Ch. 20 - Second Law of Thermodynamics

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 20 - Second Law of Thermodynamics

Problem 18

Chapter 20, Problem 18

The working substance of a certain Carnot engine is 1.0 mol of an ideal monatomic gas. During the isothermal expansion portion of this engine’s cycle, the volume of the gas doubles, while during the adiabatic expansion the volume increases by a factor of 6.2. The work output of the engine is 920 J in each cycle. Compute the temperatures of the two reservoirs between which this engine operates.

Verified step by step guidance

Identify the key processes in the Carnot cycle: isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression. For this problem, focus on the isothermal and adiabatic processes described.

For the isothermal expansion, use the ideal gas law and the formula for work done during isothermal processes: \( W_{iso} = nRT_H \ln \left( \frac{V_f}{V_i} \right) \), where \( T_H \) is the temperature of the hot reservoir, \( V_f \) and \( V_i \) are the final and initial volumes, \( n \) is the number of moles, and \( R \) is the gas constant.

For the adiabatic expansion, use the relationship between temperature and volume for an adiabatic process: \( T V^{\gamma - 1} = \text{constant} \), where \( \gamma = \frac{C_p}{C_v} \) is the adiabatic index. For a monatomic gas, \( \gamma = \frac{5}{3} \). Use this to relate the temperatures and volumes during the adiabatic expansion.

The total work output of the Carnot engine is given as 920 J. Use the efficiency formula for a Carnot engine: \( \eta = 1 - \frac{T_C}{T_H} \), where \( T_C \) is the temperature of the cold reservoir. Combine this with the work expressions to solve for \( T_H \) and \( T_C \).

Substitute the known values (\( n = 1.0 \) mol, \( V_f/V_i = 2 \) for isothermal expansion, \( V_f/V_i = 6.2 \) for adiabatic expansion, and \( W = 920 \) J) into the equations derived in the previous steps to calculate the temperatures of the two reservoirs.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Carnot Engine

A Carnot engine is an idealized thermodynamic cycle that provides the maximum possible efficiency for a heat engine operating between two temperature reservoirs. It consists of two isothermal processes (heat absorption and rejection) and two adiabatic processes (expansion and compression). The efficiency of a Carnot engine depends solely on the temperatures of the hot and cold reservoirs, making it a benchmark for real engines.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of an ideal gas through the equation PV = nRT. In the context of the Carnot engine, this law helps determine the changes in temperature and pressure during the isothermal and adiabatic processes, allowing for the calculation of the temperatures of the reservoirs based on the gas's behavior.

Work Done by a Gas

The work done by a gas during expansion or compression can be calculated using the formula W = ∫PdV, where P is the pressure and V is the volume. In the case of isothermal expansion, the work done is related to the temperature and the change in volume. Understanding how to calculate work is crucial for analyzing the performance of the Carnot engine and determining the temperatures of the reservoirs.

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Calculating Work Done on Monoatomic Gas

Related Practice

Textbook Question

One mole of monatomic gas undergoes a Carnot cycle with T_H = 350°C and T_L = 210°C. The initial pressure is 8.8 atm. During the isothermal expansion, the volume doubles. Calculate the efficiency of the cycle using Eqs. 20–1 and 20–3.

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Textbook Question

One mole of monatomic gas undergoes a Carnot cycle with T_H = 350°C and T_L = 210°C. The initial pressure is 8.8 atm. During the isothermal expansion, the volume doubles. Find the values of the pressure and volume at the points a, b, c, and d of Fig. 20–5.

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Textbook Question

(II) 1.00 mole of nitrogen (N₂) gas and 1.00 mole of argon (Ar) gas are in separate, equal-sized, insulated containers at the same temperature. The containers are then connected and the gases (assumed ideal) allowed to mix. What is the change in entropy

(a) of the system

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Textbook Question

A four-cylinder gasoline engine has an efficiency of 0.22 and delivers 180 J of work per cycle per cylinder. If the engine runs at 25 cycles per second (1500 rpm), determine the total heat input per second from the gasoline.

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Textbook Question

A particular car does work at the rate of about 7.0 kJ/s when traveling at a steady 21.8 m/s along a level road. This is the work done against friction. The car can travel 17 km on 1.0 L of gasoline at this speed (about 40 mi/gal). What is the minimum value for T_H if T_L is 25°C? The energy available from 1.0 L of gas is 3.2 x 10⁷ J.

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Textbook Question

Assume that a 65-kg hiker needs to eat 4.0 x 10³ kcal of energy to supply a day’s worth of metabolism ( = Q_H). Estimate the elevation change the person can climb in one day, using only this amount of energy. As a fun and rough prediction, treat the person as an isolated heat engine, operating between the internal temperature of 37°C (98.6°F) and the ambient air temperature of 20°C.

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