PHY1039 Properties of Matter First Law of Thermodynamics and Heat Capacity (See Finn’s Thermal Physics, Ch. 3) March 5 and 8, 2012 Lectures 9 and 10

Internal Energy, U Definition: The internal energy is the total energy of a system, equal to the sum of the kinetic and potential energies of its constituent atoms/molecules. Gases and liquids Kinetic energy: Translational motion with a velocity. Rotation Molecular vibration y z x Translational energy

Rotational energy of a diatomic molecule, e.g. N 2, O 2, H 2 Kinetic Energy of Gas Molecules Figure from “Understanding Properties of Matter” by M. de Podesta Rotational energy of a non-linear triatomic molecule, e.g. H 2 O Figure from P. Atkin’s The Elements of Physical Chemistry Three degrees of freedom (one for each type of rotation)

Symmetric and anti-symmetric stretching vibration Bending vibration Vibrational Energy of Gas Molecules Resonant frequency for a stretching vibration of an N 2 molecule is on the order of Hz. Spring constant K is about 100 N/m. Modes of Vibration in Triatomic Molecule N N

Kinetic Energy: Molecular vibration in x, y and z directions Internal Energy of Solids roro Potential Energy: Varies with distance between molecules (analogy to stretching a spring) Three degrees of freedom (one in each direction)

Internal Energy, U Internal energy depends on the state of a system through the state variables: U = U(P, V, T). We call U a “state function” because it depends on the state of the system.  U 12 will always take the same value, regardless of the path, in going from 1 to 2. The difference in value of the state functions for two different states is said to be “path independent”. U1U1 U2U2 P V Consider two different states of a system: 1 and 2. The difference in the internal energy between the two states is:

Change in Internal Energy,  U in an Adiabatic Process Adiabatic work on a system takes it from State 1 to State 2. Area under the PV curve determines the amount of work done (value is path dependent) Adiabatic work changes the internal energy of the system by  U. For the two states,  U is fixed and hence there can be only one adiabatic path between States 1 and P V Adiabatic wall F

Non-Adiabatic Path 1 2 P V 2’ What is Now consider an alternative, non-adiabatic path from 1 to 2 via 2’.  U 12 must be the same (it is path independent) as before, but W 12’2 is different than on the adiabatic path.

First Law of Thermodynamics Heat, Q, is the difference between  U and W: Q =  U – W The First Law of Thermodynamics is usually stated as:  U = Q + W Differential form of First Law: dU = dQ + dW Sign convention: Heat into system: +ve Q Heat out of system: -ve Q Units of U, Q and W are Nm (or Joules, J) The First Law is a “balance sheet” of energy. It tells us that heat into and work on a system raises its internal energy. If a system does work on its surroundings or gives off heat, then its internal energy decreases.

Heat, Q Heat is the non-mechanical exchange of energy between a closed system and its surroundings. Heat flows spontaneously across a diathermal wall when two systems at different temperatures are in contact. In contrast, work acts in a specific direction: pushing atoms together or pulling them further apart. F HotterColder Heat is “energy in motion” that moves across the system wall. Heat is stored in the system as internal energy.

Mechanisms of Heat Exchange between System and Surroundings Thermal radiation: emission of electromagnetic radiation Conduction: carried by molecular vibrations hot cold Convection: carried by molecular transport from one position to another

Radiation energy density Planck distribution law infrared UV-Vis. Spectral Distribution of Thermal Radiation Effective temperature of the Sun is 5780 K  UV-visible radiation.

Internal Energy Change in a Cycle 1 2 P V 3 In a cycle, the system goes to other states and then returns to its original state. The change in internal energy at the end of the cycle is 0.  U 1231 = 0  U 1321 = 0

Path Independence of Internal Energy Change The sign of  U depends on the direction of the path (i.e. which are the original and final states?). 1 2 P V 3  U 12 =  U 132  U 12 =  U 132 = -  U 21 = -  U 231

Meaning of the First Law (1) Work can be converted into heat or heat can be converted into work in a process when there is no change in the internal energy.  U = 0 = Q + W W = - Q (remembering the sign convention!) If the state variables return to their original position after a process, then  U = 0. (2) Adiabatic work can raise the internal energy of a system, and hence raise its temperature. Q = 0, so that  U = W Examples of mechanical work include compressing a gas, stretching a wire or member, stirring a paddle in a viscous liquid.

_imageDetail.cfm?id_image=3087 In the 19 th century, it was believed that systems contained a finite amount of heat. “Particles” of heat, called caloric, were thought to flow from hot to cold objects. Matter was thought to possess a finite amount of heat. Benjamin Thompson (Count Romford), a British soldier, observed that when boring tools did frictional work on cannons, there was a build-up of heat – enough to boil water. He reasoned that work on a system can be given off to the surroundings as heat. Mechanical Equivalence of Heat

Adiabatic Work, W, on the system: Internal Energy, U, increases: T rises

Joule’s Paddle Experiment Nm of Work raised the T of 1 g of water by 1 K = 4.2 J = 1 calorie 4.2 Nm of adiabatic work on the water is equivalent to 4.2 Nm of heat into the water in raising the internal energy.

What was the significance of to James Joule? (Discuss in SGT) The tomb of James Joule

p/chap23/page28.htm Are perpetual motion machines possible? W =  U - Q Continuous work on surroundings (-ve W) requires the continual input of heat (+ve Q) or removal of internal energy (which is finite!)

V P Two Types of Heat Capacity Isochoric Process Isobaric Process T1T1 T2T2 V P T1T1 T2T2 (V 1, P 1 ) (V 1, P 2 ) P1P1 P2P2 V1V1 V2V2 T 2 > T 1 V1V1 P1P1 (V 1, P 1 ) (V 2, P 1 )

Molar Heat Capacity, C P, of the Elements at 298 K Figure from “Understanding Properties of Matter” by M. de Podesta

Molar Heat Capacity, C P, of Monoatomic and Diatomic Gases Figures from “Understanding Properties of Matter” by M. de Podesta Diatomic Gas Monoatomic Gas