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Ch 19: The First Law of Thermodynamics
Young & Freedman Calc - University Physics 14th Edition
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Tutti i libri di testoYoung & Freedman Calc 14th EditionCh 19: The First Law of ThermodynamicsProblema 18b
Capitolo 19, Problema 18b

A cylinder contains 0.01000.0100 mol of helium at T=27.0T = 27.0°C. If instead the pressure of the helium is kept constant, how much heat is needed to raise the temperature from 27.027.0°C to 67.067.0°C? Draw a pVpV-diagram for this process.

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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. Helium is a monatomic ideal gas, so C_v = (3/2)R, where R is the ideal gas constant.
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. For a monatomic ideal gas, C_p = (5/2)R.
Step 4: To understand the difference in heat required between parts (a) and (b), note that at constant pressure, the gas does work on the surroundings as it expands, requiring additional heat input. This is why C_p > C_v.
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 depends only on the temperature change and is the same for both processes, as internal energy is a state function and does not depend on the path taken.

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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 affects the internal energy of the gas in both constant volume and constant pressure processes.
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The First Law of Thermodynamics

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 (C_v) or constant pressure (C_p), which is essential for calculating the heat needed in parts (a) and (b) of the question.
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Specific Heat & Temperature Changes

Ideal Gas Law

The Ideal Gas Law, expressed as PV = nRT, relates the pressure, volume, and temperature of an ideal gas. It is fundamental for understanding the behavior of the helium gas in the cylinder, especially when analyzing the pV-diagrams and determining how changes in temperature affect pressure and volume under different conditions.
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Ideal Gases and the Ideal Gas Law
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A cylinder contains 0.01000.0100 mol of helium at T=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 67.067.0°C while keeping the volume constant? Draw a pVpV-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 27.027.0°C to 67.067.0°C? Draw a pVpV-diagram for this process.

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A cylinder contains 0.01000.0100 mol of helium at T=27.0T = 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 67.067.0°C while keeping the volume constant? Draw a pVpV-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 27.027.0°C to 67.067.0°C? Draw a pVpV-diagram for this process.

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A cylinder contains 0.01000.0100 mol of helium at T=27.0T = 27.0°C. How much heat is needed to raise the temperature to 67.067.0°C while keeping the volume constant? Draw a pVpV-diagram for this process.

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