Temperature and Ideal Gas

Temperature and Ideal Gas • Everything is made of atoms • In gases the molecules don’t interact with each other. Simple • How does the atomic (molecular) nature of a gas explain its properties? • Air in your tires? • How hot is hot? • How cold is cold?

Temperature The quantity indicating how warm or cold an object is relative to some standard is TEMPERATURE (T). T does not depend on quantity of a substance. A cup and a thimble of boiling water both have same T. When two objects have the same temperature, they are in thermal equilibrium. Heat is the flow of energy due to a temperature difference. Heat always flows from objects at high temperature to objects at low temperature.

The Zeroth Law of Thermodynamics: If two objects are each in thermal equilibrium with a third object, then the two objects are in thermal equilibrium with each other. There is no heat flow between objects in thermal equilibrium

Temperature Scales (*) Values given at 1 atmosphere of pressure.

The temperature scales are related by: Fahrenheit/Celsius Absolute/Celsius

Example (text problem 13.3): (a) At what temperature (if any) does the numerical value of Celsius degrees equal the numerical value of Fahrenheit degrees? (b) At what temperature (if any) does the numerical value of kelvins equal the numerical value of Fahrenheit degrees?

Question • Which is smaller, a change of 1oF or 1oC? • A) 1oF • B) 1oC • C) they are the same

Molecular Picture of a Gas • Atoms and molecules are the basic units of matter We want to explain thermal Properties in terms of atoms and molecules. How many atoms?

Molecular Picture of a Gas We can specify the amount of a substance by giving its mass or by the number of molecules (or atoms) it has. If we know the mass of a molecule we can go from one description to the other A golf ball weighs 1.6 ounces. I have a 20 lb box of golf balls I have a box of 200 golf balls Totally equivalent

Molecular Picture of a Gas If a sample contains a single substance (element or compound) the number of particles in the sample is N = M/m. N equals the total mass of the sample (M) divided by the mass (m) of the atom (or molecule) The number density of particles is N/V where N is the total number of particles contained in a volume V.

It is convenient to have a standard number to facilitate this going back and forth from the two descriptions. Since we deal with human size numbers (gms) this will involve a very large number of atoms One mole of a substance contains the same number of particles as there are atoms in 12 grams of 12C. The number of atoms in 12 grams of 12C is Avogadro’s number.

Why 12 gm of Carbon? A carbon-12 atom by definition has a mass of exactly 12 atomic mass units (12 u or 12 amu). 12g = 12u NA u = 1g/(6x1023) =1.66x10-24g =1.66x10-27kg This is the conversion factor between the atomic mass unit and kg (1 u = 1.661027 kg). NA and the mole are defined so that a 1 gram sample of a substance with an atomic mass of 1 u contains exactly NA particles. A mole of O2 has mass 32 gm., of water m=18 gm

Example (text problem 13.37): Air at room temperature and atmospheric pressure has a mass density of 1.2 kg/m3. The average molecular mass of air is 29.0 u. How many air molecules are there in 1.0 cm3 of air? The total mass of air in the given volume is:

Example continued:

Question • Which contains more atoms, 5 mol. of helium • (mass He =4amu) or 1 mol of neon (m Ne =20amu) • A) Helium • B) Neon • C) both have same number of atoms

Question • Which contains more atoms, 1 mol of helium • or 1 mol of Steam (water) • A) Helium • B) water • C) both have same number of atoms

Decrease the volume Increase the pressure Constant T: P ~1/V

Increase the number of moleculesIncrease the pressure Constant V,T: P ~ N

We also know that, as you drive, tire pressure increases with T Constant V,N: P ~ T

Absolute Temperature and the Ideal Gas Law Constant P: V ~T

Absolute Temperature and the Ideal Gas Law Experiments done on dilute gases (a gas where interactions between molecules can be ignored) show that: For constant pressure Charles’ Law For constant volume Gay-Lussac’s Law

For constant temperature Boyle’s Law Avogadro’s Law For constant pressure and temperature

What Temperature do we use? There is a lowest possible T V→0

Absolute Temperature • There is a coldest possible temperature Absolute zero. • All objects will transfer heat to an object at absolute 0. • Experiment show (e.g. V→0 ) that the coldest possible T is -273.15oC. • Kelvin scale measures T from Absolute 0 in units of 1oC TK=Tc+273o

Putting all of these statements together gives the ideal gas law (microscopic form): k = 1.381023 J/K is Boltzmann’s constant The ideal gas law can also be written as (macroscopic form): R = NAk = 8.31 J/K/mole is the universal gas constant and n is the number of moles.

Example (text problem 13.41): A cylinder in a car engine takes Vi = 4.50102 m3 of air into the chamber at 30 C and at atmospheric pressure. The piston then compresses the air to one-ninth of the original volume and to 20.0 times the original pressure. What is the new temperature of the air? Here, Vf = Vi/9, Pf = 20.0Pi, and Ti = 30 C = 303 K. The ideal gas law holds for each set of parameters (before compression and after compression).

Example continued: Take the ratio: The final temperature is The final temperature is 673 K = 400 C.

Putting all of these statements together gives the ideal gas law (microscopic form): k = 1.381023 J/K is Boltzmann’s constant The ideal gas law can also be written as (macroscopic form): R = NAk = 8.31 J/K/mole is the universal gas constant and n is the number of moles.

Question When the temperature of a quantity of gas is increased A) the pressure must increase. B) the volume must increase. C) the pressure and/or the volume must increase. D) none of the above

Question A pot of water on the stove is heated from 25oC to 100oC. By what factor does the • temperature in Kelvin change? • A) T2= 4T1 • B) T2 = 1.25T1 • C) T2 = 0.80T1 • D) T2= 0.20T1

Question • Inside your air-conditioned apartment, you blow up a balloon as large as possible and then take it outside on a hot summer day. The balloon is most likely to then A) shrink. B) remain the same size. C) expand and pop.

Question The Kelvin temperature of an ideal gas is doubled and the volume is halved. How is the pressure affected? A) increases by a factor of 2 B) increases by a factor of 4 C) stays the same D) decreases by a factor of 2 E) decreases by a factor of 4

Kinetic Theory of the Ideal Gas An ideal gasis a dilute gas where the particles act as point particles with no interactions except for elastic collisions. Point particles can have only KE, no internal PE Add heat (energy) to gas, energy increases KE increases. But if we add heat, temperature also increases. T depends on KE

Kinetic Theory of Ideal Gas T~ KEav/molecule if T =absolute 0, molecules don’t move. T T T More total energy, but same T, same average KE

Temperature is related to average KE • This is true even for liquids and solids

Pressure is caused by collisions Gas particles have random motions. Each time a particle collides with the walls of its container there is a force exerted on the wall. The force per unit area on the wall is equal to the pressure in the gas.

The pressure will depend on: • The number of gas particles ~N • Frequency of collisions with the walls ~ v • Amount of momentum transferred during each collision ~ mv • P~ Nmv2 ~N KEmol

The pressure in the gas is Where <Ktr> is the average translational kinetic energy of the gas particles; it depends on the temperature of the gas.

Typical air molecule is moving more than • 1,000 Miles/hr. Some move faster some slower

The average kinetic energy also depends on the rms speed of the gas where the rms speed is

What is rms?root mean square • v2rms is the average of the square of the velocities. It is not the square of the average Average of squares rms

rms Example v1 = 4 v2 = 8 vav = 6 v12 = 16 v22 = 64

The distribution of speeds in a gas is given by the Maxwell-Boltzmann Distribution.

Example (text problem 13.60): What is the temperature of an ideal gas whose molecules have an average translational kinetic energy of 3.201020 J?

Example (text problem 13.70): What are the rms speeds of helium atoms, and nitrogen, hydrogen, and oxygen molecules at 25 C? On the Kelvin scale T = 25 C = 298 K.

Question At a given temperature, a hydrogen molecule has a speed of 800 m/s. At the same temperature, an oxygen molecule has a speed of A) 800 m/s. B) 400 m/s. C) 200 m/s. D) 100 m/s.

Question When will a real gas behave most like an ideal gas? A) at high temperatures and high pressures B) at low temperatures and high pressures C) at low temperatures and low pressures D) at high temperatures and low pressures

Temperature and Ideal Gas