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### Chapter 10

Temperature and Kinetic Theory

Definitions

- Temperature – a measure of the average kinetic energy of the the molecules making up a substance, measured in [C] or [F] or [K].
- Internal Energy (thermal energy) the combined kinetic and potential energy of the molecules of a substance measured in [Joules].
- Heat – the transfer of thermal energy measured in [Joules].

Kinetic energy – when molecules are in motion

Potential energy – when molecules have stored energy

Internal EnergyUse two points to find the equation…

Fahrenheit and Celsius ScalesFahrenheit, Celsius and Kelvin Scales

- F = 9/5 C + 32
- C = 5/9(F – 32)
- K = C + 273

Ideal Gas Law

- PV = nRT
- P = Pressure
- V = Volume
- n = number of moles
- R = univ. gas constant (8.31 J/mol ºK)
- T = Temperature in Kelvin

Things to Know

- A mole of a substance is a quantity containing 6.02 X 1023 molecules
- Standard Temperature and Pressure means p = 1 atm = 1.01 X 105 Pa,
T = 0ºC = 273 K

- Formula weight in Grams = 1 mole
- Must use Kelvin for temperature!

Special Cases of Ideal Gas Law

If the number of molecules (mass) doesn’t change:

- Constant Temperature, then p1V1 = p2V2
- Constant Pressure , then V1/T1 = V2/T2
- Constant Volume, then p1/T1 = p2/t2

Examples

- A low density gas in a rigid container is initially at 20ºC and a particular pressure, p1. If the gas is heated to 60ºC, by what factor does pressure change?

Examples

- The weather report gives the day’s high temperature as 10ºC and predicts the next day’s high temperature as 20ºC. A father tells his son that this means it will be twice as warm tomorrow, but the son says it does not mean that. Do you agree with the father or the son?

Thermal Expansion

- Solids and liquids expand or contract with changes in temperature.
- Space between molecules becomes greater or less as temperature changes.
- ΔL = αL0ΔT ; L0= original length
α =thermal coefficient of

linear expansion

Example

- A steel beam is 5.0 m long at 20ºC. On a hot day, the temperature rises to 40ºC. What is the change in the beam’s length? α = 12 X 10-6 C-1

Area and Volume Expansion

- ΔA = 2αA0ΔT Area Expands with Temperature
- ΔV = 3αV0ΔT Volume Expands with Temperature

Macroscopic vs Microscopic Ideal Gas Law

Macroscopic

pV = nRT R = 8.31 J/mol K

n = # of moles

- Microscopic
pV = NkbT kb = 1.38 X 10-23 J/K

N = # of molecules

Kinetic Theory for Monatomic Gases

- Monatomic – single atom gases.
- Diatomic – molecules contain 2 atoms.
- Monatomic gases are easy to study because atoms move without rotation or vibration
- Monatomic gases obey the laws of mechanics (recall for elastic collisions we apply Conservation of Momentum and Conservation of Energy)

Molecules (atoms) undergo perfectly elastic collisions with the walls of the container.

We assume molecules are separated by large enough distances so that molecular collisions can be neglected.

Then…

pV = 1/3 Nmv2rms

N = # of molecules

m = mass of molecule

vrms = average speed of molecule

Kinetic Theory for Monatomic Gases

Math… the walls of the container.

pV = 1/3 Nmv2rms = NkbT

So ½ mv2rms = 3/2 kbT

Or 3/2 kbT = ½ mv2rms

What does this mean?

Temperature is proportional to average KE!

Example the walls of the container.

- Find the average speed (v2rms) of a Helium atom in a 20ºC balloon at room temperature. Assume
mHe = 6.65 X 10-27 kg

Summary- Kinetic Theory for Monatomic Ideal Gas the walls of the container.

Average Kinetic Energy

- KEav = ½ m vrms2 = 3/2 kbT
kb = 1.38 X 10 -23 J/K

Total Internal Energy

- U = 3/2 NkbT = 3/2 nRT

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