#import "@preview/gentle-clues:0.3.0": note, important

= Chapter 12: Thermal Properties of Matter

== The Atomic Model of Matter

/ Gas: System in which each particle moves freely through space until it occasionally collides with another particle or the wall
/ Liquid: Weak bonds permit motion while keeping the particles close together
/ Solid: Particles are connected by stiff spring-like bonds. Solids have a definite shape and can be compressed only slightly

==== Atomic Mass and Atomic Mass Number

/ Atomic mass number: $A$ is the sum of the number of protons and neutrons in an atom
/ Atomic mass: The atomic mass scale is established by defining the mass of Carbon-12 to be exactly 12 u
/ Atomic mass unit: $1 u = 1.66 times 10^(-27)$
/ Molecular mass: Sum of the atomic masses of the atoms that form the molecule

==== Moles

/ Mole: 1 mole (abbr. mol) is $6.02 times 10^23$ basic particles

This is *Avogadro's number* ($N_A$). $N_A$ has units $"mol"^(-1)$

The number $n$ of moles in a substance containing $N$ basic particles is $N/N_A$

The basic particle depends on the substance:
- *Monatomic gas* means that the basic particles are atoms, e.g., helium
- *Diatomic gas* means that the basic particle is a two-atom diatomic molecule, e.g., $upright(O)_2$

/ Molar mass: Molar mass ($M_"mol"$) is mass in grams of 1 mol of substance:

$ n = (M "(in grams)")/M_"mol" $

=== Pressure

- Particles in a gas move around in a container, they sometimes bounce off the walls, creating a force on the walls
- The collisions with the wall create a force perpendicular to the wall

/ Pressure: The pressure of a gas is the ratio of the force to the area

$ p = F/A $

SI unit for pressure is pascal (Pa), equal to $N/m^2$

/ Standard atmosphere: Pressure from atmosphere at sea level: 1 atm = 101,300 Pa = 14.7 psi

The net pressure force is exterted only *where there is a pressure difference* between the two sides of a surface

$ F_"net" = F_2 - F_1 = p_2A - p_1A = A(p_2 - p_1) = A Delta p $

$Delta p$ is gauge pressure

/ Vacuum: Enclosed space with $p << 1 "atm"$
/ Perfect vacuum: Vacuum with $p = 0 "Pa"$, but impossible to remove every molecule
/ Absolute pressure: $p = F/A$
/ Gauge pressure: $p_g$ is the *difference* between the absolute pressure and the atmospheric pressure

== Ideal Gas Law

Version with $N$ as number of molecules:

$ p V = N k_upright(B) T$

Version with $n$ as number of moles:

$ p V = n R T $

- $p$ is absolute pressure
- $V$ is volume of sample ($upright(m)^3$)
- $n$ is number of moles in the sample/container of gas
- $R$ is *gas constant* ($R = N_upright(A) k_upright(B) = 8.31 upright(J/"mol" dot K)$)
- $T$ is temperature in Kelvin

== Ideal Gas Processes

*Ideal gas processes* have the following properties:
- Quantity of gas is fixed
- Well-defined initial state. Initial values of pressure, volume, and
  temperature written as $p_i$, $V_i$, $T_i$
- Well-defined final state where pressure, volume, and temperature are $p_f$, $V_f$, and $T_f$

Initial and final states related by:
$ (p_f V_f)/T_f = (p_i V_i)/T_i $

$p$ and $V$ can be in any units since they appear on both sides,
but $T$ must be in Kelvin since it needs to start at 0.

=== $p V$ Diagrams

In a $p V$ diagram, each point on the graph represents a single, unique state of the gas

Assuming that $n$ is known for a sealed container, we can find $T$ from the ideal-gas law
since we know $p$ and $V$

#align(center)[#image("../Images/12-pV-Diagram.png", height: 50%)]

=== Constant-Volume Processes

/ Isochoric: An isochoric process is a *constant-volume process*

Warming a gas will raise its pressure without changing its volume. This is an example of a constant-volume process

A constant-volume process appears on a $p V$ diagram as a vertical line.

#align(center)[#image("../Images/12-Constant-Volume.png", width: 50%)]

=== Constant-Pressure Processes

/ Isobaric: An isobaric process is a *constant-pressure process*

A constant-pressure process appears on a $p V$ diagram as a horizontal line.

For example, you can heat up a container so that the temperature increases,
making the pressure increase with it, but the volume wouldn't increase.

#align(center)[#image("../Images/12-Constant-Pressure.png")]

=== Constant-Temperature Processes

/ Isothermal: An isothermal process is a *constant-temperature process*

#align(center)[#image("../Images/12-Constant-Temperature-Piston.png", width: 50%)]

/ Isotherm: A graph of an isothermal process

The location of the isotherm depends on temperature.
The direction along the isotherm depends on the process.

#align(center)[#image("../Images/12-Constant-Temperature-pV.png")]

=== Thermodynamics of Ideal-Gas Processes

- Heat and work are just two different ways to add energy to a system
- When gases expand, they do work on the piston
- If the gas expands under constant pressure, pushing the piston (with area $A$) from $x_i$ to $x_f$, then the work done is:

#important(title: "Work done by a gas in a constant-pressure process")[
  $ W_"gas" = F_"gas" d = (p A)(x_f - x_i) = p(x_f A - x_i A) = p(V_f - V_i) = p Delta V $
]

To calculate work, *pressure must be in Pa and volume in $upright(m)^3$*

$x_i A$ is the initial volume and $x_f A$ is the final volume

For all ideal-gas processes, the work is the area under the $p V$ graph between $V_i$ and $V_f$:

#align(center)[#image("../Images/12-Gas-Work.png", width: 50%)]

#note[
  No work is done in a constant-volume process.
]

=== Adiabatic Processes

/ Adiabatic processes: Processes where heat is not transferred, i.e., $Q = 0$

An adiabatic expansion lowers the temperature of a gas
- Expansion means the gas does the work
- This means work is negative
- So $Delta E_"thermal"$ is negative
- Thus, temperature decreases

An adiabatic compression raises the temperature of a gas

Adiabatic processes allow you to use work rather than heat to change temperature of the gas

== Specific Heat and Heat of Transformation

/ Specific heat: The specific heat ($c$) of a substance is the amount of heat that raises the temperature of 1 kg of that substance by 1 K

Heat needed to produce a temperature change $T$ for mass $M$ with specific heat $c$:

$ Q = M c Delta T $

Substances with large $c$, like water, are slow to warm up and cool down.
Described as having a large thermal inertia.

=== Phase Changes

/ Melting point: Temperature at which solid becomes liquid
/ Freezing point: Temperature at which liquid becomes solid
/ Condensation point: Temperature at which gas becomes liquid
/ Boiling point: Temperature at which liquid becomes gas
/ Phase changes: Melting and freezing are phase changes
/ Phase equilibrium: A system at the melting point is in phase equilibrium

=== Heat of Transformation

A phase change is characterized by a change in thermal energy without a change in temperature

/ Heat of transformation: $L$ is the amount of heat energy that causes 1 kg of a substance to undergo a phase change

/ Heat of fusion: $L_f$ is the heat of transformation between a solid and a liquid
/ Heat of vaporization: $L_v$ is the heat of transformation between a liquid and a gas

Heat needed to melt/freeze mass $M$: $Q = plus.minus M L_f$ \
Heat needed to boil/condense mass $M$: $Q = plus.minus M L_v$

=== Evaporation

/ Evaporation: Process of some molecules moving from liquid to gas phase at temperatures lower than boiling point

At any temperature, some of the molecules are moving fast enough to go into the gas phase,
carrying away thermal energy and reducing the average kinetic energy (temperature) of the liquid
