A cylinder contains 0.01000.0100 0.0100 mol of helium at T=27.0T = 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.

Ch 19: The First Law of Thermodynamics

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 19: The First Law of Thermodynamics

Problema 18c

Capitolo 19, Problema 18c

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.

Guida verificata passo dopo passo

Step 1: 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 by converting the temperatures from Celsius to Kelvin and finding the difference.

Step 2: 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 by converting the temperatures from Celsius to Kelvin and finding the difference.

Step 3: To understand the difference in heat required between parts (a) and (b), note that C_p > C_v for any gas. This is because at constant pressure, the gas does work as it expands, requiring more heat input. Therefore, more heat is required in part (b).

Step 4: For part (d), the change in internal energy ΔU for an ideal gas is given by ΔU = nC_vΔT, regardless of whether the process is at constant volume or constant pressure. Calculate ΔU for both parts (a) and (b) using the same formula.

Step 5: Compare the changes in internal energy for parts (a) and (b). They should be the same because the change in internal energy for an ideal gas depends only on the temperature change, not on the process path. This illustrates the first law of thermodynamics, where the internal energy change is independent of the path taken.

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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 this problem, it helps us understand how heat added to the system affects internal energy and work done by the gas. For a constant volume process, no work is done, so all heat goes into changing the internal energy.

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The First Law of Thermodynamics

Specific Heat Capacity

Specific heat capacity is the amount of heat required to change the temperature of a substance by one degree Celsius per unit mass. For gases, it varies depending on whether the process is at constant volume (Cv) or constant pressure (Cp). This concept is crucial for calculating the heat needed in both scenarios described in the problem.

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Specific Heat & Temperature Changes

Ideal Gas Law

The Ideal Gas Law, PV = nRT, relates pressure, volume, and temperature of an ideal gas. It is essential for understanding how changes in temperature affect pressure and volume in the cylinder. In this problem, it helps determine the behavior of helium under constant volume and constant pressure conditions, influencing the heat calculations.

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Ideal Gases and the Ideal Gas Law

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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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