A P P H Y S I C S

Thermal Physics
Equations not on the equation sheet
Q
 mc

T c  specific heat, units: J/(kg·K)
Q
  mL
L  Latent Heat, units: J/kg

Every substance has a unique relationship between its mass and number of moles
Molar Mass (M) the ratio of the mass of a substance in grams to the number of moles of the substance
How do you determine Molar Mass?
the mass of 1 mole of a substance equals the atomic mass of the substance in units of grams rather than atomic mass units
Ex. What is the molar mass of O
2
?

What is the molar mass (M) of CH
4
?
What number of moles (n) are there in 40 g of methane gas?
How many molecules (N) of CH
4 does this include?
What is the mass of 24.08 x 10 23 molecules of ethanol (C
2
H
5
OH)?
Note: n  number of moles
N  number of particles

The behaviors of ideal gases at low pressures are relatively easy to describe:
The volume V is proportional to the number of moles n and thus to the number of molecules (this concept stems from
Avogadro’s Law)

Robert Boyle (1627 – 1691)
 Irish physicist and chemist who employed Robert Hooke as an assistant
(you know the Hooke’s law guy and the “cell” guy)
Boyle’s Law
The volume V varies inversely with the pressure P when temperature (T) and amount of gas (n) are constant.

Jacques Charles (1746 – 1823)
 French inventor, physicist and hot air balloonist.
Charles’ Law
The pressure P is directly proportional to the absolute temperature T (temperature in Kelvin) when volume V and amount n are constant

Combining of Boyle’s Law and Charles’ Law
Adding Avogadro’s Law yields:
R is the ideal gas constant or R = 0.08206 L ∙ atm/(mol ∙ K)

The condition called standard temperature and pressure (STP) for a gas is defined to be a temperature of 0 o C and a pressure of 1 atm (1.013 x 10 5
Pa). If you want to keep 1 mole of an ideal gas in your room at STP, how big is the Tupperware that you need?
[Answer in units of liters, 1 m 3 = 1000 L]

Assumptions of the Kinetic-Molecular Model
1.
2.
3.
4.
A container with volume V contains a very large number N of identical molecules, each with mass m. The container has perfectly rigid walls that do not move.
The molecules behave as point particles; their size is small in comparison to the average distance between particles and to the dimensions of the container.
The molecules are in constant random motion; they obey Newton’s laws. Each molecule occasionally makes a perfectly elastic collision with a wall of the container.
During collisions, the molecules exert forces on the walls of the container; these forces create the pressure that the gas exerts.

For an ideal gas, the average kinetic energy K avg per molecule is proportional to the absolute (Kelvin) temperature T.
The ratio R/N o occurs frequently in molecular theory and is known as the
Boltzmann constant k
B
.
What is the value of the Boltzmann constant including units??
Ludwig Boltzmann (1844 – 1906) was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics.

If molecules have an average kinetic energy K avg
K avg

3
2 k
B
T given by the equation: then what is their average speed???

Five ideal-gas molecules chosen at random are found to have speeds of
500, 600, 700, 800, and 900 m/s, respectively. Find the rms speed for this collection. Is it the same as the average speed of these molecules?

Starting with the typical ideal gas law (PV = nRT ) derive an expression for the gas law that includes both the Boltzmann Constant k number of molecules of an ideal gas N.
B and the

a) b)
What is the average (translational) kinetic energy of a molecule of oxygen (O
2
) at a temperature of 27 o C, assuming that oxygen can be treated as an ideal gas?
What is the total (translational) kinetic energy of the molecules in 1 mole of oxygen at this temperature?
c) Compare the root-mean-square speeds of oxygen and nitrogen molecules at this temperature (assuming that they can be treated as ideal gases).
[mass of oxygen molecule m
O2 m
N2
= 4.65 x 10 -26 kg]
= 5.31 x 10 -26 kg, mass of nitrogen molecule

DWT
Particles in a system (like an ideal gas) are in motion. Therefore:
1.
the particles have some velocity
2.
3.
the particles have some kinetic energy the system as a whole has some internal energy as a result of the individual particles’ kinetic energy
This is true of material in any phase (solid, liguid, gas, plasma).

Kinetic Energy of the individual particles
The system as a whole has some internal energy as a result of the individual particles’ kinetic energy
Kinetic energy of all the particles

DWT
Thermodynamics is the study of energy relationships that involve heat, mechanical work, and other aspects of energy and energy transfer.
There are two ways to transfer energy to an object:
1.
Heat the object
2.
do Work on the object
Both of these energy transfer methods add to the internal energy of the object.
The First Law of Thermodynamics (1LT)

Classic Thermodynamic System
Gas in a cylinder confined by a piston.

Pressure
P
V o
Work
V f
Volume
Area under the curve is the work done by the gas.
Notice the arrow denoting direction of the process.

1.
2.
3.
4.

Pressure
P a
P b
V a
For Ideal Gases:
V b
Volume
The curve represents pressure as a function of volume for an ideal gas at a single temperature. The curve is called an isotherm.
For the curve, PV is constant and is directly proportional to T (Boyle’s
Law).

Pressure
T b
T a
P
Work
V a
V b
Volume
The curve is called an isobar.
The pressure of the system
(system - constant amount of gas, n) changes as a result of heat being transferred either into or out of the system and/or work done on or by the system.
isotherms
T a
> T b

Pressure
P b
P a
The curve is called an isochor.
There is no work done in this process. All of the energy added/subtracted as heat changes the internal energy.
isotherms
V
Volume

Pressure
P b
The curve is called an adiabat.
No heat is transferred into or out of the system. (An adiabatic curve at any point is always steeper than the isotherm passing through the same point.)
P a
V a
Work
V b isotherms
Volume

Thermodynamics of Boiling Water
One gram of water (1 cm 3 ) becomes 1671 cm 3 of steam when boiled at a constant pressure of 1 atm (1.013 x 10 5 Pa). The latent heat of vaporization at this pressure is L
Compute: v
= 2.256 x 10 6 J/kg. a) b) the work done by the water when it vaporizes.
its increase in internal energy.

(Isovolumetric)
Heating Water
Water with a mass of 2.0 kg is held at constant volume in a container while 10,000 J of heat is slowly added by a flame. The container is not well insulated, and as a result 2,000 J of heat leaks out to the surroundings.
a) b)
What is the increase in internal energy?
What is the increase in temperature?
[the specific heat of water is 4186 J/kg ∙ o C]

The amount of heat Q needed for a certain temperature change Δ T is proportional to the temperature change and to the number of moles n of the substance being heated;
Q = nCΔT

 where C is a quantity, different for different materials, called the molar heat
capacity of the material.
Units of C: J/(mol ∙K)
Relation of specific heat (c) to molar heat capacity (C) is the molar mass
C = Mc
Because of 1LT the molar heat capacities are not the same for different thermodynamic processes…

The molar heat capacities are different for isochoric (constant volume) and isobaric (constant pressure) processes.
1LT  ΔU = Q+W  Q = ΔU – W
Isochoric
W = -P Δ V = 0
Isobaric
For pressure to remain constant the volume must change
Q = Δ U – o
W = -P Δ V
Q = Δ U
Q = Δ U + P Δ V
All of the heat gained/lost results directly in a change in internal energy.
Molar heat for a constant volume, C v
Some of the heat gained by the system is converted into work as the system expands.
Molar heat for a constant pressure, C p
Relationship between C v and C p
:
C p
= C v
+ R
R  universal gas constant