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The Nature of Science and Physics

Kinematics

Introduction to One-Dimensional KinematicsDisplacementVectors, Scalars, and Coordinate SystemsTime, Velocity, and SpeedVideo: One-Dimensional KinematicsAccelerationMotion Equations for Constant Acceleration in One DimensionProblem-Solving Basics for One-Dimensional KinematicsFalling ObjectsGraphical Analysis of One-Dimensional Motion

Two-Dimensional Kinematics

Dynamics: Force and Newton's Laws of Motion

Introduction to Dynamics: Newton's Laws of MotionDevelopment of Force ConceptNewton's First Law of Motion: InertiaNewton's Second Law of Motion: Concept of a SystemNewton's Third Law of Motion: Symmetry in ForcesVideo: Newton's LawsNormal, Tension, and Other Examples of ForcesProblem-Solving StrategiesFurther Applications of Newton's Laws of MotionExtended Topic: The Four Basic Forces—An Introduction

Further Applications of Newton's Laws: Friction, Drag, and Elasticity

Uniform Circular Motion and Gravitation

Introduction to Uniform Circular Motion and GravitationRotation Angle and Angular VelocityCentripetal AccelerationCentripetal ForceFictitious Forces and Non-inertial Frames: The Coriolis ForceNewton's Universal Law of GravitationVideo: GravitationSatellites and Kepler's Laws: An Argument for Simplicity

Work, Energy, and Energy Resources

Introduction to Work, Energy, and Energy ResourcesWork: The Scientific DefinitionKinetic Energy and the Work-Energy TheoremGravitational Potential EnergyVideo: Potential and Kinetic EnergyConservative Forces and Potential EnergyNonconservative ForcesConservation of EnergyPowerWork, Energy, and Power in HumansWorld Energy Use

Linear Momentum and Collisions

Rotational Motion and Angular Momentum

Introduction to Rotational Motion and Angular MomentumAngular AccelerationKinematics of Rotational MotionVideo: Rotational MotionDynamics of Rotational Motion: Rotational InertiaRotational Kinetic Energy: Work and Energy RevisitedAngular Momentum and Its ConservationVideo: Angular MomentumCollisions of Extended Bodies in Two DimensionsGyroscopic Effects: Vector Aspects of Angular Momentum

Statics and Torque

Fluid Statics

Introduction to Fluid StaticsWhat Is a Fluid?DensityPressureVariation of Pressure with Depth in a FluidPascal's PrincipleGauge Pressure, Absolute Pressure, and Pressure MeasurementArchimedes' PrincipleVideo: BuoyancyCohesion and Adhesion in Liquids: Surface Tension and Capillary ActionPressures in the Body

Fluid Dynamics and Its Biological and Medical Applications

Introduction to Fluid Dynamics and Biological and Medical ApplicationsFlow Rate and Its Relation to VelocityBernoulli's EquationVideo: Fluid FlowThe Most General Applications of Bernoulli's EquationViscosity and Laminar Flow; Poiseuille's LawThe Onset of TurbulenceMotion of an Object in a Viscous FluidMolecular Transport Phenomena: Diffusion, Osmosis, and Related Processes

Temperature, Kinetic Theory, and the Gas Laws

Heat and Heat Transfer Methods

Thermodynamics

Introduction to ThermodynamicsThe First Law of ThermodynamicsThe First Law of Thermodynamics and Some Simple ProcessesIntroduction to the Second Law of Thermodynamics: Heat Engines and Their EfficiencyCarnot's Perfect Heat Engine: The Second Law of Thermodynamics RestatedApplications of Thermodynamics: Heat Pumps and RefrigeratorsEntropy and the Second Law of Thermodynamics: Disorder and the Unavailability of EnergyStatistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation

Oscillatory Motion and Waves

Introduction to Oscillatory Motion and WavesHooke's Law: Stress and Strain RevisitedPeriod and Frequency in OscillationsSimple Harmonic Motion: A Special Periodic MotionVideo: Harmonic MotionThe Simple PendulumEnergy and the Simple Harmonic OscillatorUniform Circular Motion and Simple Harmonic MotionDamped Harmonic MotionForced Oscillations and ResonanceWavesSuperposition and InterferenceEnergy in Waves: Intensity

Physics of Hearing

- Express the ideal gas law in terms of molecular mass and velocity.
- Define thermal energy.
- Calculate the kinetic energy of a gas molecule, given its temperature.
- Describe the relationship between the temperature of a gas and the kinetic energy of atoms and molecules.
- Describe the distribution of speeds of molecules in a gas.

Figure 1 shows an elastic collision of a gas molecule with the wall of a container, so that it exerts a force on the wall (by Newton’s third law). Because a huge number of molecules will collide with the wall in a short time, we observe an average force per unit area. These collisions are the source of pressure in a gas. As the number of molecules increases, the number of collisions and thus the pressure increase. Similarly, the gas pressure is higher if the average velocity of molecules is higher. The actual relationship is derived in the Making Connections feature below. The following relationship is found:

$PV=\frac{1}{3}Nm{\overline{v^2}}\\$

, where $\overline{v^2}\\$

is the average of the molecular speed squared.What can we learn from this atomic and molecular version of the ideal gas law? We can derive a relationship between temperature and the average translational kinetic energy of molecules in a gas. Recall the previous expression of the ideal gas law:

Equating the right-hand side of this equation with the right-hand side of

$PV=\frac{1}{3}Nm{\overline{v^2}}\\$

gives $\frac{1}{3}Nm{\overline{v^2}}=NkT\\$

.Figure 2 shows a box filled with a gas. We know from our previous discussions that putting more gas into the box produces greater pressure, and that increasing the temperature of the gas also produces a greater pressure. But why should increasing the temperature of the gas increase the pressure in the box? A look at the atomic and molecular scale gives us some answers, and an alternative expression for the ideal gas law.

The figure shows an expanded view of an elastic collision of a gas molecule with the wall of a container. Calculating the average force exerted by such molecules will lead us to the ideal gas law, and to the connection between temperature and molecular kinetic energy. We assume that a molecule is small compared with the separation of molecules in the gas, and that its interaction with other molecules can be ignored. We also assume the wall is rigid and that the molecule’s direction changes, but that its speed remains constant (and hence its kinetic energy and the magnitude of its momentum remain constant as well). This assumption is not always valid, but the same result is obtained with a more detailed description of the molecule’s exchange of energy and momentum with the wall.

If the molecule’s velocity changes in the

$F=\frac{\Delta{p}}{\Delta{t}}=\frac{2mv_x}{\Delta{t}}\\$

.There is no force between the wall and the molecule until the molecule hits the wall. During the short time of the collision, the force between the molecule and wall is relatively large. We are looking for an average force; we take Δ

$\Delta{t}=\frac{2l}{v_x}\\$

, and the expression for the force becomes$\displaystyle{F}=\frac{2mv_x}{\frac{2l}{v_x}}=\frac{{mv_x}^2}{l}\\$

This force is due to $\displaystyle{F}=N\frac{m\overline{{v_x}^2}}{l}\\$

,
where the bar over a quantity means its average value. We would like to have the force in terms of the speed $\overline{v^2}=\overline{{v_x}^2}+\overline{{v_y}^2}+\overline{{v_z}^2}\\$

.
Because the velocities are random, their average components in all directions are the same:$\overline{{v_x}^2}=\overline{{v_y}^2}=\overline{{v_z}^2}\\$

.
Thus,$\overline{v^2}=3\overline{{v_x}^2}\\$

or $\overline{{v_x}^2}=\frac{1}{3}\overline{v^2}\\$

.
Substituting $\frac{1}{3}\overline{v^2}\\$

into the expression for $F=N\frac{m\overline{v^2}}{3l}\\$

.
The pressure is $\frac{F}{A}\\$

$P=\frac{F}{A}=N\frac{m\overline{v^2}}{3Al}=\frac{1}{3}\frac{Nm\overline{v^2}}{V}\\$

, where we used$PV=\frac{1}{3}Nm\overline{v^2}\\$

This equation is another expression of the ideal gas law.$\frac{1}{2}mv^2\\$

, from the left-hand side of the equation by canceling $\overline{\text{KE}}=\frac{1}{2}m\overline{v^2}=\frac{3}{2}kT\\$

The average translational kinetic energy of a molecule, $\overline{\text{KE}}\\$

, is called $\overline{\text{KE}}=\frac{1}{2}m\overline{v^2}=\frac{3}{2}kT\\$

is a molecular interpretation of temperature, and it has been found to be valid for gases and reasonably accurate in liquids and solids. It is another definition of temperature based on an expression of the molecular energy.It is sometimes useful to rearrange

$\overline{\text{KE}}=\frac{1}{2}m\overline{v^2}=\frac{3}{2}kT\\$

, and solve for the average speed of molecules in a gas in terms of temperature,$\displaystyle\sqrt{\overline{v^2}}=v_{\text{rms}}=\sqrt{\frac{3kT}{m}}\\$

where - What is the average kinetic energy of a gas molecule at 20.0ºC (room temperature)?
- Find the rms speed of a nitrogen molecule (N
_{2}) at this temperature.

$\overline{\text{KE}}=\frac{1}{2}m\overline{v^2}=\frac{3}{2}kT\\$

Before substituting values into this equation, we must convert the given temperature to kelvins. This conversion gives $\overline{\text{KE}}=\frac{3}{2}kT=\frac{3}{2}\left(1.38\times10^{-23}\text{ J/K}\right)\left(293\text{ K}\right)=6.07\times10^{-21}\text{ J}\\$

$\displaystyle\sqrt{\overline{v^2}}=v_{\text{rms}}=\sqrt{\frac{3kT}{m}}\\$

but we must first find the mass of a nitrogen molecule. Using the molecular mass of nitrogen N$m=\frac{2(14.0067)\times10^{-3}\text{ kg/mol}}{6.02\times10^{23}\text{ mol}^{-1}}=4.65\times10^{-26}\text{ kg}\\$

$\displaystyle{v}_{\text{rms}}=\sqrt{\frac{3kT}{m}}=\sqrt{\frac{3\left(1.38\times10^{-23}\text{ J/K}\right)\left(293\text{ K}\right)}{4.65\times10^{-26}\text{ kg}}}=511\text{ m/s}\\$

The distribution of thermal speeds depends strongly on temperature. As temperature increases, the speeds are shifted to higher values and the distribution is broadened.

What is the implication of the change in distribution with temperature shown in Figure 5 for humans? All other things being equal, if a person has a fever, he or she is likely to lose more water molecules, particularly from linings along moist cavities such as the lungs and mouth, creating a dry sensation in the mouth.

Identify the unknowns: We need to solve for temperature,

Determine which equations are needed. To solve for mass

$m=\frac{\text{molar mass}}{\text{number of atoms per mole}}\\$

To solve for temperature $\overline{KE}=\frac{1}{2}m\overline{v^2}=\frac{3}{2}kT\\$

or $\sqrt{\overline{v^2}}=v_{\text{rms}}=\sqrt{\frac{3kT}{m}}\\$

to yield $T=\frac{m\overline{v^2}}{3k}\\$

, where Plug the known values into the equations and solve for the unknowns.

$m=\frac{\text{molar mass}}{\text{number of atoms per mole}}=\frac{4.0026\times10^{-3}\text{ kg/mol}}{6.02\times10^{23}\text{ mol}}=6.65\times10^{-27}\text{ kg}\\$

$T=\frac{\left(6.65\times10^{-27}\text{ kg}\right)\left(11.1\times10^3\text{ m/s}\right)^2}{3\left(1.38\times10^{-23}\text{ J/K}\right)}=1.98\times10^4\text{ K}\\$

This temperature is much higher than atmospheric temperature, which is approximately 250 K (

In fact, so few have speeds above the escape velocity that billions of years are required to lose significant amounts of the atmosphere. Figure 6 shows the impact of a lack of an atmosphere on the Moon. Because the gravitational pull of the Moon is much weaker, it has lost almost its entire atmosphere. The comparison between Earth and the Moon is discussed in this chapter’s Problems and Exercises.

- Kinetic theory is the atomistic description of gases as well as liquids and solids.
- Kinetic theory models the properties of matter in terms of continuous random motion of atoms and molecules.
- The ideal gas law can also be expressed as $\text{PV}=\frac{1}{3}\text{Nm}\overline{{v}^{2}}\\$, where
*P*is the pressure (average force per unit area),*V*is the volume of gas in the container,*N*is the number of molecules in the container,*m*is the mass of a molecule, and$\overline{{v}^{2}}\\$is the average of the molecular speed squared. - Thermal energy is defined to be the average translational kinetic energy $\overline{\text{KE}}\\$of an atom or molecule.
- The temperature of gases is proportional to the average translational kinetic energy of atoms and molecules: $\overline{\text{KE}}=\frac{1}{2}m\overline{{v}^{2}}=\frac{3}{2}\text{kT}\\$or$\sqrt{\overline{{v}^{2}}}={v}_{\text{rms}}=\sqrt{\frac{3\text{kT}}{m}}\\$.
- The motion of individual molecules in a gas is random in magnitude and direction. However, a gas of many molecules has a predictable distribution of molecular speeds, known as the
*Maxwell-Boltzmann distribution*.

- How is momentum related to the pressure exerted by a gas? Explain on the atomic and molecular level, considering the behavior of atoms and molecules.

- Some incandescent light bulbs are filled with argon gas. What is
*v*_{rms}for argon atoms near the filament, assuming their temperature is 2500 K? - Average atomic and molecular speeds (
*v*_{rms}) are large, even at low temperatures. What is*v*_{rms}for helium atoms at 5.00 K, just one degree above helium’s liquefaction temperature? - (a) What is the average kinetic energy in joules of hydrogen atoms on the 5500ºC surface of the Sun? (b) What is the average kinetic energy of helium atoms in a region of the solar corona where the temperature is 6.00 × 10
^{5}K? - The escape velocity of any object from Earth is 11.2 km/s. (a) Express this speed in m/s and km/h. (b) At what temperature would oxygen molecules (molecular mass is equal to 32.0 g/mol) have an average velocity
*v*_{rms}equal to Earth’s escape velocity of 11.1 km/s? - The escape velocity from the Moon is much smaller than from Earth and is only 2.38 km/s. At what temperature would hydrogen molecules (molecular mass is equal to 2.016 g/mol) have an average velocity
*v*_{rms}equal to the Moon’s escape velocity? - Nuclear fusion, the energy source of the Sun, hydrogen bombs, and fusion reactors, occurs much more readily when the average kinetic energy of the atoms is high—that is, at high temperatures. Suppose you want the atoms in your fusion experiment to have average kinetic energies of 6.40 × 10
^{−14}J. What temperature is needed? - Suppose that the average velocity (
*v*_{rms}) of carbon dioxide molecules (molecular mass is equal to 44.0 g/mol) in a flame is found to be 1.05 × 10^{5}m/s. What temperature does this represent? - Hydrogen molecules (molecular mass is equal to 2.016 g/mol) have an average velocity
*v*_{rms}equal to 193 m/s. What is the temperature? - Much of the gas near the Sun is atomic hydrogen. Its temperature would have to be 1.5 × 10
^{7}K for the average velocity*v*_{rms}to equal the escape velocity from the Sun. What is that velocity? - There are two important isotopes of uranium—
^{235}U and^{238}U; these isotopes are nearly identical chemically but have different atomic masses. Only^{235}U is very useful in nuclear reactors. One of the techniques for separating them (gas diffusion) is based on the different average velocities*v*_{rms}of uranium hexafluoride gas, UF_{6}. (a) The molecular masses for^{235}U UF_{6}and^{238}U UF_{6}are 349.0 g/mol and 352.0 g/mol, respectively. What is the ratio of their average velocities? (b) At what temperature would their average velocities differ by 1.00 m/s? (c) Do your answers in this problem imply that this technique may be difficult?

$\overline{\text{KE}}\\$

, the average translational kinetic energy of a molecule3. (a) 1.20 × 10

5. 458 K

7. 1.95 × 10

9. 6.09 × 10