Download calculations of mechanical work in 1, 2 and 3 dimensions

PHY1039
Properties of Matter
Thermodynamic Mechanical Work:
1, 2 and 3 Dimensions
(See Finn’s Thermal Physics, Ch. 2)
27 February and March 1
Lectures 7 and 8
Types of Work
Mechanical: related to force acting through a distance: our main focus
Electrical: electrical current flowing through a resistor under an applied
potential
Capacitive: storing of electric charge under an applied electric field
Magnetic: increasing the magnetic moment of a material with an applied
magnetic field
SI Units of work: Nm (also called a Joule, J)
http://www.nearfield.com/~dan/sports/bike/river/coyote/index.htm
Examples of Mechanical Work
Actuator driven by thermal
expansion of water and air
Force acting through a distance
to do work on surroundings
Change in length, L, under
a constant (?) force
Source: www.ritsumei.ac.jp
Front cover of Scientific American, Oct. 2003
Work can be done on the system by the surroundings or done by the
system on the surroundings.
Example of negative linear expansivity being
exploited to convert chemical energy into work
A chemical reaction with methanol creates heat in the NiTi alloy wire.
At a higher temperature, the wire’s length shrinks - thus lifting a
weight and doing work on the surroundings.
V.H. Ebron et al., Science (2006) 311, 1580
Mechanical Work, W, in One Dimension
𝑑𝑊 = F𝑑𝐿
Initial State
𝐿2
𝑊=
T1
F 𝑑𝐿
𝐿1
+F
L1
Ask yourself: Is F
constant with changing L?
If yes:
𝑊 = F𝐿 = F (𝐿2 − 𝐿1 )
Final State
T2 > T1
IMPORTANT
Sign Convention:
dL
+F
L2
Usually dL is +ve
when T increases!
Positive W: work done on
system by surroundings
Negative W: work done on
surroundings by the system
Isothermal Mechanical Work in One Dimension
𝐿2
𝑊=
Initial State
𝐿1
T1
F=0
L1
F 𝑑𝐿
(or initial
value)
Ask yourself: Is F
constant with changing L?
(What does the equationof-state say?)
If yes:
Final State
𝑊 = F𝐿 = F (𝐿2 − 𝐿1 )
T1
dL
+F
If no: must integrate
over F(L).
L2
See today’s tutorial.
Weak Nano-scale Forces Can be Measured
Atomic force microscope (AFM)
The AFM probe is
exceedingly sharp so that
only a few atoms are at its
tip!
Sensitive to forces on the order of nano-Newtons.
Tip for Atomic Force Microscopy
F
The tip is on a cantilever, which typically has a
spring constant on the order of k = 10 N/m.
Radius of curvature ~ 10 nm
Ideally, one of the atoms
at the tip is slightly above
the others.
Modelled as a simple spring:
F = kz
where z is the deflection in the vertical
direction.
AFM tips from NT-MDT. See www.ntmdt.ru
Doing Work on the Nano-Scale
Force /nN
F
200
Pushing on AFM
probe tip
100
trace
retrace
0
-100
-200
Pulling on the AFM
probe tip
-300
-400
0
2
4
Distance/ m
6
8
L
Emulsion: Droplet of one liquid dispersed in another
Interface with a
tension of G
Oil
Do work on system
(e.g. by shaking)
Water
Interface with a
tension of G
Oil
Water
Water
Work to create emulsion
droplets:
𝐴
𝑊=
Γ𝑑𝐴
𝐴𝑜
Oil
See emulsion droplet formation:
http://youtu.be/9y7J3CBe1eI
Physics of a “Bubble Jet” Printer
Question for Tutorial: How much work is required to create a droplet of ink?
Isothermal Mechanical Work in Two Dimensions
Initial State
ℓ
For each surface:
T1
F
Γ=
ℓ
A1
𝑑𝑊 = F𝑑𝐿 = Γℓ𝑑𝐿
𝑑𝑊 = Γ𝑑𝐴
𝐴2
𝑊=
𝐴1
L
Final State
ℓ
A2
G𝑑𝐴
Membrane: two sides
Surface: one side only
T1
dA
dL
F
Ask yourself: Is G
constant with changing A?
If yes:
𝑊 = G𝐴 = G(𝐴2 − 𝐴1 )
Surface Tension of Molecular Layers
Langmuir trough
water
G
G is a function of A!
http://www.engineerdir.com/product/catalog/
16626/
Area adjusted with barriers
barrier
A
http://www.augsburg.edu/ppages/~stottrup/ http://www.reflec.ameslab.gov/aboutus/monolayers.html
Exptech.html
Isobaric Mechanical Work in Three Dimensions
Initial State
F
Isobaric = constant P
A
A
V1
F 𝐴𝑝𝑝𝑙𝑖𝑒𝑑 𝑓𝑜𝑟𝑐𝑒
P= =
𝐴
T1
Final State
F
A
A
V2
+dL
𝑑𝑊 = −𝑃𝑑𝑉
𝐴𝑟𝑒𝑎
Positive P is pressing
inward.
But F pressing
inwards is negative!
Only applies to
reversible
processes!
T2 > T1
𝑑𝑊 = F𝑑𝐿 = −𝑃𝐴𝑑𝐿
𝑉2
𝑊=−
𝑃𝑑𝑉
𝑉1
Ask yourself: Is P
constant with changing V?
If yes:
𝑊 = −𝑃𝑉 = −𝑃(𝑉2 − 𝑉1 )
Question: Is this result consistent with our
sign convention?
Isobaric Conditions
Earth’s Atmospheric Pressure
Height (km)
Crab Nebula
Pressure is low but roughly constant!
0
1
2
3
4
5
6
7
8
9
10
Pressure (kPa)
101.3
89.9
79.5
70.1
61.6
54.0
47.2
41.1
35.6
30.7
26.4
http://en.wikipedia.org/wiki/File:Crab_Nebula.jpg
Isochoric = constant volume. How much work is done in an isochoric process?
Isothermal Mechanical Work in Three Dimensions
Initial State
Isothermal = constant T
A
A
𝑉2
T1
𝑊=−
𝑉1
V1
Apply a pressure to
the system and
compress it to a
smaller V.
Final State
FA
A
V2
𝑃𝑑𝑉
-dL
T1
Ask yourself: Is P
constant with changing V?
(What does the equation-ofstate say?)
Work can be done on solid, liquid and gas
systems by compressing them.
Work is done on the surroundings by a
system when it expands.
CO Map of the Universe
This is the first-ever all-sky map of carbon monoxide in the cosmos. The Planck
space telescope was designed to look at the background glow in the cosmos in an
effort to understand how it formed. Coincidentally, scientists have found that it can
help to spot star-forming regions where carbon monoxide glows brightly despite its
low abundance
http://www.bbc.co.uk/news/science-environment-17027949
Adiabatic Free Expansion of a Gas
V1
V2
Vacuum
Compressive (Tensile) Force from
Thermal Expansion of Beams
Applying the “Thermodynamic Method” to
Problem Solving
Wire pulled in tension between two walls:
T1
F1
F1
L1
Question: What is the increase on the tension of the wire
when it is cooled from a temperature of T1 to T2?
(F1, L1, T1)
(F2, L1, T2)