Module 1: Temperature, Thermal Expansion, and the Physics of Gases

Introduction to Microscopic and Macroscopic Physics

• The first module investigates the relationship between temperature, thermal expansion, and gas laws.

• Small Scale Physics: This area examines the physics of solids, liquids, and gases at the microscopic level.

• Microscopic Level Definitions:

  • Focuses on the molecular or atomic level of materials.
  • Atoms and molecules are represented as a series of dots that are in constant motion.
  • Even in solids, particles are not stationary; they are constantly vibrating.
  • Interactions at this level involve principles from "Physics 1," such as forces, collisions involving the conservation of linear momentum, and the conservation of mechanical energy.

• Complexity of Microscopic Systems:

  • Microscopic descriptions are extremely complicated due to the sheer number of molecules involved.
  • A typical molar system contains a number of molecules on the order of $10^{25}$ (or 1 followed by 25 zeros).
  • While basic physics can describe the interaction of 3 to 5 atoms, describing the physics of a large number of molecules (capital $N$) is nearly impossible using classical mechanics alone.
  • Quantum Mechanics: The specific field of physics and chemistry that governs how atoms and molecules interact at the microscopic scale.

• Macroscopic Physics:

  • Describes the behavior of large-scale systems comprising many trillions of molecules.
  • Unlike microscopic physics, macroscopic physics is often straightforward to describe using a few thermodynamic variables.

Thermodynamic Variables and Units

• Pressure ($P$):

  • Relates to the collective force that molecules exert on the walls of their container.
  • Ideal gas molecules (like air) are in motion; as they strike the container walls, they impart a force per unit area.
  • Units of Pressure:
  • Metric/SI Unit: Pascal ($Pa$).
  • $1\,Pa = 1\,N/m^2$.
  • Atmospheric Pressure ($1\,atm$) is a common unit for convenience.
  • $1\,atm = 101,325\,Pa$.
  • English units mentioned: Pounds per square inch ($psi$), where atmospheric pressure is roughly $15\,psi$.

• Temperature ($T$):

  • A single number representing one aspect of the physics of a system (e.g., the air in a room).
  • It is an equilibrium property, meaning in a system at thermal equilibrium, the temperature is the same everywhere.

• Volume ($V$):

  • Represents the large-scale size of the system.

• Number of Molecules ($N$):

  • The total count of all molecules in the system.

The Concept of Moles and Molar Systems

• Moles ($n$):

  • Represents an amount of substance. The symbol for moles is lowercase $n$.
  • The number of moles is proportional to the total number of molecules ($N$).
  • Avogadro's Number ($N_A$):
  • Used to scale down the massive number of molecules into a manageable figure for macroscopic physics.
  • $N_A = 6.02 \times 10^{23}\,\text{molecules/mol}$.
  • Formula for Moles (By Count):
  • $n = \frac{N}{N_A}$

• Molar Mass ($M$):

  • The mass of one mole of a substance.
  • Formula for Moles (By Mass):
  • $n = \frac{m}{M}$, where $m$ is the mass in grams.
  • Example Calculations:
  • A single oxygen atom ($O$) has a molar mass of $16\,g/mol$ (based on 8 protons and 8 neutrons).
  • Diatomic Oxygen ($O_2$), which is the oxygen we breathe, consists of two oxygen atoms with a chemical bond. Its molar mass is $M = 32\,g/mol$.
  • Problem: If you have $100\,g$ of diatomic oxygen, how many moles are present?
  • Solution: $n = \frac{100\,g}{32\,g/mol} \approx 3.31\,mol$. This is considered a "molar system" because the result is on the order of 1.

Thermal Energy and Heat Transfer

• Thermal Energy (Internal Energy):

  • The total energy stored within a system at the microscopic level.
  • For a solid (like gold blocks), it consists of:
  1. Kinetic Energy: The motion of atoms vibrating.
  2. Potential Energy: The energy stored in chemical bonds, which act like little springs.

• Mechanism of Heat ($Q$):

  • Heat is the transfer of energy between systems due to a temperature difference.
  • Energy naturally flows from a high-temperature system ("hot") to a low-temperature system ("cold").

• Thermal Equilibrium:

  • Consider two identical gold blocks ($A$ and $B$) in an isolated container where no energy can enter or leave.
  • Initial state: $T_{A, initial} > T_{B, initial}$, meaning Thermal Energy $E_{A, initial} > E_{B, initial}$.
  • When connected, heat flows from $A$ to $B$. Block $A$ cools down as it loses energy; block $B$ heats up as it gains energy.
  • Eventually, they reach thermal equilibrium, where:
  • $T_{A, final} = T_{B, final}$.
  • For identical blocks: $E_{A, final} = E_{B, final}$.

• Key Distinction Between Energy and Temperature:

  • Energy is additive: $E_{total} = E_{A} + E_{B}$.
  • Temperature is not additive: $T_{final} \neq T_{A} + T_{B}$. Temperature is an equilibrium property of the whole system.

Temperature Scales and Calibration

• Liquid Expansion Thermometers:

  • Most liquids, solids, and gases expand when heated.
  • A crude thermometer can be built with a water-filled cylinder and a bulb.
  • By observing the linear relationship between the height of the liquid and the energy (heat) added over time, a degree-based scale can be established.

• Celsius Scale ($^{\circ}C$):

  • Set arbitrarily based on water properties at atmospheric pressure:
  • Water Freezes: $0^{\circ}C$.
  • Water Boils: $100^{\circ}C$.

• The Kelvin Scale ($K$) and Absolute Zero:

  • Experiments with constant volume chambers show that for any gas (e.g., $O_2$, $N_2$, $CO_2$), pressure ($P$) is directly proportional to temperature ($T$).
  • If you plot $P$ vs. $T$ ($^{\circ}C$), the lines for different gases all converge at the same point when pressure reaches zero.
  • Absolute Zero: The coldest temperature possible, occurring at zero pressure.
  • Value: $-273.15^{\circ}C$ (roughly $-273^{\circ}C$ for most problems).
  • Conversion Formula: $T(K) = T(^{\circ}C) + 273$.
  • Note: In Kelvin, we do not use the word "degrees"; we simply say "Kelvin."

• Fahrenheit Scale ($^{\circ}F$):

  • Commonly used in the US.
  • Conversion Formula: $T(^{\circ}F) = \frac{9}{5}T(^{\circ}C) + 32$.

Temperature Changes ($\Delta T$)

• Change in Temperature ($\Delta T$): Defined as $T_{final} - T_{initial}$.
• Conversions for $\Delta T$:
  • Celsius to Kelvin: Sine the "size" of a Kelvin is identical to the size of a degree Celsius, $\Delta T(K) = \Delta T(^{\circ}C)$. Do not add 273 when converting a change.
  • Celsius to Fahrenheit: The additive constant (+32) cancels out during subtraction.
  • Formula: $\Delta T(^{\circ}F) = \frac{9}{5} \Delta T(^{\circ}C)$.

Thermal Expansion in Solids

• Linear Expansion:

  • When a metal beam is heated, its length increases proportionally to its initial length and the change in temperature.
  • Formula: $\Delta L = \alpha L_{0} \Delta T$.
  • Total New Length: $L = L_{0} + \Delta L$.
  • Alpha ($\alpha$): The coefficient of linear expansion. It varys based on the specific material.

• Volume Expansion:

  • Solids and liquids expand in three dimensions ($x, y, z$).
  • Formula: $\Delta V = \beta V_{0} \Delta T$.
  • Beta ($\beta$): The coefficient of volume expansion. For solids, $\beta \approx 3\alpha$.

• Stress and Strain:

  • Strain ($\epsilon$): Describes the deformation of a material. It is dimensionless.
  • $\epsilon = \frac{\Delta L}{L_{0}} = \alpha \Delta T$
  • Stress ($\sigma$): Represents the force per unit area exerted during deformation.
  • Units: $N/m^2$ (same as Pressure).
  • Young's Modulus ($Y$): A constant representing the material's stiffness. It relates stress and strain linearly.
  • Formula: $\sigma = Y \epsilon$
  • Thermal Stress Formula: $\sigma = Y \alpha \Delta T$. This indicates how much internal force is generated by changes in temperature.

Demonstrations and Experimental Observations

• Balloons in Liquid Nitrogen:

  • Liquid nitrogen boils at $77\,K$ (extremely cold compared to room temperature at $300\,K$).
  • Cooling balloons reduces their volume ($V$) and pressure ($P$).
  • Since the balloons are sealed, the number of moles ($n$) remains constant.   

• Constant Volume Chamber:

  • A rigid sphere prevents volume change ($V_{constant}$).
  • Cooling the sphere in liquid nitrogen significantly drops the atmospheric pressure ($15\,psi$) to lower levels (e.g., $10\,psi$).
  • Heating the sphere in hot water quickly increases the pressure back to or above atmospheric levels.

• Bimetallic Strip:

  • A strip made of two different metals bonded together with different coefficients of linear expansion ($\alpha$).
  • When heated with a propane torch, one side expands faster than the other, causing the strip to bend.
  • When cooled in liquid nitrogen, the strip bends in the opposite direction due to different contraction rates.

• Ball and Ring Expansion:

  • Illustrates expansion in more complex geometries.
  • A metal ball may pass through a ring at room temperature but will be too large to pass through after heating due to uniform expansion in all dimensions.