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.

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 18a

Chapter 19, Problem 18a

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.

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 gives T1 = 300.15 K and T2 = 340.15 K.

Step 2: For part (a), use the formula for heat transfer at constant volume, Q = n * Cv * ΔT, where n is the number of moles, Cv is the molar heat capacity at constant volume for helium (approximately 12.5 J/mol·K), and ΔT is the change in temperature (T2 - T1).

Step 3: For part (b), use the formula for heat transfer at constant pressure, Q = n * Cp * ΔT, where Cp is the molar heat capacity at constant pressure for helium (approximately 20.8 J/mol·K). Calculate the heat required using the same ΔT as in part (a).

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, which requires additional heat. This is why Cp > Cv.

Step 5: For part (d), calculate the change in internal energy using ΔU = n * Cv * ΔT for both parts (a) and (b). Since internal energy change depends only on temperature change and not on the process, the values will be the same for both parts. The difference in heat is due to the work done in part (b).

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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 differs at constant volume (C_v) and constant pressure (C_p), which is essential for calculating the heat needed in parts (a) and (b) of the question, as these processes occur under different conditions.

Ideal Gas Law

The Ideal Gas Law, expressed as PV = nRT, relates the pressure, volume, and temperature of an ideal gas. This law helps in understanding the behavior of the helium gas under different conditions, such as constant volume and constant pressure, and is fundamental for drawing the pV-diagrams and analyzing the changes in internal energy and heat transfer.

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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. 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.

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Textbook Question

During an isothermal compression of an ideal gas, $410$ J of heat must be removed from the gas to maintain constant temperature. How much work is done by the gas during the process?

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

When water is boiled at a pressure of $2.00$ atm, the heat of vaporization is $2.20$\times$10^6$ J/kg and the boiling point is $120$ °C. At this pressure, $1.00$ kg of water has a volume of $1.00$\times$10^{-3}$ m³, and $1.00$ kg of steam has a volume of $0.824$ m³. Compute the increase in internal energy of the water.

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Textbook Question

When water is boiled at a pressure of $2.00$ atm, the heat of vaporization is $2.20 \times 10^{6}$ J/kg and the boiling point is $120$ °C. At this pressure, $1.00$ kg of water has a volume of $1.00 \times 10^{- 3}$ m³, and $1.00$ kg of steam has a volume of $0.824$ m³. Compute the work done when $1.00$ kg of steam is formed at this temperature.

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