Honors Physics

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Honors Physics

• CHAPTER 17
• Phases and Phase Changes
• Teacher Luiz Izola

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Chapter Preview

1. Ideal Gases
2. Kinetic Theory
3. Solids and Elastic Deformation
4. Phase Equilibrium and Evaporation
5. Latent Heats
6. Phase Changes and Energy Conservation

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Learning Objectives

• Discuss the phases of matter (solid, liquid,
  gas) in detail.
• Show that ideal gas (one that has no
  interactions between its molecules) is a good
  approximation to real gases.
• Show the relation amongst kinetic theory of
  gases, temperature of a substance, and kinetic
  energy of its molecules.
• Explore the relationship between a force applied
  to a solid and the resulting deformation.
• Finally, the behavior of a substance when it
  changes phases.

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Ideal Gases

• Ideal gases are gases where intermolecular
  interactions are almost non-existent.
• The pressure of an ideal gas depends on the
  following
• Temperature (T)
• Number of Molecules (N)
• Volume (V)
• There three combinations
• Constant V, N ? P (constant)T
• Constant V, T ? P (constant)N
• Constant N, T ? P (constant) / V

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Ideal Gases

• Combining the three previous observations, the
  equation of state, a relationship between the
  thermal properties of a substance, for an ideal
  gas was defined.
• Equation of State for an Ideal Gas
• PV kNT
• K is known as Boltzmann Constant k 1.38 x
  10-23 J/K
• Ex A persons lungs might hold 6.0L (1L 10-3
  m3) of air at body temperature (310K) and
  atmospheric pressure of (101 kPa). Given that
  the air is 21 oxygen, find the number of oxygen
  molecules in the lung.

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Ideal Gases

• Another way to write the ideal gas equation of
  state is in terms of the number of moles in a gas
  opposed to the number of molecules, N.
• A mole is the amount of substance that contains
  as many elementary entities as there are atoms in
  12g of carbon-12.
• The number of atoms in a mole of carbon-12 is
  know as Avogadros Number (NA).
• Avogadros Number (NA)
• NA 6.022 x 1023 molecules/mol
• Now, if we call n number of moles in a gas. We
  can calculate the number of molecules as
• N n x NA

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Ideal Gases

• Then, we can rewrite the ideal-gas equation of
  state as
• PV nNAkT
• The constants NA, k are combined to form the
  universal gas constant R.
• Universal Gas Constant (R)
• R NA k (6.022 x 1023 )(1.38 x 10-23 ) 8.31
  J/ (mol.K)
• Thus, another way to write the ideal-gas
  equation is
• PV nRT

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Ideal Gases

• Ex How many moles of air are in an inflated
  basketball? Assume that the P 171kPa, the
  temperature is 293K, and the diameter of the
  ball is 30cm.
• A mole has precisely the same number (NA) of
  particles. What differs from substance to
  substance is the mass in each mole. Therefore, we
  define atomic or molecular mass (M) as
• The mass in grams of 1 mole of that substance.
• Notice that, if we measure a mass of copper equal
  to 63.546g, we have, in effect, counted out NA
  atoms of copper. Therefore, the mass of an
  individual copper atom, m, is
• m M / NA

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Ideal Gases

• Ex Find the mass of (a) a copper atom, (b) a
  molecule of oxygen, O2. Atomic masses are listed
  in Appendix E.
• ISOTHERMS
• Robert Boyle established the fact that the
  pressure of a gas varies inversely with volume
  as long as the temperature and the number of
  molecules are held constant. This is known as
  Boyles Law.
• Boyles Law
• PiVi PfVf

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Ideal Gases

• Ex A cylindrical flask of cross-sectional area A
  is fitted with an airtight piston. Contained
  within the flask is an ideal gas. The initial
  pressure is 130kPa and the piston height above
  the base of the flask is 25cm. When more mass is
  added, the pressure goes to 170kPa. Assuming the
  temperature is always 290K, what is the new
  height of the piston?

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Ideal Gases

• Charless Law states that the volume of a gas
  divided by its temperature is constant, as long
  as the pressure and the number of molecules are
  constant.
• Charless Law
• Vi / Ti Vf / Tf
• Ex Consider the example from previous slide. In
  this case the temperature is changed from an
  initial value of 290K to a final value of 330K.
  The pressure remains constant at 130kPa, and the
  initial height of the piston is 25cm. Find the
  final height of the piston.

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Kinetic Theory of Gases

• In kinetic theory, we imagine a gas to be made up
  of a collection of molecules moving inside a
  container of volume V. We assume the following
• Container holds a very large number N of
  identical molecules. Each molecule has a mass m.
  and behaves as a point particle.
• The molecules obey Newtons laws of motion at
  all times.
• When the molecules hit the walls or collide
  with one another, they bounce elastically.

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Kinetic Theory of Gases

• Imagine a cube container with side L. Its volume
  is L3. Consider a molecule moving in the negative
  direction x towards the wall. If its initial
  speed is vx, its initial momentum is pi,x -mvx.
  After bouncing off the wall (elastically), it
  moves with the same spped in the direction,
  therefore pf,x mvx. As a result, the molecules
  change in momentum is
• ?px pf,x - pf,x mvx (- mvx) 2mvx
• The time required for a 2L roundtrip is ?t
  2L/vx
• By 2nd Newtons Law, we have
• F ?px / ?t 2mvx / (2L/vx) mvx2 / L
• F mvx2 / L

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Kinetic Theory of Gases

• The average pressure exerted by the this wall is
  simply the force divided by the area. Since the
  area is A L2, we have
• P F / A
• P mvx2 / V

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Kinetic Theory of Gases

• Kinetic Energy and Pressure
• The pressure of a gas is proportional to the
  number of molecules and inversely proportional to
  the volume.
• The pressure of a gas is directly proportional
  to the average kinetic energy of its molecules.
• P 1/3 (N/V)(2Kavg) 2/3 (N/V) (1/2mv2)avg
• Kinetic Energy and Temperature
• Kavg (1/2mv2) avg (3/2)kT

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Kinetic Theory of Gases

• Ex Find the average kinetic energy of airs
  oxygen. Assume the air is at temperature 210C.
• Solving (1/2mv2)avg (3/2)kT for v, we have a
  special kind of velocity called root mean
  square
• Vrms ( (3kT)/m)1/2
• In terms of molecular mass, it becomes
• Vrms ( (3RT)/M)1/2
• Ex The atmosphere is composed of N2 (78), O2
  (21). (a) Is the rms of N2 (28g/mol) gt, lt, or
  equal the rms of O2 (32g/mol)? (b) Find the rms
  of N2 and O2 at 293K.

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Kinetic Theory of Gases

• The internal Energy of an Ideal Gas
• It is equals to the sum of all its potential and
  kinetic energies. As we know, in an ideal gas,
  there is no molecular interactions, other than
  perfectly elastic collisions hence, there is no
  potential energy.
• The total energy of the system is the sum of
  kinetic energy of each one of its molecules.
• Internal Energy of a Monatomic Ideal Gas (U)
• U 3/2(nRT) (Joules)
• Ex A basketball at 290K holds 0.95 mo of air
  molecules. What is the internal energy in the
  ball?

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Solids and Elastic Deformation

• The shape of a solid can be changed if it is
  acted by a force.
• Changing the Length of a Solid
• Pulling a solid rod with a force F, for example,
  will cause a intermolecular spring in the
  direction of the force to expand by an amount
  proportional to the force F.
• The stretch ?L is proportional to the initial
  length L0. Also, the amount of stretch for a
  given force F is inversely proportional to the
  cross-sectional area A of the rod.
• The proportionality constant ?, called Youngs
  Modulus, is related to each specific material.
  See table 17.1, pg 546

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Solids and Elastic Deformation

• Changing the Length of a Solid
• Based on the previous observations, the
  calculation of such force and the resulting
  change in length is given by
• F ? (?L/ L0)A

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Solids and Elastic Deformation

• Changing the Length of a Solid
• There is a straightforward connection between the
  previous formula and Hookes Law. Notice that the
  force required to cause a certain stretch is
  proportional to the stretch just as in Hookes
  Law. Therefore we can write their relation as
• F (?.A / L0).?L kx
• Ex1 A person carries a 21-kg suitcase in one
  hand. Assuming the upper arm bone supports the
  entire weight of the case, find the amount by
  which it stretches. (The upper arm is about 33cm
  with a cross-section 5.2x10-4m2).
• Ex2 A rock climber hangs freely from a nylon
  rope that is 12m long and has a diameter of
  5.5mm. If the rope stretches 4.7cm, what is the
  mass of the climber?

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Solids and Elastic Deformation

• Changing the Shape of a Solid
• This kind of deformation is referred as shear
  deformation.
• Consider the figure below. A force F is applied
  to the right of the top cover and static friction
  applies a force to the left of the bottom cover.
  The result is that the book remains at rest but
  becomes slanted by the amount ?x. Table 17-2,
  page 548 has the shear modulus of some materials.

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Solids and Elastic Deformation

• Changing the Shape of a Solid
• The force required to cause a given amount of
  slant is proportional to ?x, inversely
  proportional to the thickness of the book L0, and
  proportional to the surface area A of the books
  cover. Defining the formula, we have
• F S.(?x/ L0).A
• The constant of proportionality is the shear
  modulus (S).
• Be aware of the differences between Youngs and
  Shear modulus. The term L0 in Youngs refers to
  the solids length in the direction of the
  applied force. In shear, refers to the thickness
  of the of the solid measured in a direction
  perpendicular to the applied force. What about
  the area?

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Solids and Elastic Deformation

• Changing the Shape of a Solid
• Ex1 A horizontal force of 1.2N is applied to the
  top of a stack of pancakes 13cm in diameter and
  9cm high. The result is a shear deformation of
  2.5cm. What is the shear modulus of these
  pancakes?
• Ex2 A lead brick (see below) rests on a solid
  surface. A 2400N force is applied. (a) What is
  the height change? (b) What is the shear
  deformation?

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Solids and Elastic Deformation

• Changing the Volume of a Solid
• Figure below shows a spherical solid whose volume
  decreases by the amount ?x when the pressure
  acting on it increases by ?P. Experiments show
  that the pressure difference required to cause a
  change in volume ,?v, is proportional to and
  inversely proportional to the initial volume V0.

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Solids and Elastic Deformation

• Changing the Volume of a Solid
• ?P -B(?P/V0)
• The constant of proportionality is called bulk
  modulus (B).
• Since B is positive, we have to add a minus sign
  in front of the equation. If the pressure
  increases (?P gt 0), the volume will decrease (?V
  lt 0) and the final quantity will be positive.
• Table 17-3, page 549, gives you a list of Bulk
  Modulus for some materials.

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Solids and Elastic Deformation

• Changing the Volume of a Solid
• Ex1 A gold doubloon ( cylinder shape) 6.1cm in
  diameter and 2mm thick is dropped over the side
  of a pirate ship. When it comes to rest on the
  ocean floor at a depth of 770m, how much has its
  volume changed?
• Ex2 The deepest place in all the oceans is the
  Marianas Trench, where the depth is 10.9km and
  the pressure is 1.10x108Pa. If a copper ball
  10.0cm in diameter is taken to the bottom of the
  trench, by how much does its volume decrease?
• Ex3 In 1934, Charles Beebe broke the diving
  record. He went 923m below the surface. He dove
  in a bathysphere, a steel sphere 4.75ft in
  diameter. How much did the volume of the sphere
  change at the record depth?