Introducing%20the%20P-V%20diagram

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Introducing the P-V diagram

• P-V (pressure versus volume) diagrams can be very
  useful.
• What are the units resulting from multiplying
  pressure in kPa by volume in liters?

Rank the four states shown on the diagram based
on their absolute temperature, from greatest to
least.
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Introducing the P-V diagram

• P-V (pressure versus volume) diagrams can be very
  useful.
• What are the units resulting from multiplying
  pressure in kPa by volume in liters?

Rank the four states shown on the diagram based
on their absolute temperature, from greatest to
least. Temperature is proportional to PV, so rank
by PV 2 gt 13 gt 4.
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Isotherms

• Isotherms are lines of constant temperature.
• On a P-V diagram, isotherms satisfy the equation
• PV constant

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Thermodynamics

• Thermodynamics is the study of systems involving
  energy in the form of heat and work.
• Consider a cylinder of ideal gas, at room
  temperature.
• When the cylinder is placed in a
• container of hot water, heat is
• transferred into the cylinder.
• Where does that energy go?
• The piston is free to move up
• or down without friction.

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Thermodynamics
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The First Law of Thermodynamics

• Some of the added energy goes into raising the
  temperature of the gas (we call this raising the
  internal energy). The rest of it does work,
  raising the piston. Conserving energy
• (the first law of thermodynamics)
• Q is heat added to a system (or removed if it is
  negative)
• is the internal energy of the system (the
  energy associated with the motion of the atoms
  and/or molecules), so is the change in
  the internal energy, which is proportional to the
  change in temperature.
• W is the work done by the system.
• The First Law is often written as

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Work

• We defined work previously as
•     (true if the force is constant)
• F PA, so
• At constant pressure the work done by the system
  is the pressure multiplied by the change in
  volume.
• If there is no change in volume, no work is done.
  
• In general, the work done by the system is the
  area under the P-V graph. This is why P-V
  diagrams are so useful.

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Work the area under the curve

• The net work done by the gas is positive in this
  case, because the change in volume is positive,
  and equal to the area under the curve.

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A P-V diagram question
An ideal gas initially in state 1 progresses to a
final state by one of three different processes
(a, b, or c). Each of the possible final states
has the same temperature. For which process is
the change in internal energy larger? 1. a
2. b 3. c 4. Equal for all three 5. We cant
determine it
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A P-V diagram question

• Because the change in temperature is the same,
  the change in internal energy is the same for all
  three processes.

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Another P-V diagram question
An ideal gas initially in state 1 progresses to a
final state by one of three different processes
(a, b, or c). Each of the possible final states
has the same temperature. For which process is
more heat transferred into the ideal gas? 1.
a 2. b 3. c 4. Equal for all three 5. We
cant determine it
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Another P-V diagram question

• The heat is the sum of the change in internal
  energy (which is the same for all three) and the
  work (the area under the curve), so whichever
  process involves more work requires more heat.

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Another P-V diagram question

• The heat is the sum of the change in internal
  energy (which is the same for all three) and the
  work (the area under the curve), so whichever
  process involves more work requires more heat.
• Process c involves more
• work, and thus requires
• more heat.

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Constant volume vs. constant pressure
We have two identical cylinders of ideal gas.
Piston 1 is free to move. Piston 2 is fixed so
cylinder 2 has a constant volume. We put both
systems into a reservoir of hot water and let
them come to equilibrium. Which statement is
true? 1. Both the heat Q and the change in
internal energy will be the same for the two
cylinders 2. The heat is the same for the two
cylinders but cylinder 1 has a larger change in
internal energy. 3. The heat is the same for the
two cylinders but cylinder 2 has a larger change
in internal energy. 4. The changes in internal
energy are the same for the two cylinders but
cylinder 1 has more heat. 5. The changes in
internal energy are the same for the two
cylinders but cylinder 2 has more heat.
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Constant volume vs. constant pressure

• Each cylinder comes to the same temperature as
  the reservoir. How do the changes in internal
  energy compare?
• Which cylinder does more work?

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Constant volume vs. constant pressure

• Each cylinder comes to the same temperature as
  the reservoir. How do the changes in internal
  energy compare?
• The same number of moles of the same gas
  experience the same temperature change, so the
  change in internal energy is the same.
• Which cylinder does more work?
• Cylinder 2 does no work, so cylinder 1 does more
  work.
• By the first law, cylinder 1 requires more heat
  to produce the same change in temperature as
  cylinder 2. The heat required depends on the
  process.

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Solving thermodynamics problems

• A typical thermodynamics problem involves some
  process that moves an ideal gas system from one
  state to another.
• Draw a P-V diagram to get some idea what the
  work is.
• Apply the First Law of Thermodynamics (this is a
  statement of conservation of energy).
• Apply the Ideal Gas Law.
• the internal energy is determined by the
  temperature
• the change in internal energy is determined by
  the change in temperature
• the work done depends on how the system moves
  from one state to another (the change in internal
  energy does not)

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Constant volume (isochoric) process

• No work is done by the gas W 0. The P-V
  diagram is a vertical line, going up if heat is
  added, and going down if heat is removed.
• Applying the first law
• For a monatomic ideal gas

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Constant pressure (isobaric) process

• In this case the region on the P-V diagram is
  rectangular, so its area is easy to find.
• For a monatomic ideal gas

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Heat capacity

• For solids and liquids
• For gases , where C, the heat
  capacity, depends on the process.
• For a monatomic ideal gas
• Constant volume
• Constant pressure
• In general

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Constant temperature (isothermal) process

• No change in internal energy
• The P-V diagram follows the isotherm.
• Applying the first law, and
• using a little calculus

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Zero heat (adiabatic) process

• Q 0. The P-V diagram is an interesting line,
  given by
• For a monatomic ideal gas
• Applying the first law

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