Thermodynamic Processes

Lecturer: Professor Stephen T. Thornton

If you add some heat to a substance,

is it possible for the temperature of

the substance to remain unchanged?

A) yes

B) no

C) depends on Q

D) depends on W

ReadingReading Quiz Quiz

If you add some heat to a substance,

is it possible for the temperature of

the substance to remain unchanged?

A) yes

B) no

C) depends on Q

D) depends on W

Yes, it is indeed possible for the temperature to stay the same. This is precisely what occurs during a phase change – the added heat goes into changing the state of the substance (from solid to liquid or from liquid to gas) and does not go into changing the temperature! Once the phase change has been accomplished, then the temperature of the substance will rise with more added heat.

ReadingReading Quiz Quiz

Follow-up:Follow-up: Does that depend on the substance?

Last Time

HeatInternal energySpecific heatLatent heatFirst Law of Thermodynamics

Today

Thermodynamic processesIsothermal, adiabatic

Specific heat of gases Equipartition

Conceptual Quiz:A system absorbs heat Q and has an equal amount of work done on it. What is the change in the internal energy of the system? A) Q System gains heat Q positive

B) 2Q System loses heat Q negative

C) -2Q Work done by system W positive

D) zero Work done on system W negative E) Q/2

E Q W

Answer: B

System absorbs heat Q so it is positive. System has work of equal value done on it, so it is negative.

W = -Q

E = Q – W = Q – (-Q) = 2Q

A Constant-Pressure Process

System does work to push piston in cylinder at constant pressure. Volume expands.

Area of graph = work done by system

F PA

W F x PA x P V

P V W

Area of graph is W

In a general problem like this example, the area under the curve is equal to the work done by the system.

Area here is work

W P dV

We add heat to a system at constant volume. What is the work done? W= PV = 0 Because volume doesn’t change, the work done W must be zero.

0W

E Q W Q

Isotherms on a PV diagram

process.

is constant.T

Isothermal

An Isothermal Expansion constant

constant

hyperbola

PV nRT

PV

= area under curve. We use calculus to show

ln lnf f

i i

W

V VW nRT NkT

V V

dVnRTV

W P dV

In an adiabatic process, the system is well insulated thermally, and no heat flows (Q = 0).When the piston compresses the volume, the pressure and temperature must both go up.

Adiabatic Heating

If we push down quickly, there is no time for heat to flow, and the process is adiabatic. Temperature rises quickly.

Do demo

When the piston moves up, the volume expands, and the pressure and temperature decrease.

Adiabatic process occurs often when the process is rapid, and there is no time for heat to flow.

Conceptual QuizConceptual Quiz

A) the cooler one

B) the hotter one

C) both the same

Two equal-mass liquids, initially at the

same temperature, are heated for the same

time over the same stove. You measure

the temperatures and find that one liquid

has a higher temperature than the other.

Which liquid has a higher specific heat?

Both liquids had the same increase in internal energy,

because the same heat was added. But the cooler liquidcooler liquid

had a lower temperaturelower temperature change.

Because QQ = = mcmcTT, if QQ and mm are both the same and TT is

smaller, then cc (specific heat) must be bigger.

Conceptual QuizConceptual Quiz

A) the cooler one

B) the hotter one

C) both the same

Two equal-mass liquids, initially at the

same temperature, are heated for the same

time over the same stove. You measure

the temperatures and find that one liquid

has a higher temperature than the other.

Which liquid has a higher specific heat?

The specific heat of concrete is

greater than that of soil. A baseball

field (with real soil) and the

surrounding parking lot are warmed

up during a sunny day. Which would

you expect to cool off faster in the

evening when the sun goes down?

A) the concrete parking lot

B) the baseball field

C) both cool off equally fast

Conceptual QuizConceptual Quiz

The specific heat of concrete is

greater than that of soil. A baseball

field (with real soil) and the

surrounding parking lot are warmed

up during a sunny day. Which would

you expect to cool off faster in the

evening when the sun goes down?

A) the concrete parking lot

B) the baseball field

C) both cool off equally fast

The baseball field, with the lower specific heat, will change

temperature more readily, so it will cool off faster. The high specific

heat of concrete allows it to “retain heat” better and so it will not cool

off so quickly—it has a higher “thermal inertia.”

Conceptual QuizConceptual Quiz

Monatomic Gas. One and one-half moles of an ideal monatomic gas expand adiabatically, performing 7500 J of work in the process. What is the change in temperature of the gas during this expansion?

Conceptual Quiz:An ideal gas is heated so that it expands at constant pressure. The gas does work W. What heat is added to the gas? Hint: E = Q – W  A) Q = W B) Q = -WC) Q = 0 D) Q > WE) Q < W

Answer: D

Because the gas is heated, the temperature will increase. Therefore the internal energy E > 0. W > 0, so if E = Q – W > 0, then Q = ΔE +W > W.

An ideal gas is heated so that it expands at constant pressure. The gas does work W. What heat is added to the gas?

Thermodynamic Processes and Their Characteristics

Constant pressure W = PV Q = Eint + PV

Constant volume W = 0 Q = Eint

Isothermal (constant temperature)

W = Q Eint = 0

Adiabatic (no heat flow)

Eint = Q – W

W = – Eint

Q = 0

Conceptual Quiz:A gas at point A compresses isothermally along curve (ii). If the gas compresses adiabatically, what curve does it follow?

A) Curve (i)

B) Curve (ii)

C) Curve (iii)

Answer: C

In the isothermal process, some heat flows out of the system in order to keep the temperature constant. In the adiabatic process, no heat can flow out, so the temperature must rise. For the same volume, therefore the pressure must rise, compared to the isothermal case. Curve (iii) must be correct.

Conceptual Quiz:What happens if we compress a cylinder that is thermally isolated? E = Q – W A) W > 0, Q > 0B) W < 0, Q > 0, E = 0C) W > 0, Q = 0, E > 0D) W < 0, Q = 0, E > 0

Answer: D

The system is thermally isolated, so the heat flow Q = 0. An external agent pushes the piston down and does work on the system. Therefore the system does negative work, W < 0.

E = Q – W > 0.

Conceptual QuizConceptual Quiz

Water has a higher specific

heat than sand. Therefore,

on the beach at night,

breezes would blow:

A) from the ocean to the beach

B) from the beach to the ocean

C) either way, makes no difference

Conceptual QuizConceptual Quiz

DaytimeDaytime sun heats both the beach and the watersun heats both the beach and the water

» beach heats up fasterbeach heats up faster

» warmer air above beach riseswarmer air above beach rises

» cooler air from ocean moves in underneathcooler air from ocean moves in underneath

» breeze blows ocean breeze blows ocean land land

ccsandsand < < ccwaterwater

NighttimeNighttime sun has gone to sleepsun has gone to sleep

» beach cools down fasterbeach cools down faster

» warmer air is now above the oceanwarmer air is now above the ocean

» cooler air from beach moves out to the oceancooler air from beach moves out to the ocean

» breeze blows land breeze blows land ocean ocean

Water has a higher specific

heat than sand. Therefore,

on the beach at night,

breezes would blow:

A) from the ocean to the beach

B) from the beach to the ocean

C) either way, makes no difference

Why do we almost always feel a breeze at the beach?See if this works this summer!

Water has high thermal conductivity!

Work Done by Thermal Systems

We have looked at specific heat for solids and liquids. Let’s now look at specific heat for gases. We have two important cases: constant volume and constant pressure. Let heat flow Q into the system at constant volume and the temperature rises by T.

, constant volume

is molar specific heat = /

, compare with

V V V

V V V V

V V P P

Q mc T c

C Mc c C M

Q nC T Q nC T

P PC Mc

int

int int

int

First law,

For constant volume, 0, and

3, and because ,

23

2

3so

2

This is molar specific heat for a monatomic ideal gas at constant volume.

V

V V

V

Q E W

W

Q E E nRT

Q E nR T nC T

C R

Now let’s determine CP at constant pressure. Consider an ideal gas.

int

3 5

2 2

5

2

And we have

P

P

P V

W P V nR T

Q E W

nR T nR T nR T

C R

C C R

CP – CV for Various Gases

Helium (monatomic) 0.995 R

Nitrogen (diatomic) 1.00 R

Oxygen 1.00 R

Argon 1.01 R

Carbon Dioxide (triatomic) 1.01 R

Methane 1.01 R

Experimental Values

Seems to be true for real gases, including diatomic gases, not only monatomic gases.

A diatomic molecule can do more than just move – it can rotate or vibrate, and both contribute to its total kinetic energy.

The equipartition theorem states that each degree of freedom contributes kT/2 to the average energy of a molecule. We have degrees of freedom for

• velocity in x, y, and z - translation

• rotation about two different axes

• vibration, which includes two contributions

For s degrees of freedom, we have

2sE kT

int 2sE N E NkT

For N molecules, we have

3

2kT

2

2kT

2

2kT

The average kinetic energy of translation (3 dimensions) is

The addition of rotations about two axes adds two more factors of ½ kT:

Finally, the addition of vibration contributes two more factors of ½ kT, one for the motion of the atoms and one for the energy in the “spring”:

72

E kT

52

E kT

for each dimension2

32

kTE kT

2sE kT

For molecular hydrogen:

/V

C R

Copyright © 2009 Pearson Education, Inc.

In this table, we see that the molar specific heats for gases with the same number of molecules are almost the same, and that the difference CP – CV is almost exactly equal to 2 (R) in all cases.

Specific Heat of Gas. The specific heat at constant volume of a particular gas is0.182 kcal/kg·K at room temperature, and its molecular mass is 34. (a) What is its specific heat at constant pressure? (b) What do you think is the molecular structure of this gas?

Piston & Cylinder. When 6.3 x 105 J of heat is added to a gas enclosed in a cylinder fitted with a light frictionless piston maintained at atmospheric pressure, the volume is observed to increase from 2.2 m3 to 4.1 m3. Calculate (a) the work done by the gas, and (b) the change in internal energy of the gas. (c) Graph this process on a PV diagram.