A cylinder contains 0.01000.0100 0.0100 mol of helium at T=27.0T = 27.0T=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 67.067.067.0°C while keeping the volume constant? Draw a pVpVpV-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 27.027.027.0°C to 67.067.067.0°C? Draw a pVpVpV-diagram for this process.

Ch 19: The First Law of Thermodynamics

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 19: The First Law of Thermodynamics

Problem 18d

Chapter 19, Problem 18d

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.

Verified step by step guidance

Step 1: Convert the initial and final temperatures from Celsius to Kelvin by adding 273.15 to each temperature. This is necessary because thermodynamic calculations require temperatures in Kelvin.

Step 2: For part (a), use the formula for heat transfer at constant volume: Q = nC_vΔT, where n is the number of moles, C_v is the molar heat capacity at constant volume for helium, and ΔT is the change in temperature. Calculate ΔT using the temperatures in Kelvin.

Step 3: For part (b), use the formula for heat transfer at constant pressure: Q = nC_pΔT, where C_p is the molar heat capacity at constant pressure for helium. Again, calculate ΔT using the temperatures in Kelvin.

Step 4: To understand the difference in heat required between parts (a) and (b), note that C_p > C_v for any gas. The additional heat in part (b) is used to do work on the surroundings as the gas expands at constant pressure.

Step 5: For part (d), calculate the change in internal energy using ΔU = nC_vΔT for both parts (a) and (b). The change in internal energy is the same for both processes because it depends only on the temperature change and not on the path taken (constant volume or constant pressure).

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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 the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. This principle is crucial for understanding how heat transfer and work affect the internal energy of the gas in both constant volume and constant pressure processes.

Specific Heat Capacities

Specific heat capacity is the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius. For gases, specific heat capacity varies depending on whether the process is at constant volume (Cv) or constant pressure (Cp). This concept helps explain why different amounts of heat are required in parts (a) and (b) of the question.

Ideal Gas Law

The Ideal Gas Law, expressed as PV = nRT, relates the pressure, volume, and temperature of an ideal gas. It is essential for understanding how changes in temperature affect pressure and volume in the given scenarios, and for drawing the pV-diagrams. This law also aids in calculating changes in internal energy and heat transfer for the helium gas.

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Related Practice

Textbook Question

A cylinder contains $0.0100$ mol of helium at $T = 27.0$ °C. 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 $p V$ -diagram for this process.

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Textbook Question

A cylinder contains $0.0100$ mol of helium at $T = 27.0$ °C. What accounts for the difference between your answers to parts (a) and (b)? In which case is more heat required? What becomes of the additional heat?

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

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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.

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Textbook Question

A cylinder contains $0.0100$ mol of helium at $T equals 27.0$ °C. How much heat is needed to raise the temperature to $67.0$ °C while keeping the volume constant? Draw a $p V$ -diagram for this process.

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Textbook Question

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?

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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?

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