Thermodynamics, Ideal Gas Law, and Kinetic Theory: Study Notes- lecture 29

The Ideal Gas Law and Avogadro's Number

Avogadro's Number and the Mole

The mole (mol) is the SI unit for measuring the amount of substance, defined as the number of atoms in 12 grams of carbon-12. This number is known as Avogadro's Number:

• Avogadro's Number (N_A): $N_A = 6.022 \times 10^{23}$ atoms/mol

• Number of moles ($n$) in a sample: $n = \frac{N}{N_A}$, where $N$ is the number of particles.

• Alternatively, $n = \frac{M_{\text{sample}}}{M}$, where $M_{\text{sample}}$ is the sample mass and $M$ is the molar mass.

Additional info: The image humorously represents Avogadro's Number by writing it on an avocado, reinforcing the concept for students.

The Ideal Gas Law

The Ideal Gas Law describes the relationship between pressure, volume, temperature, and the number of moles in a gas:

• $PV = nRT$

• $R$ is the universal gas constant: $R = 8.31 \frac{J}{mol \cdot K}$

• Alternate form: $PV = NkT$, where $N$ is the number of particles and $k$ is the Boltzmann constant ($k = 1.38 \times 10^{-23} \frac{J}{K}$)

• The law is derived from Boyle's, Charles', Avogadro's, and Gay-Lussac's Laws.

Example: Volume Change with Temperature

When mass and pressure are constant, the ratio of volume to temperature remains constant:

• $\frac{V}{T} = \text{constant}$

• Final volume: $V_f = V_0 \frac{T_f}{T_0}$

• Example: If $V_0 = 10\ m^3$, $T_0 = 100\ K$, $T_f = 50\ K$, then $V_f = 5\ m^3$

Kinetic Theory of Gases

Distribution of Speeds

In a gas, particles move at a range of speeds. The distribution of these speeds depends on temperature:

• Higher temperature increases both the most probable speed and the spread of speeds.

• Statistical methods are used to describe the behavior of large numbers of particles.

Additional info: The image shows the Maxwell speed distribution for 300 K and 1200 K, illustrating how temperature affects molecular speed.

Root-Mean-Square Speed

The root-mean-square speed ($v_{rms}$) is a statistical measure of the speed of particles in a gas:

• $v_{rms} = \sqrt{\langle v^2 \rangle}$

• $\langle v^2 \rangle$ is the average of the squared speeds.

• Note: $\langle v^2 \rangle \neq \langle v \rangle^2$

Monatomic Ideal Gas

For a monatomic ideal gas (single atoms, not molecules):

• Three degrees of freedom (motion in x, y, z directions)

• Average kinetic energy: $KE = \frac{1}{2} m v_{rms}^2 = \frac{3}{2} kT$

• Internal energy: $U = \frac{3}{2} NkT = \frac{3}{2} nRT$

Thermodynamics

Basic Concepts

Thermodynamics is the study of heat, work, and energy transfer. A system is the collection of objects under study, and the environment is everything else. Energy transfer from the environment to the system is considered positive in physics.

Walls: Diathermal and Adiabatic

Systems are separated from their environment by walls:

• Diathermal walls: Allow heat to flow between systems.

• Adiabatic walls: Do not allow heat flow; systems are insulated.

Additional info: The image illustrates the difference between adiabatic and diathermal walls, showing how heat transfer is blocked or allowed.

The Zeroth Law of Thermodynamics

The zeroth law establishes the concept of thermal equilibrium:

• If two objects are at the same temperature, no net heat flows between them.

• If A is in equilibrium with C, and C with B, then A is in equilibrium with B.

Thermal Processes

The First Law of Thermodynamics

The first law relates internal energy change to heat and work:

• $\Delta U = Q - W$

• Work is positive when done by the system, negative when done on the system.

Types of Thermal Processes

Four main quasi-static processes are considered:

• Isobaric: Constant pressure ($W = P \Delta V$; $\Delta U = Q - P(V_f - V_0)$)

• Isochoric: Constant volume ($W = 0$; $\Delta U = Q$)

• Isothermal: Constant temperature ($W = nRT \ln \frac{V_f}{V_0}$; $\Delta U = 0$; $Q = nRT \ln \frac{V_f}{V_0}$)

• Adiabatic: No heat transfer ($Q = 0$; $W = \frac{3}{2} nR(T_0 - T_f)$; $\Delta U = -W = \frac{3}{2} nR(T_f - T_0)$)

Complex Processes and PV Graphs

For processes not fitting the four main types, the work done can be found from the area under a pressure-volume (PV) graph:

• Work done = Area under the PV curve

Additional info: The image shows a PV graph, illustrating how work is calculated as the area under the curve.

Thermodynamics Example: Energy Replacement

Example calculation for energy lost during exercise:

• Power output: $P = 170\ W$

• Time: $t = 1\ hour = 3600\ s$

• Work done: $W = Pt = 170 \times 3600 = 6.1 \times 10^5\ J$

• Heat lost: $Q = -4.2 \times 10^5\ J$

• Internal energy lost: $\Delta U = Q - W = -10.3 \times 10^5\ J$

• Energy per banana: $100\ Calories = 4.186 \times 10^5\ J$

• Bananas needed: $N = \frac{10.3 \times 10^5}{4.186 \times 10^5} \approx 2.5$ bananas

Process	Constant	Work Done	Internal Energy Change
Isobaric	Pressure	$W = P \Delta V$	$\Delta U = Q - P \Delta V$
Isochoric	Volume	$W = 0$	$\Delta U = Q$
Isothermal	Temperature	$W = nRT \ln \frac{V_f}{V_0}$	$\Delta U = 0$
Adiabatic	No heat transfer	$W = \frac{3}{2} nR(T_0 - T_f)$	$\Delta U = -W$

Additional info: Table summarizes the main thermal processes, their constants, work done, and internal energy changes.