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 27a

Chapter 20, Problem 27a

What is the coefficient of performance of an ideal heat pump that extracts heat from 6°C air outside and deposits heat inside a house at 24°C?

Verified step by step guidance

Understand the concept: The coefficient of performance (COP) for a heat pump is a measure of its efficiency. For an ideal heat pump, the COP is given by the formula:

COP = \frac{T_{h}}{T_{h}}

, where

T_{h}

is the temperature of the hot reservoir (inside the house) and

T_{c}

is the temperature of the cold reservoir (outside air).

Convert the temperatures from Celsius to Kelvin: Since the formula requires temperatures in Kelvin, use the conversion

T = t + 273.15

, where

t

is the temperature in Celsius. For the hot reservoir,

T_{h} = 24 + 273.15

. For the cold reservoir,

T_{c} = 6 + 273.15

Substitute the converted temperatures into the COP formula: Replace

T_{h}

and

T_{c}

in the formula

COP = \frac{T_{h}}{T_{h}}

Simplify the expression: Perform the subtraction in the denominator and simplify the fraction to find the COP.

Interpret the result: The COP value represents how many units of heat energy are delivered to the house for each unit of work input. A higher COP indicates a more efficient heat pump.

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Key Concepts

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

Coefficient of Performance (COP)

The Coefficient of Performance (COP) is a measure of the efficiency of a heat pump, defined as the ratio of heat output to the work input. For heating applications, it is calculated as the heat delivered to the warm space divided by the energy consumed by the pump. A higher COP indicates a more efficient heat pump, meaning it can transfer more heat for each unit of energy consumed.

Heat Pump Operation

A heat pump operates by transferring heat from a cooler area to a warmer area, utilizing the principles of thermodynamics. It absorbs heat from the outside environment (in this case, 6°C air) and releases it indoors (at 24°C). The process involves a refrigerant that circulates through the system, changing states from liquid to gas and back, which allows it to absorb and release heat effectively.

Thermodynamic Temperature Scale

The thermodynamic temperature scale, often measured in Kelvin, is crucial for calculating the efficiency of heat pumps. Temperatures in Celsius must be converted to Kelvin by adding 273.15, as thermodynamic equations require absolute temperatures. This conversion is essential for accurately determining the COP and understanding the energy transfer processes involved in heat pump operation.

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Related Practice

Textbook Question

An ideal heat pump is used to maintain the inside temperature of a house at Tᵢₙ = 22°C when the outside temperature is Tₒᵤₜ. Assume the heat pump does work at a rate of 1700 W. Also assume that the house loses heat via conduction through its walls and other surfaces at a rate given by ( 650 W/C°) (Tᵢₙ - Tₒᵤₜ). If the outside temperature is less than you just calculated, what happens?

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Textbook Question

(II) What is the temperature inside an ideal refrigerator–freezer that operates with a COP = 7.0 in a 22°C room?

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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. 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

If 0.45 kg of water at 100°C is changed by a reversible process to steam at 100°C, determine the change in entropy of the surroundings.

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Textbook Question

How much less per year would it cost a family to operate a heat pump that has a coefficient of performance of 2.9 than an electric heater that costs \$2100 to heat their home for a year? If the conversion to the heat pump costs \$15,000, how long would it take the family to break even on heating costs? How much would the family save in 20 years?

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