Energy in Molecules

• Energy


  A quantitative measure of the ability to supply heat or do work
• Kinetic (Ek)


  Energy associated with motion
• Potential (Ep)


  Energy stored based on location in a field
• Thermal energy


  The kinetic energy of molecular motion, which we measure by finding the temperature of an object
• Heat


  The amount of thermal energy transferred from one object to another as a result of the difference in temperature between them
• Zeroth law: Thermal energy flows along energy gradient (high T to low T) DT = 0 = thermal equilibrium
• Kinetic Molecular Theory


  • A gas consists of tiny particles, either atoms or molecules, in constant random motion
  • Particles << Vtotal (most of the volume of the gas is empty space)
  • Particles act independently of one another (there are no attractive or repulsive forces between the particles)
  • Collisions amongst particles or with container are elastic—the total kinetic energy is constant at a constant temperature
• Internal Energy (U)


  The sum of Translational, Rotational, Vibrational and Electronic Energies of all molecules
• Molecular energy can be subdivided into

  • Thermal (Kinetic) energy: Energy due to random movement
  • Intra- and inter-molecular energy (potential energy): associated with molecular and atomic interactions
• Types of Thermal (Kinetic) energies

  • Translational Energy (ETrans): EK associated with random motion of molecules—3 degrees of freedom (x, y, z)
  • Rotational Energy (ERot): EK associated with rotation about axis of inertia
  • Vibrational Energy (EVib): EK associated with relative motion of atoms in a molecule
  • Electronic Energy (EElec): EK of electrons in the molecule
• For an ideal gas where no bonds are formed or broken (Ep = 0): U = SE = Ek + Ep = Etrans + Erot + Evib + Eelec ≈ Etrans + Erot + Evib
• Over a limited range of temperatures, above and below room temperature, the rise in temperature is directly proportional to heat supplied, for most substances.
• Temperature rise depends on

  (i) The amount of heat added, q (ii) The amount of substance present, m (iii) The specific heat, c (iv) The conditions under which heat is added
• Heat capacity (C)


  The proportionality constant, C, is called the "heat capacity" of the system and has units J K-1 or Cal °C-1
• Specific heat capacity (c)

  Heat capacity per unit mass, which has units: J K-1 kg-1 or Cal °C-1 g-1
• Molar heat capacity (cm)


  The amount of energy required to raise the temp. of 1g of substance by 1°C or 1K
• Heat capacity (C)


  Depends on how the heating is carried out: At constant volume (dV = 0), no expansion work vs At constant pressure (dP = 0), only PV work
• Cp > Cv for gases, because at constant pressure the gas expands when heated and some extra heat has to be supplied to do the work of expansion against the atmosphere (p ΔV) in raising the temperature of the gas by 1 K.
• Cv values for perfect gases: Cv = (3/2)R for monatomic gases, Cv = (5/2)R for diatomic gases, Cv = (3R) for polyatomic gases
• Substances have different specific and molar heat capacities, suggesting different internal distributions of energy occur in different states of matter.
• Heat capacities are usually a bit larger than the heat capacities at constant volume
• Cp > Cv
• Values of Cv for some gases

  • He
  • H2
  • O2
  • Cl2
  • N2
  • H2O
  • CO2
• Points of note from the table of Cv values

  • Significant variation of values - in terms of the nature of the molecule and in terms of temperature
  • Except He, Cv increases with temp
  • Cv of Diatomic and polyatomic > monoatomic and shows greatest variation
• Lecture 2 Energy in Molecules Dr. Mark Lawrence Office: Room 31 (outside the back entrance of the lab) CHEM1910
• Predicting Cv using classical mechanics

  1. Consider change in momentum of molecules hitting one wall of a cubic container

  2. Obtain the actual mean square velocity

  3. Assume molecular motions in x, y and z directions are equally probably for large N (no net flow in a particular direction)

  4. Derive expression for Cv in terms of R and N
• The actual mean square velocity: <formula>3RT/m</formula>
• Pressure exerted by N molecules on one wall: <formula>NmvxA/V</formula>
• Pressure of a perfect gas: <formula>PV = nRT</formula>
• Review – Lecture 1 Heat capacity – its absolute value and its variation with temperature and with the chemical identity of molecules - appears to give some idea about the distribution of energy in a collection of molecules. We used a simple model - the "Kinetic Model of Gases" – that describes a perfect gas – and found that it gives good predictions of the heat capacity (at constant volume) of gases like He(g) which are close to being perfect. Perfect here means that the gas meets the assumptions made in the model: Having molecules/atoms that are in ceaseless random motion, Having molecules that are small compared to free space in the gas (i.e. distance separating the molecules), and Having molecules that interact only in elastic collisions with each other and with the container.
• Translational kinetic energy per mole: <formula>3/2(R/NA)T</formula>
• Equipartition Theorem


  Equipartition into "Quadratic Components" or "Degrees of Freedom"
• Cv = 3R/2 for a perfect gas
• Excellent agreement between predicted and measured Cv for the simplest gas - a monoatomic inert (noble) gas
• The estimate/prediction clearly does not apply to diatomic and tri-atomic molecular gases
• Heat capacity


  The amount of heat required to raise the temperature of a substance by 1 degree
• Predicted Heat Capacities


  Compared to Measured Heat Capacities
• Equipartition into "Quadratic Components" or "Degrees of Freedom"

  1. The kinetic energy for a mole of perfect gas is given by:

  2. Recall:

  3. The result can be interpreted to suggest that the kinetic energy of 'N' molecules is made up of three quadratic (squared) terms:

  4. Each component contributes equally (isotropic) to the overall kinetic energy.

  5. Each squared term represents 1 degree of freedom (DoF).

  6. For 1 mole of a monatomic gas three terms contribute to the overall kinetic energy; i.e. 3 degrees of freedom.
• Specific heat capacity


  The heat capacity per unit mass of a substance
• Degree of Freedom


  The number of independent ways that molecules can possess energy