Thermodynamics Internal energy of a system can be increased either by adding energy to the system or by doing work on the system Remember internal energy.

Thermodynamics

• Internal energy of a system can be increased either by adding energy to the system or by doing work on the system

• Remember internal energy of a system is the sum of the kinetic energies and the potential energy at the molecular level

• Adding heat is transferring energy

Work done on a gas

• Consider a gas at equilibrium, pressure and temperature is the same throughout.

• The gas is in a piston with area A.

• Work done on the gas W = -Fy = -PA y

• As Ay = Volume then W = -PV

• Note as volume is decreased V is negative and work W is positive. If V is positive work is done on the environment.

Pressure Diagram

• The equation above is only valid when the pressure is constant during the work process

• Such a process is called an isobaric process.

• The area under a P vs V graph represents the work done. The arrow represents +ve or -ve work p

V

First Law of Thermodynamics

• If a system undergoes a change from its initial state to a final state, where Q is the energy transferred to the system by heat and W is the work done on the system, the change of internal energy of the system

U = Uf- Ui = Q + W

• Q is positive when heat is added to the system, W is positive when work is done on the system.

Internal energy of a gas• Change in internal energy of an ideal gas is,

U = 3/2nRT

The molar specific heat of a monatomic gas is

Cv = 3/2R

Therefore change in internal energy of the ideal gas is U = nCv T

The larger the Cv the more energy needed to change the temperature.

Degrees of Freedom

• Each different way a molecule can store energy is called a degree of freedom

• Each degree of freedom contributes 1/2R to the molar specific heat.

• Monatomic molecules can move in three directions thus Cv = 3/2R

• A diatomic molecule can move three and tumble two ways thus Cv = 5/2R

Isobaric Processes

• Pressure remains constant throughout

• Expanding gas works on the environment

• When gas works it losses internal energy

• Temperature decreases as energy decreases

• If volume increases and temperature decreases then thermal energy (heat) must be added to gas to maintain constant pressure

Isobaric Processes

• From U = Q + W then Q = U -W

Q = U -P V

From P V = nRT and U = 3/2nRT

then Q = 3/2 nRT + nRT = 5/2 nRT

Another way of expressing heat transfer is

Q = nCp T where Cp = 5/2R

Cp is the molar heat capacity Cp = Cv + R

Adiabatic Processes

• In adiabatic processes no energy enters or leaves the system by heat (insulated System

• A rapid system is considered adiabatic, there is no time for heat energy transfer.

• As Q = 0 then U = W The work done is the change in internal energy. Work can calculated from a PV diagram Pvy =constant

• y = Cp/Cv called adiabatic index

Isovolumetric Processes

• Sometimes called an isochoric process, it occurs at a constant volume, shown as a vertical line in a PV diagram.

• As volume does not change no work is done by or on the system, thus W= 0

U = Q The change in internal energy of the system is equal to the transfer of heat to the system. Q = nCv T

Isothermal Processes

• During the process the temperature of the system does not change. As U depends on temperature, U= 0 as T = 0 and W = -Q

• The work done on the system is equal to the negative thermal energy transferred to the system.

• For work done on the environment

Wenv = nRT ln(Vf/Vi)

Second Law of Thermodynamics

• Heat engines take in heat energy and convert it to other forms of energy, electrical and mechanical. Ex. Coal burnt, heat converts water to steam, steam turns a turbine, turbine drives a generator. In general a heat engine carries a substance through a cyclic process.

• Entropy tendency to greater disorder

Heat Engines cont.

• The Weng done by a heat engine equal the net work absorbed by the engine. The initial and final internal energies are equal.

U= 0 therefore Qnet = -W = Weng • where Qnet = Qh - Qc

• The work done by an engine for a cyclic process is the area enclosed by the curve of PV diagram

Thermal Efficiency

• The thermal efficiency of a heat engine e is defined as the work done by the engine, Weng, divided by the energy absorbed through one cycle.

• E = Weng/ Qh = (Qh-Qc)/Qh = 1- Qc/Qh

• The values of Q are absolute values

Refrigerators and Heat Pumps

• Heat engines operate in reverse, energy is injected into the engine. Work is done by the system where heat is removed from a cool reservoir to a hot reservoir. A compressor reduces the volume of a gas increasing its temperature, the gas later expands requiring heat to do so thus drawing heat from the cool refrigerator

Coefficients of Performance

• Coefficient of performance for a refrigerator is equal to the magnitude of extracted energy divided by the work performed.

• COP (cooling) = Qc /W the larger the ratio the better the performance

• For a heat pump operating in heat mode

COP (heating) = Qh /W

Second Law of Thermodynamics

• No heat engine operating in a cycle can absorb energy from a reservoir and use it entirely for the performance of an equal amount of work. (e<1) e = Weng/ Qh

• Some energy Qc is always lost to the environment.

The Carnot Engine

• The Carnot Cycle is the operation of an ideal reversible cycle, using two energy reservoirs.

• Carnot’s theorem states that no real engine operating between two energy reservoirs can be more efficient than a Carnot engine operating between the same two reservoirs.

• The Carnot engine is only theoretical and would have to run infinitely slowly, thus having zero power output.

Carnot cont.

• The Carnot cycle contains an ideal gas in a thermally nonconductive cylinder with a moveable piston at one end. The cycle passes through four stages, two isothermal and two adiabatic. During the adiabatic stages the gas temperatures range between Tc and Th. All cycles are reversible.

Stage 1 Isothermal Expansion

• The base of the cylinder consists of a hot energy reservoir at Th.

• Gas in the cylinder absorbs heat energy Qh from a reservoir, thus doing work by raising the piston. The gas expands isothermally at temperature Th.

• W = -Q

Stage 2 Adiabatic Expansion

• Base of the cylinder is replaced with a thermally insulated base.

• The gas continues to expand, this time adiabatically. ( no energy enters or leaves the system by heat.

• The expanding gas does work on the piston, raising it further while the gas temperature decreases from Th to Tc. W = ΔU

Stage 3 Isothermal Compression

• The cylinder base is replaced with a cold reservoir at Tc.

• The gas is compressed at the temperature of Tc and during this time expels energy to the reservoir, Qc.

• Work is done on the gas.

Stage 4 Adiabatic Compression

• The base is again replaced with a thermally non-conducting wall.

• The gas is compressed adiabatically increasing its temperature to Th thus doing work on the gas.

• This 4 stage cycle is constantly repeated.• Ec = 1 – Tc/Th T is in Kelvin• This is used to rate engine efficiency

Entropy S

• Entropy is the state of disorder(randomness) • The change in entropy ΔS = Qr/TWhere Qr is the energy absorbed or expelled

during a reversible process. T is constant in Kelvin. r means reversible.

• A change in entropy occurs between two equilibrium states. The path taken is not important

Entropy cont.

• If the laws of nature are allowed to operate without interference it is more likely to have a disorderly arrangement than an orderly one. Using probability Boltzmann found that S = kBlnW where kB is Boltzmann’s constant and W is a number proportional to the probability of a specific occurrence. The second law states what is most likely not what will happen.