Ch 19: The First Law of Thermodynamics

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 19: The First Law of Thermodynamics

Problem 21

Chapter 19, Problem 21

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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Start by recalling the first law of thermodynamics, which states that the change in internal energy (ΔU) of a system is equal to the heat added to the system (Q) minus the work done by the system (W): ΔU = Q - W.

For a monatomic ideal gas, the change in internal energy can be expressed as ΔU = (3/2) n R ΔT, where n is the number of moles, R is the ideal gas constant, and ΔT is the change in temperature.

Since the pressure is constant, the work done by the gas during expansion can be calculated using the formula W = P ΔV, where P is the pressure and ΔV is the change in volume.

Using the ideal gas law, PV = nRT, express the change in volume ΔV in terms of the change in temperature ΔT: ΔV = nRΔT/P.

Substitute the expression for ΔV into the work formula to find W in terms of ΔT: W = nRΔT. Now, to find the fraction of heat energy used for work, divide the work done by the total heat added: Fraction = W/Q = (nRΔT)/Q. Simplify this expression to find the fraction of heat energy used for expansion work.

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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. In mathematical terms, ΔU = Q - W, where ΔU is the change in internal energy, Q is the heat added, and W is the work done by the system. This principle is crucial for understanding how energy is conserved and transformed in thermodynamic processes.

Work Done by a Gas at Constant Pressure

When a gas expands at constant pressure, the work done by the gas is given by W = PΔV, where P is the constant pressure and ΔV is the change in volume. This concept is essential for calculating the work done during the expansion of a gas, as it directly relates the pressure and volume change to the energy used in the process.

Heat Capacity at Constant Pressure (Cp)

The heat capacity at constant pressure, Cp, is the amount of heat required to raise the temperature of a substance by one degree while maintaining constant pressure. For a monatomic ideal gas, Cp is related to the specific heat at constant volume, Cv, by Cp = Cv + R, where R is the ideal gas constant. Understanding Cp helps determine how much of the heat added to the system is used to increase the internal energy versus doing work.

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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

Propane gas (C₃H₈) behaves like an ideal gas with $g equals 1.127$ . Determine the molar heat capacity at constant volume and the molar heat capacity at constant pressure.

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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

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

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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Heat QQ 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?

Verified video answer for a similar problem:

Key Concepts

First Law of Thermodynamics

Work Done by a Gas at Constant Pressure

Heat Capacity at Constant Pressure (Cp)

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?