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KYA 212

Thermodynamics Assignment 1

Set 22.09.16

Due 30.09.16

Question 5 is an extension question for anyone who wants to apply some basic thermodynamics in a

real world situation. It will be treated in the same manner as the challenge questions in

KYA101/102 and worth 10% of the marks on the assignment, but marked out of a larger number.

1.

Using a diagram, explain what is meant by the critical point of a fluid system.

2.

Find expressions for v and T for a van der Waals gas at the critical point and hence evaluate β

1# ∂v&

,)

at the critical point (where β is the isobaric coefficient expansivity of a fluid β = %

v $ ∂ T (‘ p

3.

a.

Describe the principle of operation of a constant volume gas thermometer and

explain why it is important in the development of thermodynamic theory.

b.

The table below gives the pressure p (in mm Hg) of the gas in a constant volume gas

thermometer at an unknown temperature T for two values of the pressure pt at the triple

point of water. Determine the limiting value of the ratio p/pt as pt → 0 . Hence find the

unknown temperature T.

pt 100.0 300.0

p 127.9 385.5

4.

In the diagram, consider the system S1 to

comprise the water, of volume V, the paddles,

and the heater, R. Assume (A1) that the walls,

top and bottom of the container are adiabatically

isolating walls.

a. What is meant by assumption A1 above?

M

heater, R

h

I

b. What would be the effect on the system of

lighting the bunsen burner BB

underneath the container?

Now let the mass M of 600 kg fall through a

height h = 50 m, turning the paddles.

E

BB

c. How much work has been done on the

system S1.

After some time, the temperature of S1 is measured and the rise in temperature ΔT calculated.

d.

Why is it necessary to wait ‘some time’ before measuring the new temperature of S1?

From your own experience, about how long would you wait?

[Continued Over Page…]

e.

What modification is necessary to A1 in order to measure the new temperature of S1?

f.

What is the change in internal energy of the system S1? Why?

Instead of allowing the mass M to fall to produce the ΔT, a current could be passed through

the resistor.

g.

For how long must a current I = 4A at E = 250 V be passed?

h.

How much heat is thereby transferred to S1?

Now consider the system S1 to be divided into two parts, in thermal contact with each other:

S2 comprising solely the resistor and S3 comprising the rest of S1. If the current is passed

through the resistor as in (g) above,

5.

i.

What work is done on S2?

j.

k.

About how much heat is transferred to S2? (Write down an algebraic expression of the

First Law of Thermodynamics and suppose that the state of S2 is altered only a negligible

amount by the passage of I for the calculated time).

How much heat is transferred to S3?

l.

What work is done on S3?

The left-hand figure below gives an overview of the complexity of the large-scale energy

exchanges that contribute to the Earth’s energy budget. The right-hand figure shows sea level

rise measurements over the last century, which have averaged 2 mm per year over the last

century. Use the information on the next page and information you source yourself to

estimate the following :

a.

Estimate the rate at which the energy stored in the world’s oceans is increasing (give

your answer in Joules per year).

b.

Estimate the mean temperature increase in the world’s oceans (give your answer in

degrees per year).

c.

Estimate the fractional change in the energy output of the Sun would be required to

produce the observed changes.

In answering the questions above you should clearly state what simplifying assumptions you

have made. For information not given with the assignment you should clearly identify the

source.

Figures from http://www.nasa.gov/images/content/57911main_Earth_Energy_Budget.jpg and

http://en.wikipedia.org/wiki/Current_sea_level_rise

2

(Potentially) useful information for Question 5

The Earth and Sun :

Surface area of the oceans : 3.6 x 108 km2

Approximate volume of the oceans : 1.3 x 109 km3

Average depth of oceans : 3790 m

Average temperature at a depth of 1000 m is 4°C

Average energy from the Sun at the Earth 1366 Wm-2

Approximate volume of ice in the Greenland iceshelf 2.85 x 106 km3

Properties of water

Density at 4°C (maximum density) 1000 kg m-3

Specific heat water 4.187 kJ kg-1 K-1

Specific heat ice 2.108 kJ kg-1 K-1

Specific heat water vapour 1.996 kJ kg-1 K-1

Specific heat dry air 1.006 kJ kg-1 K-1

Volume expansion coefficient 2.14 x 10-4 K-1

From http://www.engineeringtoolbox.com/water-thermal-properties-d_162.html

3

KYA 212

Thermodynamics Assignment 1

Set 24.09.15

Due 02.10.15

Question 5 is an extension question for anyone who wants to apply some basic thermodynamics in a

real world situation. It will be treated in the same manner as the challenge questions in

KYA101/102 and worth 10% of the marks on the assignment, but marked out of a larger number.

1.

Using a diagram, explain what is meant by the critical point of a fluid system.

Find expressions for v and T for a van der Waals gas at the critical point and hence evaluate β

1# ∂v&

,)

at the critical point (where β is the isobaric coefficient expansivity of a fluid β = %

v $ ∂ T (‘ p

2.

3.

a.

Describe the principle of operation of a constant volume gas thermometer and

explain why it is important in the development of thermodynamic theory.

b.

The table below gives the pressure p (in mm Hg) of the gas in a constant volume gas

thermometer at an unknown temperature T for two values of the pressure pt at the triple

point of water. Determine the limiting value of the ratio p/pt as pt → 0 . Hence find the

unknown temperature T.

pt 100.0 300.0

p 127.9 385.5

4.

In the diagram, consider the system S1 to

comprise the water, of volume V, the paddles,

and the heater, R. Assume (A1) that the walls,

top and bottom of the container are adiabatically

isolating walls.

a. What is meant by assumption A1 above?

M

heater, R

h

I

b. What would be the effect on the system of

lighting the bunsen burner BB

underneath the container?

Now let the mass M of 600 kg fall through a

height h = 50 m, turning the paddles.

E

BB

c. How much work has been done on the

system S1.

After some time, the temperature of S1 is measured and the rise in temperature ΔT calculated.

d.

Why is it necessary to wait ‘some time’ before measuring the new temperature of S1?

From your own experience, about how long would you wait?

[Continued Over Page…]

e.

What modification is necessary to A1 in order to measure the new temperature of S1?

f.

What is the change in internal energy of the system S1? Why?

Instead of allowing the mass M to fall to produce the ΔT, a current could be passed through

the resistor.

g.

For how long must a current I = 4A at E = 250 V be passed?

h.

How much heat is thereby transferred to S1?

Now consider the system S1 to be divided into two parts, in thermal contact with each other:

S2 comprising solely the resistor and S3 comprising the rest of S1. If the current is passed

through the resistor as in (g) above,

5.

i.

What work is done on S2?

j.

k.

About how much heat is transferred to S2? (Write down an algebraic expression of the

First Law of Thermodynamics and suppose that the state of S2 is altered only a negligible

amount by the passage of I for the calculated time).

How much heat is transferred to S3?

l.

What work is done on S3?

The left-hand figure below gives an overview of the complexity of the large-scale energy

exchanges that contribute to the Earth’s energy budget. The right-hand figure shows sea level

rise measurements over the last century, which have averaged 2 mm per year over the last

century. Use the information on the next page and information you source yourself to

estimate the following :

a.

Estimate the rate at which the energy stored in the world’s oceans is increasing (give

your answer in Joules per year).

b.

Estimate the mean temperature increase in the world’s oceans (give your answer in

degrees per year).

c.

Estimate the fractional change in the energy output of the Sun would be required to

produce the observed changes.

In answering the questions above you should clearly state what simplifying assumptions you

have made. For information not given with the assignment you should clearly identify the

source.

Figures from http://www.nasa.gov/images/content/57911main_Earth_Energy_Budget.jpg and

http://en.wikipedia.org/wiki/Current_sea_level_rise

2

(Potentially) useful information for Question 5

The Earth and Sun :

Surface area of the oceans : 3.6 x 108 km2

Approximate volume of the oceans : 1.3 x 109 km3

Average depth of oceans : 3790 m

Average temperature at a depth of 1000 m is 4°C

Average energy from the Sun at the Earth 1366 Wm-2

Approximate volume of ice in the Greenland iceshelf 2.85 x 106 km3

Properties of water

Density at 4°C (maximum density) 1000 kg m-3

Specific heat water 4.187 kJ kg-1 K-1

Specific heat ice 2.108 kJ kg-1 K-1

Specific heat water vapour 1.996 kJ kg-1 K-1

Specific heat dry air 1.006 kJ kg-1 K-1

Volume expansion coefficient 2.14 x 10-4 K-1

From http://www.engineeringtoolbox.com/water-thermal-properties-d_162.html

3

1.

Using a diagram, explain what is meant by the critical point of a fluid system.

ANSWER

As a liquid is expanded isothermally at temperature T, its

pressure falls, then it boils and finally expands as a vapour.

At a high temperature TH, the fluid behaves as a gas.

The critical isotherm TC divides these two behaviours, and

the critical point, on this isotherm, is when the “boiling”

region has zero length.

Mathematically, the critical point is a horizontal inflection

on the critical isotherm:

” !2 p%

” ! p%

$# ! v ‘& = 0 = $# ! v 2 ‘& .

T

T

4

Critical point

p

TH

TC

T

V

2.

Find expressions for v and T for a van der Waals gas at the critical point and hence evaluate !

1# “v&

,)

at the critical point (where ! is the isobaric coefficient expansivity of a fluid ! = %

v $ ” T (‘ p

SOLUTION

The van der Waals fluid has equation of state

a

( p + 2 )(v ! b) = RT .

v

To find the critical point values vC and TC requires that the van der Waals equation be expressed

to give pressure as: p = RT (v ! b)!1 ! av !2 . The critical point is the point on an isothermal for

” !2 p%

” ! p%

which $

=0=$ 2’ .

# ! v ‘&

# !v &

T

T

Evaluating these and solving for v and T gives

“p

!

= # RT (V # b)#2 + 2aV #3 = 0 (1) and

“V

“2 p

! 2 = 2RT (V # b)#3 # 6aV #4 = 0 (2)

“V

From (1)

2a(V # b)2

(3)

RV 3

Substituting (3) into (2)

T=

2R

2a(V # b)2 6a

$

# 4 =0

(V # b)3

RV 3

V

!

4a

6a

# 4 =0

3

(V # b)V

V

!

4a

6a

= 4

3

(V # b)V

V

! 4V = 6V # 6b

!V = 3b (at the critical point)

Substituting this back into (3) gives

T=

2a(3b # b)2

8a

=

3

27Rb

R(3b)

vC = 3b, and TC = 8a/27Rb.

” !V %

To see what happens to ! at the critical point we first need to find $

# !T ‘& p

5

Arranging the Van der Waal’s equation in terms of T and differentiating at constant pressure,

” !T %

1

(3

(2

$# ! v ‘& = R ((2av )(v ( b) + ( p + av ) .

P

(

)

Algebraic rearrangement, substituting for (p+av -2 ), shows that

” !T %

RTv 3 ( 2a(v ( b)2

=

.

3

$# ! v ‘&

Rv

(v

(

b)

P

(1

” !T %

” !v %

Since $

=

,

# ! v ‘& P $# !T ‘& P

then

!=

( )

RTv -2a ( v-b )

Rv 2 v-b

3

2

Substituting our values of temperature and volume at the critical point we can see that denominator

goes to 0, so from this it follows that at the critical point ! # $

8a

RTV 3 ! 2a(V ! b) =

” (3b)3 ! 2a(3b ! b)2

27Rb

8a

=

” 27b3 ! 8ab2

27b

= 8ab2 ! 8ab2

6

4.

In the diagram, consider the system S1 to

comprise the water, of volume V, the paddles,

and the heater, R. Assume (A1) that the walls,

top and bottom of the container are adiabatically

isolating walls.

(a) What is meant by assumption A1 above?

No heat transfer occurs between S1 and the

surroundings.

M

heater, R

(b) What would be the effect on the system of

lighting the bunsen burner BB

underneath the container?

None, the system is thermally isolated.

h

I

E

BB

Now let the mass M of 600 kg fall through a

height h = 50 m, turning the paddles.

(c) How much work has been done on the

system S1.

W=mgh=600×9.8×50=294 KJ

After some time, the temperature of S1 is measured and the rise in temperature “T calculated

(d) Why is it necessary to wait ‘some time’ before measuring the new temperature of S1?

From your own experience, about how long would you wait?

Thermodynamic equilibrium needs to be established before the temperature of the system can

defined. The time to establish thermodynamic equilibrium would depend upon the size of the

system, but something on the order of minutes would probably be sufficient in most situations.

(e) What modification is necessary to A1 in order to measure the new temperature of S1?

A thermometer must be placed into S1 and brought into thermal equilibrium with S1, this

means that heat will be exchanged between the thermometer and S1, which violates the

adiabatic assumption. The amount of heat exchanged is likely to be small, in which case the

difference between the actual and adiabatic behaviour is negligible. An alternative approach

would be to use the change in the resistance of the heater element to measure the temperature,

however, that will also effect the system as to measure the resistance a voltage is applied and

the current which flows measured.

(f) What is the change in internal energy of the system S1? Why?

U=Q+W=0+294 = 294KJ

Instead of allowing the mass M to fall to produce the “T, a current could be passed through

the resistor.

(g) For how long must a current I = 4A at E = 250 V be passed?

P=IV=4×250=1000W ; P=W/t so t = W/P=294000/1000 = 294s

(h) How much heat is thereby transferred to S1?

0. The current through the element is a macroscopically ordered action and by convention is

considered to be doing work on S1, not transferring heat to it.

8

Now consider the system S1 to be divided into two parts, in thermal contact with each other:

S2 comprising solely the resistor and S3 comprising the rest of S1. If the current is passed

through the resistor as in (g) above,

(i) What work is done on S2?

294KJ

(j) About how much heat is transferred to S2? (Write down an algebraic expression of the

First Law of Thermodynamics and suppose that the state of S2 is altered only a negligible

amount by the passage of I for the calculated time).

U=Q+W ; negligible change of state implies change in U = 0, so Q=-W=-294KJ

(k) How much heat is transferred to S3?

294KJ

(l) What work is done on S3?

0

9

5.

The left-hand figure below gives an overview of the complexity of the large-scale energy

exchanges that contribute to the Earth’s energy budget. The right-hand figure shows sea level

rise measurements over the last century, which have averaged 2 mm per year over the last

century. Use the information on the next page and information you source yourself to

estimate the following :

a.

Estimate the rate at which the energy stored in the world’s oceans is increasing (give

your answer in Joules per year).

b.

Estimate the mean temperature increase in the world’s oceans (give your answer in

degrees per year).

c.

Estimate the fractional change in the energy output of the Sun would be required to

produce the observed changes.

In answering the questions above you should clearly state what simplifying assumptions you

have made. For information not given with the assignment you should clearly identify the

source.

Figures from http://www.nasa.gov/images/content/57911main_Earth_Energy_Budget.jpg and

http://en.wikipedia.org/wiki/Current_sea_level_rise

SOLUTION

a.

There are two potential causes of sea level rise, the thermal expansion of the water as it heats

and an increase in the water volume in the oceans due to melting ice. We haven’t been told

which is the more important, or their relative contributions, so lets look at each in turn and see

what it implies :

Thermal expansion :

The volume expansion coefficient for water is “=2.14 x 10-4 K-1. A sea level rise of 2mm

over the entire ocean area of 3.6 x 108 km2 (3.6 x 1014 m2) corresponds to a volume increase

of Vinc = 3.6 !1014 ! 2 !10″3 = 7.2 !1011 m 3 . Assuming that both the volume expansion

coefficient and the specific heat are approximately constant with temperature over the range

we are interested in (a reasonable approximation), then we can calculate the volume of ocean

which would have to have a 1°C to account for the volume change and multiply by the

specific heat of water to determine the amount of energy required to produce change.

!U = Vice “ice Lice = 7.85#1011 # 917 # 335#103 = 2.41#1020 J

V1! C =

Vinc 7.2 “1011

=

= 3.36 “1015 m 3

! 2.14 “10#4

$ %U = C %T &V1! C = 4.186 “103 “1”1000 ” 3.36 “1015

=1.41″1022 J

Let’s just do a quick sanity check and see how the volume increase we are talking about

compares to the volume of the ocean’s as a whole. The total volume of the oceans is

10

estimated as 1.3 x 109 km3 = 1.3 x 1018 m3. So a volume increase of 7.2 x 1011 m3

corresponds to a fractional increase of 5.5 x 10-7 (i.e. not very much).

Melting of ice

The total volume of the Greenland iceshelf is 2.85 x 1015 m3. So the volume increase

corresponding to the observed sealevel rise corresponds to about 0.025% of the Greenland

iceshelf. So it is plausible that it could be a major contributor. The latent heat of Fusion of

ice is 335 kJ kg-1. Now since ice is less dense than water 7.2 x 1011 m3 of water is produced

by

!

1000

Vice = Vinc water = 7.2 “1011 ”

= 7.85″1011 m 3

!ice

917

The total energy required to melt this much ice is

!U = Vice “ice Lice = 7.85#1011 # 917 # 335#103 = 2.41#1020 J

So significantly less energy is required to produce the volume increase through ice melt than

it is through thermal expansion.

According to the Wikipedia article on “Current Sea Level Rise”, ice melt contributes 0.2 – 0.4

mm per year to the sea level rise (i.e. 10-20%). If we estimate that 15% of the sea level rise

comes from ice melt and the remainder from thermal expansion then

!U = 1.41″1022 ” 0.85+ 2.41″1020 ” 0.15

= 1.20 “1020 J

So our best estimate of the rate at which the energy stored in the world’s oceans is increasing

is 1.2 x 1020 J yr-1. For comparison, in 2005 Australia’s total electricity consumption was

2.198 x 108 MWh (7.91 x 1017 J) (i.e. 15000 times less)

b.

The total ocean volume is 1.3 x 1018 m3, so the energy increase per unit volume

averaged over the whole ocean is

!U 1.41″1022 ” 0.85

=

= 9.219 kJm #3

V

1.3″1018

!U 9.219 “103

$

=

= 9.219 Jkg #3

m

1000

!T !U 9.219

$

=

=

= 0.0022! C

m mC 4187

Now this would only be the case if the oceans fully mixed on timescales of a year

(which isn’t the case, it takes many thousands), so the actual temperature increase will

be mainly in the upper levels of the ocean.

c.

The Earth presents a cross-sectional area of

(

A = ! RE2 = ! ” 6.378″106

)

2

= 1.278″1014 m 2 , so the total energy per second incident

on the Earth’s surface is E = 1.278!1014 !1366 = 1.746 !1017 J . So the additional

energy we estimate to be stored annually in the oceans corresponds to 68740 seconds

(0.796 days) worth of energy from the Sun. This is 0.2% of the annual solar energy

output, so an increase in solar output of 0.2% could account for the observed increase

NOTE: The best evidence that we have is that the cause of the observed sea level rise is

anthropogenic climate change, not variation in solar output, however, the calculation we have

done above demonstrates what an important overall role the Sun plays in the climate. At the

present time the Sun has just come out of one of the quietest periods (as measured by Sunspot

activity) for 100 years, which means if anything in recent years we would expect solar activity

to be lower than average rather than higher.

11

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