Lecture 7 - Thermodynamics, Energy, and Entropy Study Notes

Lecture 7: Thermodynamics, Energy, and Entropy

III. Temperature, Entropy, and the Second Law of Thermodynamics

• Meaning of Temperature

  • Temperature is a measure that reflects the average kinetic energy of particles in a substance.

• Kinetic Theory

  • Explains the microscopic relationship between temperature and kinetic energy:

  • Kinetic energy, $E_k$, is related to temperature, $T$.

  • The average kinetic energy of the particles increases with temperature.

• Equipartition Principle

  • A method to calculate kinetic energy based on temperature.

  • States that at thermal equilibrium, each degree of freedom in molecular motions has an average energy of:

    Ek = (1/2)*kB*T

  • Where $k_B$ is the Boltzmann constant.

• Degrees of Freedom of Molecular Motions

  • Refers to the different ways in which a molecule can store energy through translational, rotational, and vibrational motions.

• Number of Modes in Molecular Motions

  • The total energy, Etot of a molecule is the sume of all these different types of energy Etot = Etranslational + Erotational + Evibrational.

• Statistical Thermodynamics

  • Introduction of statistical thermodynamics leads to the definition of entropy, $S$.

  • The entropy is a state property that helps predict the direction of spontaneous change, where probability determines the direction of change.

Kinetic Theory and Maxwell-Boltzmann Distribution

• Maxwell-Boltzmann Distribution

  • Describes the distribution of molecular speeds in a gas, represented by the distribution function, $f(v)$.

  • Functions similarly to a histogram: it tells the probability or number of molecules as a function of speed.

• Main Implications of Distribution

  • The fraction of molecules with a certain speed can be derived from the distribution.

  • Energy is transferred between molecules via collisions.

  • In an ensemble of molecules, after many collisions, the Maxwell-Boltzmann distribution of speeds is observed, where the average molecular speed, $\bar{v}$, is approximately $10^{14}$ collisions/mL.

  • Heating the gas increases the spread of the distribution, extending speeds to higher values and increasing temperature proportionally:

    $T ext{ is proportional to } E_{ ext{avg}}$.

• Absolute Zero

  • Absolute zero is defined as $0 K$, at which point molecular motion ceases.

  • For a monatomic gas, the average energy relationship is given by:

    $E<em>{ ext{avg}} = rac{3}{2} k</em>B T$

  • Where the Boltzmann constant is defined as: $k<em>B = rac{R}{N</em>A}$

  • $R$ is the gas constant, and $N_A$ is Avogadro's number.

Equipartition Principle Explained

• Definition

  • The equipartition principle indicates that energy in a thermal system is equally distributed among all degrees of freedom.

• Energy Distribution

  • At thermal equilibrium, the average energy associated with each mode for a system is equal.

  • Each energy term acquires energy described by:

    $E<em>{ ext{term}}= rac{1}{2} k</em>B T$ (or $rac{1}{2} RT$ per mole).

  • Types of Motions

  • Translational Motions:

    $E_{ ext{Trans}} = rac{1}{2} mv^2$

  • Rotational Motions:

    $E_{ ext{Rot}} = rac{1}{2} I heta^2$

  • Vibrational Motions: described by harmonic oscillators:

    $E_{ ext{Vib}} = rac{1}{2} mv^2 + rac{1}{2} kR^2$.

Degrees of Freedom in Molecules

• Counting Degrees of Freedom

  • Let $N$ represent the number of atoms in a molecule.

  • Monatomic gases (e.g., He) are identified as:

  • $N = 1$ leading to 3 translational modes.

  • For diatomic molecules (e.g., O2):

  • $N = 2$, resulting in 5 modes (3 translational, 2 rotational).

  • Plus vibrational modes: $(3N - 5)$ for linear molecules; for $O_2$: $3(2) - 5 = 1$ vibrational mode.

  • For nonlinear triatomic molecules (e.g., $H_2O$):

  • $N = 3$, yielding 6 modes (3 translational, 3 rotational).

  • Plus vibrational modes: $(3N - 6)$ for nonlinear molecules; for $H_2O$: $3(3) - 6 = 3$ vibrational modes.

Example Calculations for Total Energy

• These calculations show how the total energy stored increases with more degrees of freedom:

• Monatomic Gases (He, Ar, etc.)

  • Total energy, $E$ (only 3 translational modes):

    $E = 3 imes rac{1}{2} k<em>B T = rac{3}{2} k</em>B T$

• Diatomic Molecules ($O_2$ usually implies some vibrational contribution):

  • Total energy, $E$: (3 translational, 2 rotational, 1 vibrational mode activated which contributes $k_B T$)

    $E = (3 imes rac{1}{2} k<em>B T) + (2 imes rac{1}{2} k</em>B T) + (1 imes k<em>B T) = rac{3}{2} k</em>B T + k<em>B T + k</em>B T = rac{7}{2} k_B T$

• Triatomic Molecules ($H_2O$):

  • Total energy, $E$: (3 translational, 3 rotational, 3 vibrational modes each contributing $k_B T$)

    $E = (3 imes rac{1}{2} k<em>B T) + (3 imes rac{1}{2} k</em>B T) + (3 imes k<em>B T) = rac{3}{2} k</em>B T + rac{3}{2} k<em>B T + 3 k</em>B T = 6 k_B T$

Comparative Total Energy across Molecules

• Comparison between Molecules

  • Total energy stored in diatomic and triatomic molecules is greater than that in corresponding monoatomic atoms due to additional potential energy in bonds.

  • Total energy of two monoatomic atoms:

    $2 imes rac{3}{2} k<em>B T = 3 k</em>B T$

  • Total energy of diatomic molecule:

    $rac{7}{2} k_B T$

Extra Problems for Analysis

• Determine Total Energy for Various Molecules:

  • Analyze molecules such as n-hexane and cyclohexane with respect to their structures and heat capacities.

  • Explore storage of thermal energy based on molecular structure; incorporate Lewis structures.

  • Investigate differences in heat capacities ($C_p$) among the molecules at $400 K$:

    • n-hexane: $113.5 ext{ J/mol/K}$

    • cyclohexane: $148.6 ext{ J/mol/K}$

    • benzene: $181.5 ext{ J/mol/K}$

• Energy Equations:

  Without work,

  $\Delta E = q + w$

  If $w=0$ then $\Delta E = q$ where $dq = C dT$