bartleby

Concept explainers

bartleby

Videos

Question
Book Icon
Chapter 16, Problem 62P

(a)

To determine

To show: The fraction change in density for a change in temperature is given by Δρρ=βΔT .

(a)

Expert Solution
Check Mark

Answer to Problem 62P

The fraction change in density for a change in temperature is Δρρ=βΔT .

Explanation of Solution

Given info: The density of the liquid is ρ , the change in the temperature is ΔT , the maximum density of fresh water at 4.0°C is 1.0000g/cm3 , the density of fresh water at 10.0°C is 0.9997g/cm3 and the density of water at 0°C is 0.9999g/cm3 .

The expression for the coefficient of the volume expansion is,

βΔT=ΔVV

Here,

β is the coefficient of the volume expansion.

The formula for the density is,

ρ=mV

Here,

m is the mass of the water.

V is the volume of the water.\

Differentiate the above equation with respect to volume.

dρ=mV2dV

For very small change in volume and density the expression is,

Δρ=mV2ΔV=mVΔVV

Substitute ρ for mV and βΔT for ΔVV in above equation.

Δρ=ρβΔTΔρρ=βΔT

Conclusion:

Therefore, the fraction change in density for a change in temperature is Δρρ=βΔT .

(b)

To determine

The significance of the negative sign.

(b)

Expert Solution
Check Mark

Answer to Problem 62P

the negative sign signifies that when the temperature increases, the density decreases.

Explanation of Solution

Given info: The density of the liquid is ρ , the change in the temperature is ΔT , the maximum density of fresh water at 4.0°C is 1.0000g/cm3 , the density of fresh water at 10.0°C is 0.9997g/cm3 and the density of water at 0°C is 0.9999g/cm3 .

The expression for the fraction of the density is,

Δρρ=βΔT (1)

The negative sign in the above expression shows that as the temperature increases, the density of the water is decreases.

Conclusion:

Therefore, the negative sign signifies that when the temperature increases, the density decreases.

(c)

To determine

The value of β for the temperature interval 4.0°C to 10.0°C .

(c)

Expert Solution
Check Mark

Answer to Problem 62P

The value of β for the temperature interval 4.0°C to 10.0°C is 5×105(°C)1 .

Explanation of Solution

Given info: The density of the liquid is ρ , the change in the temperature is ΔT , the maximum density of fresh water at 4.0°C is 1.0000g/cm3 , the density of fresh water at 10.0°C is 0.9997g/cm3 and the density of water at 0°C is 0.9999g/cm3 .

Recall the equation (1) and rearrange for β .

Δρρ=βΔTβ=ΔρρΔT

Substitute (0.9997g/cm31.0000g/cm3) for Δρ , 1.0000g/cm3 for ρ and (10.0°C4.0°C) for ΔT in above equation to find β .

β=(0.9997g/cm31.0000g/cm3)(1.0000g/cm3)((10.0°C4.0°C))=5×105°C1

Conclusion:

Therefore, the value of β for temperature interval 4.0°C to 10.0°C is 5×105(°C)1 .

(d)

To determine

The value of β for the temperature interval 0°C to 4.0°C .

(d)

Expert Solution
Check Mark

Answer to Problem 62P

The value of β for temperature interval 0°C to 4.0°C is 2.5×105(°C)1 .

Explanation of Solution

Given info: The density of the liquid is ρ , the change in the temperature is ΔT , the maximum density of fresh water at 4.0°C is 1.0000g/cm3 , the density of fresh water at 10.0°C is 0.9997g/cm3 and the density of water at 0°C is 0.9999g/cm3 .

The expression for β is,

Δρρ=βΔTβ=ΔρρΔT

Substitute (1.0000g/cm30.9999g/cm3) for Δρ , 1.0000g/cm3 for ρ and (4.0°C0°C) for ΔT in above equation to find β .

β=(1.0000g/cm30.9999g/cm3)(1.0000g/cm3)((4.0°C0.0°C))=2.5×105°C1

Conclusion:

Therefore, the value of β for the temperature interval 0°C to 4.0°C is 2.5×105(°C)1 .

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Students have asked these similar questions
A liquid has a density r. (a) Show that the fractional change in density for a change in temperature ΔT is Δρ/ρ = -β ΔT. (b) What does the negative sign signify? (c) Fresh water has a maximum density of 1.000 0 g/cm3 at 4.0°C. At 10.0°C, its density is 0.999 7 g/cm3. What is β for water over this temperature interval? (d) At 0°C, the density of water is 0.999 9 g/cm3. What is the value for β over the temperature range 0°C to 4.00°C?
The average kinetic energy of the molecules in a gas sample depends only on the temperature, ?. However, given the same kinetic energies, a lighter molecule will move faster than a heavier molecule, as shown in the equation for rms speed. where ?=8.314 J/(mol⋅K) and ℳ is molar mass in kilograms per mole. Note that a joule is the same as a kilogram‑meter squared per second squared (kg·m2/s2). What is the rms speed of Cl2 molecules at 401 K? What is the rms speed of He atoms at 401401 K?
The volume of an automobile tire is 2.5x10^-2m3. The pressure of the air in the tire is 3.3 atm and the temperature is 20°C. What is the mass of air in grams? The mean molecular mass of air is 29g. 1atm=1.01x10^5Pa

Chapter 16 Solutions

Bundle: Principles of Physics: A Calculus-Based Text, 5th + WebAssign Printed Access Card for Serway/Jewett's Principles of Physics: A Calculus-Based Text, 5th Edition, Multi-Term

Ch. 16 - Prob. 5OQCh. 16 - Prob. 6OQCh. 16 - What would happen if the glass of a thermometer...Ch. 16 - Prob. 8OQCh. 16 - A gas is at 200 K. If we wish to double the rms...Ch. 16 - Prob. 10OQCh. 16 - Prob. 11OQCh. 16 - A rubber balloon is filled with 1 L of air at 1...Ch. 16 - Prob. 13OQCh. 16 - An ideal gas is contained in a vessel at 300 K....Ch. 16 - Prob. 15OQCh. 16 - Prob. 16OQCh. 16 - Prob. 17OQCh. 16 - A sample of gas with a thermometer immersed in the...Ch. 16 - Prob. 19OQCh. 16 - Prob. 1CQCh. 16 - Prob. 2CQCh. 16 - Prob. 3CQCh. 16 - A piece of copper is dropped into a beaker of...Ch. 16 - Prob. 5CQCh. 16 - Prob. 6CQCh. 16 - Prob. 7CQCh. 16 - Prob. 8CQCh. 16 - Prob. 9CQCh. 16 - Prob. 10CQCh. 16 - Prob. 11CQCh. 16 - Prob. 12CQCh. 16 - Prob. 13CQCh. 16 - Prob. 1PCh. 16 - Convert the following to equivalent temperatures...Ch. 16 - Prob. 3PCh. 16 - Prob. 4PCh. 16 - Prob. 5PCh. 16 - Prob. 6PCh. 16 - Prob. 7PCh. 16 - Prob. 8PCh. 16 - Prob. 9PCh. 16 - A sample of a solid substance has a mass m and a...Ch. 16 - Each year thousands of children are badly burned...Ch. 16 - Prob. 12PCh. 16 - Prob. 13PCh. 16 - Prob. 14PCh. 16 - The active element of a certain laser is made of a...Ch. 16 - Prob. 16PCh. 16 - Prob. 17PCh. 16 - Prob. 18PCh. 16 - Prob. 19PCh. 16 - Prob. 20PCh. 16 - Prob. 21PCh. 16 - Prob. 22PCh. 16 - Prob. 23PCh. 16 - Prob. 24PCh. 16 - Prob. 25PCh. 16 - Prob. 26PCh. 16 - Prob. 27PCh. 16 - Prob. 28PCh. 16 - The mass of a hot-air balloon and its cargo (not...Ch. 16 - Prob. 30PCh. 16 - A popular brand of cola contains 6.50 g of carbon...Ch. 16 - Prob. 32PCh. 16 - At 25.0 m below the surface of the sea, where the...Ch. 16 - To measure how far below the ocean surface a bird...Ch. 16 - Prob. 35PCh. 16 - Prob. 36PCh. 16 - Prob. 37PCh. 16 - Prob. 38PCh. 16 - Prob. 39PCh. 16 - A cylinder contains a mixture of helium and argon...Ch. 16 - Prob. 41PCh. 16 - Prob. 42PCh. 16 - Prob. 43PCh. 16 - (a) How many atoms of helium gas fill a spherical...Ch. 16 - Fifteen identical particles have various speeds:...Ch. 16 - From the MaxwellBoltzmann speed distribution, show...Ch. 16 - Prob. 47PCh. 16 - Helium gas is in thermal equilibrium with liquid...Ch. 16 - Prob. 49PCh. 16 - Prob. 50PCh. 16 - Prob. 51PCh. 16 - Prob. 52PCh. 16 - A mercury thermometer is constructed as shown in...Ch. 16 - A liquid with a coefficient of volume expansion ...Ch. 16 - A clock with a brass pendulum has a period of...Ch. 16 - A vertical cylinder of cross-sectional area A is...Ch. 16 - Prob. 57PCh. 16 - Prob. 58PCh. 16 - Prob. 59PCh. 16 - The rectangular plate shown in Figure P16.60 has...Ch. 16 - In a chemical processing plant, a reaction chamber...Ch. 16 - Prob. 62PCh. 16 - Prob. 63PCh. 16 - Two concrete spans that form a bridge of length L...Ch. 16 - A 1.00-km steel railroad rail is fastened securely...Ch. 16 - Prob. 66PCh. 16 - Prob. 67PCh. 16 - Prob. 68PCh. 16 - Consider an object with any one of the shapes...Ch. 16 - Prob. 70PCh. 16 - Prob. 71PCh. 16 - Prob. 72PCh. 16 - Prob. 73PCh. 16 - A cylinder that has a 40.0-cm radius and is 50.0...Ch. 16 - Prob. 75P
Knowledge Booster
Background pattern image
Physics
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.
Similar questions
SEE MORE QUESTIONS
Recommended textbooks for you
Text book image
Principles of Physics: A Calculus-Based Text
Physics
ISBN:9781133104261
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Text book image
University Physics Volume 2
Physics
ISBN:9781938168161
Author:OpenStax
Publisher:OpenStax
Text book image
Physics for Scientists and Engineers: Foundations...
Physics
ISBN:9781133939146
Author:Katz, Debora M.
Publisher:Cengage Learning
Text book image
Physics for Scientists and Engineers, Technology ...
Physics
ISBN:9781305116399
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Text book image
College Physics
Physics
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Cengage Learning
Text book image
An Introduction to Physical Science
Physics
ISBN:9781305079137
Author:James Shipman, Jerry D. Wilson, Charles A. Higgins, Omar Torres
Publisher:Cengage Learning
A Level Physics – Ideal Gas Equation; Author: Atomi;https://www.youtube.com/watch?v=k0EFrmah7h0;License: Standard YouTube License, CC-BY