and the constant volume.

Step 4: Calculate the number of moles (n) of the gas using the ideal gas law at state 1: p₁V₁ = nRT₁. Use the given pressure, volume, and temperature (100°C = 373 K) to solve for n.

Step 5: Substitute the values for n, C_p, C_v, and ΔT into the respective formulas for Q in processes 1→2 and 2→3 to determine the heat transfer for each process. Ensure units are consistent throughout the calculations.

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Key Concepts

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

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 the context of thermodynamic processes, it relates the change in internal energy of a system to the heat added to the system and the work done by the system. This principle is crucial for calculating the heat transfer in processes like those shown in the question.

Isothermal Process

An isothermal process occurs at a constant temperature, meaning that the internal energy of an ideal gas remains unchanged. For a monatomic ideal gas, this implies that any heat added to the system is used to do work, as there is no change in internal energy. Understanding this concept is essential for analyzing the heat required during the transition from state 1 to state 2 in the given question.

Work Done by a Gas

The work done by a gas during expansion or compression can be calculated using the formula W = PΔV, where P is the pressure and ΔV is the change in volume. In the context of the processes described in the question, calculating the work done during the transitions is necessary to determine the heat transfer, as it directly influences the energy balance in accordance with the First Law of Thermodynamics.

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