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Physics 151: Lecture 38 Todayâ€™s Agenda

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- Todayâ€™s Topics (Chapter 20)
- Internal Energy and Heat
- Heat Capacity
- First Law of Thermodynamics
- Special Processes

- An aluminum plate has a circular hole cut in it. A copper ball (solid sphere) has exactly the same diameter as the hole when both are at room temperature, and hence can just barely be pushed through it. If both the plate and the ball are now heated up to a few hundred degrees Celsius, how will the ball and the hole fit ?

(a) ball wonâ€™t fit(b) fits more easily(c) same as before

(b) fits more easily

Before

After (higher T)

PV = N kBT

- Equation of state for an ideal gas

R is called the universal gas constant

PV = nRT

In SI units, R =8.315 J / molÂ·K

kBiscalled the Boltzmannâ€™s constant

kB = R/NA = 1.38 X 10-23 J/K

B = rTo V g

rTVg

mg

To

- The mass of a hot-air balloon and its cargo (not including the air inside) is 200 kg. The air outside is at 10.0Â°C and 101 kPa. The volume of the balloon is 400 m3. To what temperature must the air in the balloon be heated before the balloon will lift off ?
(Air density at 10.0Â°C is 1.25 kg/m3.)

V, T

m

T = 472 K !

- Internal energy is all the energy of a system that is associated with its microscopic components
- These components are its atoms and molecules
- The system is viewed from a reference frame at rest with respect to the center of mass of the system

- Internal energy does include kinetic energies due to:
- Random translational motion (not motion through space)
- Rotational motion
- Vibrational motion
- Potential energy between molecules

Animation

- Heat is defined as the transfer of energy across the boundary of a system due to a temperature difference between the system and its surroundings
- The term heat will also be used to represent the amount of energy transferred by this method
- Units of Heat : historically-> the calorie
- One calorie is the amount of energy transfer necessary to raise the temperature of 1 g of water from 14.5oC to 15.5oC

- In the US Customary system, the unit is a BTU (British Thermal Unit)
- One BTU is the amount of energy transfer necessary to raise the temperature of 1 lb of water from 63oF to 64oF

- The SI uinits are Joules, as we used before !

- Both heat and work can change the internal energy of a system
- The internal energy can be changed even when no energy is transferred by heat, but just by work
- Example, compressing gas with a piston
- Energy is transferred by work

- James Joule in 1843 established the equivalence between mechanical energy and internal energy
- His experimental setup is shown at right
- The loss in potential energy associated with the blocks equals the work done by the paddle wheel on the water

- The amount of mechanical energy needed to raise the temperature of water from 14.5oC to 15.5oC is 4.186 J

1 cal = 4.186 J

- The heat capacity (C) of a particular sample is defined as the amount of energy needed to raise the temperature of that sample by 1oC
- If energy Q produces a change of temperature of DT, then
Q = CDT

- Specific heat (c) is the heat capacity per unit mass

- The Nova laser at Lawrence Livermore National Laboratory in California is used in studies of initiating controlled nuclear fusion. It can deliver a power of 1.60 x 1013 W over a time interval of 2.50 ns. Compare its energy output in one such time interval to the energy required to make a pot of tea by warming 0.800 kg of water from 20.0oC to 100oC.
- Which one is larger ?

- One technique for measuring specific heat involves heating a material, adding it to a sample of water, and recording the final temperature
- This technique is known as calorimetry
- A calorimeter is a device in which this energy transfer takes place

- The system of the sample and the water is isolated
- Conservation of energy requires that the amount of energy that leaves the sample equals the amount of energy that enters the water
- Cons. of Energy :Qcold= -Qhot

- A phase change is when a substance changes from one form to another. Two common phase changes are
- Solid to liquid (melting)
- Liquid to gas (boiling)

- An ice cube (m=0.070 kg) is taken from a freezer ( -10o C) and dropped into a glass of water at 0o C. How much of water will freeze ? (C(ice) = 2,000 J/kg K; L(water) =334 kJ/kg)

m =4.19 g

- State variables describe the state of a system
- In the macroscopic approach to thermodynamics, variables are used to describe the state of the system
- Pressure, temperature, volume, internal energy
- These are examples of state variables

- The macroscopic state of an isolated system can be specified only if the system is in thermal equilibrium internally

- Transfer variables are zero unless a process occurs in which energy is transferred across the boundary of a system
- Transfer variables are not associated with any given state of the system, only with changes in the state
- Heat and work are transfer variables
- Example of heat: we can only assign a value of the heat if energy crosses the boundary by heat

- Work can be done on a deformable system, such as a gas
- Consider a cylinder with a moveable piston
- A force is applied to slowly compress the gas
- The compression is slow enough for all the system to remain essentially in thermal equilibrium
- This is said to occur quasi-statically

Therefore, the work done on the gas is dW = -P dV

- The state of the gas at each step can be plotted on a graph called a PV diagram
- This allows us to visualize the process through which the gas is progressing

- The work done on a gas in a quasi-static process that takes the gas from an initial state to a final state is the the area under the curve on the PV diagram, evaluated between the initial and final states

- This is true whether or not the pressure stays constant
- The work done does depend on the path taken

- Each of these processes has the same initial and final states
- The work done differs in each process
- The work done depends on the path

W= â€¦

W = -Pf (Vf â€“ Vi)

W = -Pi (Vf â€“ Vi)

- The First Law of Thermodynamics is a special case of the Law of Conservation of Energy
- It takes into account changes in internal energy and energy transfers by heat and work

- Although Q and W each are dependent on the path, Q + W is independent of the path
- The First Law of Thermodynamics states that DEint= Q + W
- All quantities must have the same units of measure of energy

- One consequence =>> there must exist some quantity known as internal energy which is determined by the state of the system

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Which statement below regarding the First Law of Thermodynamics is most correct ?

a.A system can do work externally only if its internal energy decreases.

b. The internal energy of a system that interacts with its environment must change.

c. No matter what other interactions take place, the internal energy must change if a system undergoes a heat transfer.

d. The only changes that can occur in the internal energy of a system are those produced by non-mechanical forces.

e. The internal energy of a system cannot change if the heat transferred to the system is equal to the work done by the system.

- An adiabatic process is one during which no energy enters or leaves the system by heat
- Q = 0
- This is achieved by:
- Thermally insulating the walls of the system
- Having the process proceed so quickly that no heat can be exchanged

- Since Q = 0, DEint = W
- If the gas is compressed adiabatically, W is positive so DEint is positive and the temperature of the gas increases
- If the gas expands adiabatically, the temperature of the gas decreases

- An isothermal process is one that occurs at a constant temperature
- Since there is no change in temperature, DEint = 0
- Therefore, Q = - W
- Any energy that enters the system by heat must leave the system by work

Isothermal Expansion

for an ideal gas :

PV = nRT and

- An isobaric process is one that occurs at a constant pressure
- The values of the heat and the work are generally both nonzero
- The work done is W = P (Vf â€“ Vi) where P is the constant pressure

- Identify processes A-D in the pV diagram below:

In an adiabatic free expansion :

a.no heat is transferred between a system and its surroundings.

b. the pressure remains constant.

c. the temperature remains constant.

d. the volume remains constant.

e. the process is reversible.

- A cyclic process is one that starts and ends in the same state
- On a PV diagram, a cyclic process appears as a closed curve

- The change in the internal energy must be zero since it is a state variable
- If DEint = 0, Q = -W
- In a cyclic process, the net work done on the system per cycle equals the area enclosed by the path representing the process on a PV diagram

- An ideal gas is carried through a thermodynamic cycle consisting of two isobaric and two isothermal processes as shown in Figure .
- What is the work done in this cycle, in terms of p1, p2, V1, V2 ?

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- Chap. 20:
- Internal Energy and Heat
- Heat Capacity
- First Law of Thermodynamics
- Special Processes