Unit 2B: Thermodynamics

Internal Energy

• the sum of all the molecular kinetic and potential energies

• can change by adding heat or doing work

  • w = fdcosθ

Work in thermodynamics

• consider the volume of a gas in thermodynamics equilibrium:

  • piston does work on the gas

  • w = F(-Δx) = PA(-Δx)

    • work done on the gas by external forces

    • 

    • Change in volume: W = P(-ΔV)

    • if a gas expands, ΔV is positive and work done is negative because energy is lost

    • if a gas compresses, ΔV is negative and work is positive because energy is gained

  • W = -W_env

    • W_env is the work done by the gas

• Example: a gas in a cylinder is at pressure 1.01×10^5 Pa and the piston has an area 0.100 m². as energy is slowly added to the gas by heat, the piston is pushed 0.04m. Calculate the work done by expanding the gas on its surroundings, W_env.

  • W = -PΔV = -(1.01×10^5)(0.100-0.04) = -404 J

• PV Graph

  • 

  • area = work = -PΔV = -P(V_f-V_i)

    • compression: volume shifts left

    • expansion: volume shifts right

First Law of Thermodynamics

• energy conservation in relation to changes in internal energy, U, due to heat/work

• changes in state— internal energy: P, T, V

• ΔU = U_f - U_i = Q + W

  • Q is energy transfer

    • +Q = heat absorbed

    • -Q = heat released

  • W is work done

    • +W = work done on the environment (by the gas)

    • -W = work done on the system

• Internal Energy of Monatomic Ideal Gas

  • ΔU = 3/2NK_BΔT = 3/2nRΔT

  • C_v = 3/2R

    • C_v is the molar-specific heat at constant volume (isovolumetric)

  • ΔU = nC_vΔT

  • Diatomic: C_v = 5/2R

4 Basic Types of Thermal Processes

1. Isobaric - constant pressure

2. Isovolumetric - constant volume

3. Isothermal - constant temperature

4. Adiabatic - no energy transferred by heat

Isobaric Process

• ΔU = 3/2NK_BΔT = 3/2nRΔT = 3/2PΔV

• Q = ΔU - W = 3/2nRΔT + (+PΔV) = 5/2nRΔT

  • C_p = constant pressure = 5/2R

• W = -PΔV

Isovolumetric Process

• ΔU = 3/2NK_BΔT = 3/2nRΔT = 3/2ΔPV

• Q = CU = nC_vΔT

  • C_v = 3/2R

• W = 0

Isothermal Process

• ΔU = 0

• Q = - W

• W = -nRTln(V_f/V_i)

Adiabatic Process

• ΔU = W

• Q = 0

• W = ΔU

  • `P(V)^γ = const.

  • γ = C_p/C_v = adiabatic index

    • monatomic: 5/3

    • diatomic: 7/3

Cyclic Process

• a system goes through a series of processes to return to the same initial state

• internal energy = 0

  • ΔU = 0

• total/net work done in a cyclic process equals the area enclosed in a PV diagram

  • 

  • clockwise cycles = -w done on the gas (compression)

  • counterclockwise cycles = +w done on gas (expansion)

• heat engines

  • takes in energy by heat and converts it to other forms of energy

  • work done by the eng = W_eng

  • energy is expelled by the engine by heat to a source at a lower temperature

    • Q = -W

  • |W_eng| = |Q_h| - |Q_c|

    • Q_h = heat absorbed

    • Q_c = heat lost

    • work done by heat engine = area

• Thermal Efficiency of Heat Engine

  • e = W_eng/|Q_h| = 1 - |Q_h|/|Q_c|

Second Law of Thermodynamics

• clausius statement: heat flows naturally from a hot object to a cold object but it will not flow spontaneously from a cold object to a hot object

• kelvin-Planck statement: no process is possible in which the sole result is to transform a given amount of heat completely into work

  • e < 1

  • Some engine is always lost to a cold re4servoir

  • can’t break even

• reversible processes are an idealization: every state along the path is an equilibrium state, so the system can return to its initial conditions by following this path in reverse

• Irreversible processes: not possible— most real natural processes

  • if a real process occurs slowly enough, it can be considered to be almost reversible

Carnot Engine/Cycle

• most efficient → ideal reversible cycle

• ideal gas contained in a cylinder with a moveable piston at one end

  • the temperature of a gas varies from T_h to T_c

• 2 adiabatic and 2 isothermal reversible processes

• Carnot’s Theorem: no real engine operating between two energy reservoirs can be more efficient than a Carnot engine operating

• T_c/T_h = Q_c/Q_h

• Thermal efficiency: e_c = (T_h - T_c)/T_h = 1 - (T_c/T_h)

• 

3rd Law of Thermodynamics

• e_c can only be 1 if T_c = 0k

• Nernsts Theorem: It is impossible to decrease the temperature of a system tro absolute 0 in a finite number of operations

• highest multiplicity = highest entropy = highest disorder

Disorder or Multiplicity

• large amount of chance in natural processes

• disorderly/random arrangements of objects are more probable than orderly ones

• isolated systems tend toward greater disorder and entropy is a measure

• greater probability/multiplicity = more entropy

• entropy is a measure of multiplicity

Entropy in second law: cyclic processes → increase/remain the same

• ΔS = Q_r/T [J/k]

• ΔS: change in entropy during any constant temperature between 2 states

• Q_r: energy absorbed/expelled during a reversible, constant temperature process

• the entropy of the universe increases in all natural processes

  • total entropy of a system and environment increases

  • decreases: gain heat for one object, but will increase for another

• entropy in a reversible adiabatic process: ΔS = 0

• defines the direction of time

• energy available for work decreases → leads to heat death