Textbook Question
A precise steel tape measure has been calibrated at 14°C. At 37°C, what will be the percentage error?
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Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179
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Table of contents
- Ch. 01 - Introduction, Measurement, Estimating10
- Ch. 02 - Describing Motion: Kinematics in One Dimension30
- Ch. 03 - Kinematics in Two or Three Dimensions; Vectors15
- Ch. 04 - Dynamics: Newton's Laws of Motion16
- Ch. 05 - Using Newton's Laws: Friction, Circular Motion, Drag Forces22
- Ch. 06 - Gravitation and Newton's Synthesis10
- Ch. 07 - Work and Energy21
- Ch. 08 - Conservation of Energy33
- Ch. 09 - Linear Momentum26
- Ch. 10 - Rotational Motion41
- Ch. 11 - Angular Momentum; General Rotation27
- Ch. 12 - Static Equilibrium; Elasticity and Fracture27
- Ch. 13 - Fluids28
- Ch. 14 - Oscillations14
- Ch. 15 - Wave Motion35
- Ch. 16 - Sound5
- Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law40
- Ch. 18 - Kinetic Theory of Gases18
- Ch. 19 - Heat and the First Law of Thermodynamics31
- Ch. 20 - Second Law of Thermodynamics32
- Ch. 21 - Electric Charge and Electric Field7
- Ch. 23 - Electric Potential20
- Ch. 24 - Capacitance, Dielectrics, Electric Energy, Storage15
- Ch. 25 - Electric Current and Resistance32
- Ch. 26 - DC Circuits22
- Ch. 27 - Magnetism21
- Ch. 28 - Sources of Magnetic Field24
- Ch. 29 - Electromagnetic Induction and Faraday's Law21
- Ch. 30 - Inductance, Electromagnetic Oscillations, and AC Circuits42
- Ch. 31 - Maxwell's Equations and Electromagnetic Waves25
- Ch. 32 - Light: Reflection and Refraction42
- Ch. 33 - Lenses and Optical Instruments21
- Ch. 34 - The Wave Nature of Light: Interference and Polarization26
- Ch. 35 - Diffraction24
- Ch. 36 - The Special Theory of Relativity29
- Ch. 38 - Quantum Mechanics1
- Ch. 40 - Molecules and Solids6
- Ch. 43 - Elementary Particles3
- Ch. 44 - Astrophysics and Cosmology9
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
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Giancoli Douglas 5th editionCh. 17 - Temperature, Thermal Expansion, and the Ideal Gas LawProblem 72b
Giancoli Douglas 5th editionCh. 17 - Temperature, Thermal Expansion, and the Ideal Gas LawProblem 72bChapter 17, Problem 72b
Show that the bulk modulus (Section 12–5) for an ideal gas held at constant temperature is B = P, where P is the pressure.
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The bulk modulus (B) is defined as the negative ratio of the change in pressure (ΔP) to the fractional change in volume (ΔV/V). Mathematically, B = -ΔP / (ΔV/V).
For an ideal gas at constant temperature (isothermal process), the ideal gas law applies: PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is the temperature.
Rearrange the ideal gas law to express pressure as P = nRT / V. Since temperature (T) is constant, nRT is also constant.
Differentiate P = nRT / V with respect to V to find the relationship between pressure and volume. This gives dP/dV = -nRT / V². Substitute this into the bulk modulus formula.
Using B = -ΔP / (ΔV/V) and substituting dP/dV = -nRT / V², simplify the expression to show that B = P. This demonstrates that the bulk modulus for an ideal gas at constant temperature equals the pressure.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Bulk Modulus
The bulk modulus (B) is a measure of a material's resistance to uniform compression. It is defined as the ratio of the change in pressure to the fractional change in volume. For an ideal gas, the bulk modulus indicates how much pressure is required to compress the gas by a certain volume, reflecting its compressibility.
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Example 1
Ideal Gas Law
The ideal gas law is a fundamental equation in thermodynamics that relates pressure (P), volume (V), temperature (T), and the number of moles (n) of a gas. It is expressed as PV = nRT, where R is the ideal gas constant. This law assumes that gas particles do not interact and occupy no volume, making it applicable under certain conditions.
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Isothermal Process
An isothermal process is a thermodynamic process that occurs at a constant temperature. For an ideal gas, this means that any change in pressure or volume must occur in such a way that the product of pressure and volume remains constant. In this context, the relationship between pressure and volume during compression or expansion is crucial for deriving the bulk modulus.
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Related Practice
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Assume that in an alternate universe, the laws of physics are very different from ours and that “ideal” gases behave as follows: At constant pressure, the volume varies directly with the 2/3 power of the temperature.
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Assume that in an alternate universe, the laws of physics are very different from ours and that “ideal” gases behave as follows: At constant temperature, pressure is inversely proportional to the square of the volume.
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