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Section summary

  • Thermal expansion is the increase, or decrease, of the size (length, area, or volume) of a body due to a change in temperature.
  • Thermal expansion is large for gases, and relatively small, but not negligible, for liquids and solids.
  • Linear thermal expansion is
    Δ L = αL Δ T , size 12{ΔL=αLΔT} {}
    where Δ L size 12{ΔL} {} is the change in length L size 12{L} {} , Δ T size 12{ΔT} {} is the change in temperature, and α size 12{α} {} is the coefficient of linear expansion, which varies slightly with temperature.
  • The change in area due to thermal expansion is
    Δ A = 2 αA Δ T , size 12{ΔA=2αAΔT} {}
    where Δ A size 12{ΔA} {} is the change in area.
  • The change in volume due to thermal expansion is
    Δ V = βV Δ T , size 12{ΔV=βVΔT} {}
    where β size 12{β} {} is the coefficient of volume expansion and β size 12{β approx 3α} {} . Thermal stress is created when thermal expansion is constrained.

Conceptual questions

Thermal stresses caused by uneven cooling can easily break glass cookware. Explain why Pyrex®, a glass with a small coefficient of linear expansion, is less susceptible.

Water expands significantly when it freezes: a volume increase of about 9% occurs. As a result of this expansion and because of the formation and growth of crystals as water freezes, anywhere from 10% to 30% of biological cells are burst when animal or plant material is frozen. Discuss the implications of this cell damage for the prospect of preserving human bodies by freezing so that they can be thawed at some future date when it is hoped that all diseases are curable.

One method of getting a tight fit, say of a metal peg in a hole in a metal block, is to manufacture the peg slightly larger than the hole. The peg is then inserted when at a different temperature than the block. Should the block be hotter or colder than the peg during insertion? Explain your answer.

Does it really help to run hot water over a tight metal lid on a glass jar before trying to open it? Explain your answer.

Liquids and solids expand with increasing temperature, because the kinetic energy of a body’s atoms and molecules increases. Explain why some materials shrink with increasing temperature.

Problems&Exercises

The height of the Washington Monument is measured to be 170 m on a day when the temperature is 35 . 0 º C size 12{"35" "." 0°C} {} . What will its height be on a day when the temperature falls to 10 . 0 º C size 12{–"10" "." 0°C} {} ? Although the monument is made of limestone, assume that its thermal coefficient of expansion is the same as marble’s.

169.98 m

How much taller does the Eiffel Tower become at the end of a day when the temperature has increased by 15 º C size 12{"15"°C} {} ? Its original height is 321 m and you can assume it is made of steel.

What is the change in length of a 3.00-cm-long column of mercury if its temperature changes from 37 . 0 º C size 12{"37" "." 0°C} {} to 40 . 0 º C size 12{"40" "." 0°C} {} , assuming the mercury is unconstrained?

5 . 4 × 10 6 m size 12{5 "." 4 times "10" rSup { size 8{ - 6} } " m"} {}

How large an expansion gap should be left between steel railroad rails if they may reach a maximum temperature 35 . 0 º C size 12{"35" "." 0°C} {} greater than when they were laid? Their original length is 10.0 m.

You are looking to purchase a small piece of land in Hong Kong. The price is “only” $60,000 per square meter! The land title says the dimensions are 20 m × 30 m . size 12{"20"" m "` times "30 m" "." } {} By how much would the total price change if you measured the parcel with a steel tape measure on a day when the temperature was 20 º C size 12{"20"°C} {} above normal?

Because the area gets smaller, the price of the land DECREASES by ~ $ 17 , 000 . size 12{ "~" $"17","000" "." } {}

Global warming will produce rising sea levels partly due to melting ice caps but also due to the expansion of water as average ocean temperatures rise. To get some idea of the size of this effect, calculate the change in length of a column of water 1.00 km high for a temperature increase of 1 . 00 º C . size 12{1 "." "00"°C "." } {} Note that this calculation is only approximate because ocean warming is not uniform with depth.

Show that 60.0 L of gasoline originally at 15 . 0 º C size 12{"15" "." 0°C} {} will expand to 61.1 L when it warms to 35 . 0 º C, size 12{"35" "." 0°"C,"} {} as claimed in [link] .

V = V 0 + Δ V = V 0 ( 1 + β Δ T ) = ( 60 . 00 L ) 1 + 950 × 10 6 / º C 35 . 0 º C 15 . 0 º C = 61 . 1 L

(a) Suppose a meter stick made of steel and one made of invar (an alloy of iron and nickel) are the same length at 0 º C size 12{0°C} {} . What is their difference in length at 22 . 0 º C size 12{"22" "." 0°C} {} ? (b) Repeat the calculation for two 30.0-m-long surveyor’s tapes.

(a) If a 500-mL glass beaker is filled to the brim with ethyl alcohol at a temperature of 5 . 00 º C, size 12{5 "." "00"°"C,"} {} how much will overflow when its temperature reaches 22 . 0 º C size 12{"22" "." 0°C} {} ? (b) How much less water would overflow under the same conditions?

(a) 9.35 mL

(b) 7.56 mL

Most automobiles have a coolant reservoir to catch radiator fluid that may overflow when the engine is hot. A radiator is made of copper and is filled to its 16.0-L capacity when at 10 . C . size 12{"10" "." 0°C "." } {} What volume of radiator fluid will overflow when the radiator and fluid reach their 95 . C size 12{"95" "." 0°C} {} operating temperature, given that the fluid’s volume coefficient of expansion is β = 400 × 10 6 / º C size 12{β="400"´"10" rSup { size 8{ +- 6} } /°C} {} ? Note that this coefficient is approximate, because most car radiators have operating temperatures of greater than 95 . 0 º C . size 12{"95" "." 0°C "." } {}

A physicist makes a cup of instant coffee and notices that, as the coffee cools, its level drops 3.00 mm in the glass cup. Show that this decrease cannot be due to thermal contraction by calculating the decrease in level if the 350 cm 3 size 12{"350"" cm" rSup { size 8{3} } } {} of coffee is in a 7.00-cm-diameter cup and decreases in temperature from 95 . 0 º C size 12{"95" "." 0°C} {} to 45 . 0 º C . size 12{"45" "." 0°C "." } {} (Most of the drop in level is actually due to escaping bubbles of air.)

0.832 mm

(a) The density of water at 0 º C size 12{0°C} {} is very nearly 1000 kg/m 3 size 12{"1000"" kg/m" rSup { size 8{3} } } {} (it is actually 9 99 . 84 kg/m 3 size 12{9"99" "." "84 kg/m" rSup { size 8{3} } } {} ), whereas the density of ice at 0 º C size 12{0°C} {} is 9 17 kg/m 3 size 12{9"17 kg/m" rSup { size 8{3} } } {} . Calculate the pressure necessary to keep ice from expanding when it freezes, neglecting the effect such a large pressure would have on the freezing temperature. (This problem gives you only an indication of how large the forces associated with freezing water might be.) (b) What are the implications of this result for biological cells that are frozen?

Show that β , size 12{β»3α,} {} by calculating the change in volume Δ V size 12{ΔV} {} of a cube with sides of length L . size 12{L "." } {}

We know how the length changes with temperature: Δ L = αL 0 Δ T size 12{ΔL=αL rSub { size 8{0} } ΔT} {} . Also we know that the volume of a cube is related to its length by V = L 3 size 12{V=L rSup { size 8{3} } } {} , so the final volume is then V = V 0 + Δ V = L 0 + Δ L 3 size 12{V=V rSub { size 8{0} } +ΔV= left (L rSub { size 8{0} } +ΔL right ) rSup { size 8{3} } } {} . Substituting for Δ L size 12{DL} {} gives

V = L 0 + αL 0 Δ T 3 = L 0 3 1 + α Δ T 3 . size 12{ V= left (L rSub { size 8{0} } +αL rSub { size 8{0} } ΔT right ) rSup { size 8{3} } =L rSub { size 8{0} rSup { size 8{3} } } left (1+αΔT right ) rSup { size 8{3} } "." } {}

Now, because α Δ T size 12{αΔT} {} is small, we can use the binomial expansion:

V L 0 3 1 + 3αΔT = L 0 3 + L 0 3 Δ T . size 12{V approx L rSub { size 8{0} } rSup { size 8{3} } left (1+3αΔT right )=L rSub { size 8{0} rSup { size 8{3} } } +3αL rSub { size 8{0} } rSup { size 8{3} } ΔT} {}

So writing the length terms in terms of volumes gives V = V 0 + Δ V V 0 + V 0 Δ T , size 12{V=V rSub { size 8{0} } +ΔV approx V rSub { size 8{0} } +3αV rSub { size 8{0} } ΔT,} {} and so

Δ V = βV 0 Δ T V 0 Δ T , or β . size 12{ΔV=βV rSub { size 8{0} } ΔT approx 3αV rSub { size 8{0} } ΔT,`"or"`β approx 3α} {}

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Concepts of physics. OpenStax CNX. Aug 25, 2015 Download for free at https://legacy.cnx.org/content/col11738/1.5
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