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Physics 111Lecture 37 (Walker: 17.1-2)

Thermal Properties of Ideal GasKinetic Theory

May 4, 2009

Quiz (Chaps. 14 & 16) on Wed. May 6

RadiationIf you are sitting in a place that is too cold, your body radiates and loses to convection more heat than it is producing. You will start shivering and your metabolic rate will increase unless you put on clothing that has good insulation and/or low emissivity (“space blanket”.)

Emissivity also determines how well a surface absorbsradiant energy.

e = 1 perfect absorber (perfect black body)

e = 0 perfect reflector (mirror)

An object at temperature T in surroundings of temperature Ts will both radiate and absorb power. The net radiated power is

Pnet = eσA(T4 - Ts4)

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Example• Person wants to “burn off” 400 Calories (1.7 x 106 J) by

standing naked in ice cave at -10°C. How long will it take if cooling is by radiation only? Take e=0.9; A=1.5m2; T = 37°C = 310K; Ts = -10°C = 263K

Pnet = eσA(T4 - Ts4)

=(0.9)(5.7x10-8W/m2K4)(1.5m2)[(310K)4-(263K)4]

= 340W

Q = Pt, so t=1.7x106J/340W= 4900s = 1.4hr

Absorption of Radiation Energy

If you are in the sunlight, the Sun’s radiation will warm you. The intensity of solar radiation is 1000 W/m2. In general, you will not be perfectly perpendicular to the Sun’s rays, and will absorb energy at the rate:

SeasonsThis cos θ effect is also responsible for the seasons.

Heat Transfer: Radiation

Thermography – the detailed measurement of radiation from the body – can be used in medical imaging. Warmer areas may be a sign of tumors or infection; cooler areas on the skin may be a sign of poor circulation.

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Ideal GasesGases are the easiest state of matter to describe, as all ideal gases exhibit similar behavior.

An ideal gas is one that has low enough density, and is far enough away from condensing to liquid, that the interactions between molecules can be ignored.

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If the volume of an ideal gas is held constant, we find that the pressure increases with temperature:

Ideal Gases

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If the volume and temperature are kept constant, but more molecules N of gas are added (such as in inflating a tire or basketball), the pressure will increase:

Ideal Gases

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Finally, if the temperature is constant and the volume decreases, the pressure increases:

Ideal Gases

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Ideal GasesCombining all three observations, we write

where k is called the Boltzmann constant:

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Ideal GasesRearranging gives us the equation of state for an ideal gas:

Instead of counting molecules, we can count moles. A mole is the amount of a substance that contains Avogadro’s number NA of molecules.

n moles of gas will contain nNA molecules.

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Ideal GasesAvogadro’s number and the Boltzmann constant can be combined to form the universal gas constant and an alternative equation of state:

where n is the number of moles of gas present.

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Ideal Gases

The atomic or molecular mass of a substance is the mass, in grams, of one mole of that substance. For example,

Helium:

Copper:

Furthermore, the mass of an individual atom is given by the atomic mass divided by Avogadro’s number:

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Problem Solving with the Ideal Gas LawUseful facts and definitions:

• Standard temperature and pressure (STP)

• Volume of 1 mol of an ideal gas at STP is 22.4 L

• If the amount of gas does not change:

• Always measure T in kelvins

• P must be the absolute pressure

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Example

• We have 1 mole of ideal gas at a temperature of 40°C at atmospheric pressure (101 kPa). Volume?

• T = 313K• PV = nRT so V = nRT/P

V = (1mol)(8.31 J/mol-K)(313K)/(1.01x105Pa)= 2.58 x 10-2 m3 = 25.8 liters

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Kinetic TheoryKinetic theory of gases relates microscopic quantities (position, velocity) to macroscopic ones (pressure, temperature). Assumptions:

• N identical molecules of mass m are inside a container of volume V; each acts as a point particle.

• Molecules move randomly and always obey Newton’s laws.

• Collisions with other molecules and with the walls are elastic.

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Kinetic TheoryPressure is the result of collisions between the gas molecules and the walls of the container.

It depends on the mass and speed of the molecules, and on the container size:

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Kinetic TheoryNot all molecules in a gas will have the same speed; their speeds are represented by the Maxwell distribution, and depend on the temperature and mass of the molecules.

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Kinetic TheoryWe replace the speed in the previous expression for pressure with the average speed:

Including the other two directions,

Therefore, the pressure in a gas is proportional to the average kinetic energy of its molecules.

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Kinetic TheoryComparing this expression with the ideal gas law allows us to relate average internal kinetic energy and temperature:

The square root of is called the root mean square (rms) speed.

The average translational kinetic energy of the molecules in an ideal gas is directly proportional to the temperature of the gas.

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Kinetic Theory

Solving for the rms speed gives:

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ExampleWhat is the RMS speed of atoms of helium gas

at T = 300K?mass of He atom = (4g/mol)/(6.02x1023/mol)

= 6.6x10-27kg

kgxKKJxvRMS 27

23

106.6)300)(/1038.1(3

−

−= = 1370 m/s

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Total Internal Kinetic Energy - Ideal GasThe internal kinetic energy of an ideal gas is the sum of the kinetic energies of all its molecules. In the case where each molecule consists of a single atom, this may be written:

Since there is no internal potential energy in an ideal gas, the total internal energy U is just the internal kinetic energy.

= (3/2)nRT

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ExampleWhat is the total internal kinetic energy of a

mole of monatomic gas at T=300K?

U = (3/2)nRT = (3/2)(8.31J/mol-K)(300K)= 3740 J

This much energy would have to be removed from the gas to cool it to absolute zero.

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Internal Energy U:The internal energy U of an object is the sum total of all the microscopic and random kinetic and potential energies of all theatoms in the object.

The amount of internal energy depends on …

has more internal energy than

… the temperature

..the number of molecules, and the internal potential energy associated with interactions.

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End of Lecture 37• For Wednesday, May 6, read Walker 17.2, 17.4.

• Homework Assignment 17a is due at 11:00 PM on Wednesday, May 6.

• Quiz (Chaps. 14 & 16) on Wed. May 6