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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?
[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
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?
[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
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.
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?
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
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