Specific HeatsCyclic Processes

Todays Lecture

Second Law of ThermodynamicsCarnot Cycle

How do you order them in terms of heat transferred to the gas?

Things to remember:

U depends on T only!

= PdVWWUQ +=

Four Processes Between Two Isotherms

The largest work done by the gas will require the most heat transfer!

What is the heat transferred to the gas in process 3?

= PdVWWUQ +=

TnCU v=

Summary of Temperature Changes

Process 1 Adiabatic Q = 0 => U = -W => T1 < TaProcess 2 IsochoricW = 0 => Q = nCvT => T2 < TaProcess 3 IsobaricP3 = Pa => Q = nCpT => T3 > TaProcess 4 IsothermalT4 = Ta => U = 0 => T4 = Ta

Adiabatic ProcessesWU =0=Q

This result is consistent with W = -U as it had to be!

We can also find the work done by an ideal gas from:

For an adiabatic expansion PV = const. Hence the work integral is:

TiTf

Vi Vf

From the ideal gas law the initial volume is:

For isothermal expansion the volume increases by 2.5 or Vf = 1.21L.For adiabatic expansion:

For He = 5/3.

Example: Isothermal and Adiabatic ExpansionConsider expanding .05 mol of He from 2.5atm to 1atm. Starting at T=295K. Find the final temperature and volume for both an adiabatic and isothermal process.

Specific Heats of an Ideal Gas

For n moles the internal energy dueto KE is:

This means we can find Cv :

For inert gases we have:

From kinetic theory we showed the average (translational) kinetic energy per molecule is:

However for diatomic gases, nitrogen, oxygen, etc

Whats up??

Internal energy may be more than just translational kinetic energy!

Equipartition Theorem:In thermodynamic equilibrium, the average energy per molecule is 1/2kTfor each degree of freedom

This means Cv=R/2 for each degree of freedomFor a diatomic molecule there are 5 degrees of freedom

Specific Heats of an Ideal Gas

Summary of the First Law of Thermodynamics First Law of Thermodynamics

Thermodynamic processes,Quasi-static - work is given by

Isothermal constant temperatureIsochoric constant volumeIsobaric constant pressureAdiabatic no heat transfer

Equipartition theorem 1/2kT average molecular energy for each degree of freedom

Cyclic ProcessesWhat is so special about them?

The system returns to the same point (state)

V

P

A B

What can we say about change U, Q and W?Since the system returns to the same state and U is a function of state only, U = 0!By the 1st law U = Q W and we have Q = W.

That is the system gets heat Q from the outside and does an equal amount of work W.

Or the other way around! Both Q and W may be negative!

What simple thermodynamic process are cyclic processes similar to?

In what are they different?

Since U = 0, all we have to do is calculate W. How do we do that?In the simplest case we can divide the process into two parts: expansion A B, with positive work done;contraction B A with negative work done.

Cyclic Processes

WAB > 0

WBA < 0

Examine this cycle one process at a time and them sum them up.

Determine the work done by anIdeal gas in the cycle ACB.

Work Done in a Cycle

Example: Work by a Gas with Isothermal Expansion

At point B an ideal gas under a pressure of 250kPa has a volume of 1L. The gas expands along an Isotherm to a volume of 5L. Assume =1.4.

Find the work done by the gas in the cycle BACB. First the work done along BA,

The work done by the gas along AC,

Since the work along CB is zero the total work during the cycle is

Example: Work by a Gas with Adiabatic Expansion

At point B an ideal gas under a pressure of 250kPa has a volume of 1L. The gas expands along an adiabat to a volume of 5L. Assume =1.4.

First we need the pressure at A.

The work along BA is

The net work is

An Adiabatic, Isothermal Cyclic Process A gas occupies 4L at 300K and 100kPa. It is compressed adiabatically to 1L (red) and then cooled at constant volume to 300K (black). Then it is allowed to expand isothermally to 4L (green). How much work is done by the gas in the cyclic process ABCA?

Adiabatic compression:

Isothermal expansion:

The total work done by the gas in the cycle is: Why is it negative??

A gas occupies 4L at 300K and 100kPa. It is compressed adiabatically to 1L (red) and then cooled at constant volume to 300K (black). Then it is allowed to expand isothermally to 4L (green). How much work is done by the gas in the cyclic process ABCA?

An Adiabatic, Isothermal Cyclic Process

An ideal gas is taken through acircular cyclic process, shown.

(a) How much work does the gas do?

(b) If there are 1.3 moles of gas, what are the max and min temperatures?

The expressions for the pressure and volume are:

The maximum and minimum temperatures occur the largest and smallest distance from the origin. This occurs when = /4 and 5/4 respectively or:

A Circular Cyclic Process

12

Question:The gas goes from the state i to the state f. What is the minimum amount of work that the gas can do?

Any other suggestions?

We can start at i , go to 1, make, say 10 circles, go to 2 and only then continue to f.

What if the pressure is not allowed to drop below Pf ?

12

With 10 loops, where the work is negative, the net work is certain to be negative!

10 loops

021 > fiW010 21121

To convert internal energy or heat into work?We build a heat engine.

WQU = nRTPV = = PdVWIsothermal engine

0=U QW =

=

1

2lnVVnRTW

In principle one can get an unlimited amount of work

BUT it will require an infinitely large expansion!What are we going to do after the gas expands? Run it back?

100% of the heat transferred to the system is converted to work.

= PdVWIsothermal Engine

=

1

2lnVVnRTW

As the system expands all the heat transferred to the system is converted to work.

W < 0

W > 0

As the system contracts back, though, the same amount of work is done by the surroundings and all the energy is returned to the reservoir.

= PdVWAdiabatic Engine

The positive work is now limited by the internal energy of the insulated system.

But again, no net work is done if you go back and forth along the same adiabat.

UW = 0=Q

W < 0

W > 0

TiTf

Vi Vf

We need an engine working in cycles and converting heat supplied from the outside into mechanical work with a possibly high efficiency

How efficient can it be?The isothermal engine could convert 100% heat into work, but did not work cyclically.

Can we match this performance with an engine operating in cycles? Any fundamental law prohibiting it?

The Second Law of Thermodynamics (Kelvin-Plank statement):

It is impossible to construct a heat engine operating in a cycle that extracts heat from a reservoir and delivers and equal amount of work.

It is impossible to construct a heat engine operating in a cycle that extracts heat from a reservoir and delivers and equal amount of work

That would be an ideal heat engineWhat is a real heat engine doing?

Works between two temperatures, -a hot reservoir and a cold reservoir. (Hot side and cold side.)

Gets some heat Qh (obtained from, say, burning a fuel) from the hot side

Rejects some heat Qc to the cold side.

Does work W = Qh - Qc

Works in a cycle, so that the internal energy does not change, U=0.

Has an efficiency e = W/Qh h

ch

h QQQ

QWe ==

Carnot Cycle

http://www.cs.sbcc.net/~physics/flash/heatengines/Carnot%20cycle.html

hc

h

c

TT

QQ =

h

ch

h

ch

h TTT

QQQ

QWe ===Efficiency

Carnot Cycle

A

CD

Since ThVB1=TcVC-1and ThVA-1=TcVD-1

We have VB / VA = VC / VD and

What is the net work and efficiency of a Carnot cycle?

B

CDABtotal WWW +=

Efficiency

In a Carnot cycle all of the heat exchange takes placeonly between the highest and lowest temperatures, hence the Carnot cycle is the most efficient cycle.

Irreversible engines are necessarily less efficient,but so are many reversible engines!

Carnot Cycle

h

ch

h

ch

h TTT

QQQ

QWe ===

Carnot Cycle

h

ch

h

ch

h TTT

QQQ

QWe ===

Efficiency

For highest efficiency we would want to run our engine between a very hot and a very cold reservoirs.

Large temperature difference, Th-Tc, and low temperature of the cold reservoir, Tc, are very helpful.

Efficiency can in principle reach 100% for Tc = 0, but we normally do not have such reservoirs available

Example: Carnot Cycle

During one cycle a Carnot engine extracts 890J from a 550K reservoir and rejects 470J to a cooler reservoir.

(a) How much work does the engine do during each cycle?

(b) What is the efficiency of the engine?

(c) What temperature is the cool reservoir?

A Carnot engine operates betweenHeliums melting point and its boiling point at 4.25K. It has an efficiency of 77.7%.

At what temperature does Helium melt?

Example: Carnot Cycle

http://www.cs.sbcc.net/~physics/flash/heatengines/Carnot%20cycle.html

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