# 11.2 Thermal expansion of solids and liquids  (Page 6/10)

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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 $\text{35}\text{.}0\text{º}\text{C}$ . What will its height be on a day when the temperature falls to $–\text{10}\text{.}0\text{º}\text{C}$ ? 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 $\text{15}\text{º}\text{C}$ ? 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 $\text{37}\text{.}0\text{º}\text{C}$ to $\text{40}\text{.}0\text{º}\text{C}$ , assuming the mercury is unconstrained?

$5\text{.}4×{\text{10}}^{-6}\phantom{\rule{0.25em}{0ex}}\text{m}$

How large an expansion gap should be left between steel railroad rails if they may reach a maximum temperature $\text{35}\text{.}0\text{º}\text{C}$ 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 $\text{20}\phantom{\rule{0.25em}{0ex}}\text{m}\phantom{\rule{0.20em}{0ex}}×\phantom{\rule{0.20em}{0ex}}\text{30 m}\text{.}$ 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 $\text{20}\text{º}\text{C}$ above normal?

Because the area gets smaller, the price of the land DECREASES by $\text{~}\text{17},\text{000}\text{.}$

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 $1\text{.}\text{00}\text{º}\text{C}\text{.}$ Note that this calculation is only approximate because ocean warming is not uniform with depth.

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

$\begin{array}{lll}V& =& {V}_{0}+\text{Δ}V={V}_{0}\left(1+\beta \text{Δ}T\right)\\ & =& \left(\text{60}\text{.}\text{00 L}\right)\left[1+\left(\text{950}×{\text{10}}^{-6}/\text{º}\text{C}\right)\left(\text{35}\text{.}0\text{º}\text{C}-\text{15}\text{.}0\text{º}\text{C}\right)\right]\\ & =& \text{61}\text{.}1\phantom{\rule{0.25em}{0ex}}\text{L}\end{array}$

(a) Suppose a meter stick made of steel and one made of invar (an alloy of iron and nickel) are the same length at $0\text{º}\text{C}$ . What is their difference in length at $\text{22}\text{.}0\text{º}\text{C}$ ? (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 $5\text{.}\text{00}\text{º}\text{C,}$ how much will overflow when its temperature reaches $\text{22}\text{.}0\text{º}\text{C}$ ? (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 $\text{10}\text{.}0º\text{C}\text{.}$ What volume of radiator fluid will overflow when the radiator and fluid reach their $\text{95}\text{.}0º\text{C}$ operating temperature, given that the fluid’s volume coefficient of expansion is $\beta =\text{400}×{\text{10}}^{–6}/\text{º}\text{C}$ ? Note that this coefficient is approximate, because most car radiators have operating temperatures of greater than $\text{95}\text{.}0\text{º}\text{C}\text{.}$

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 $\text{350}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{3}$ of coffee is in a 7.00-cm-diameter cup and decreases in temperature from $\text{95}\text{.}0\text{º}\text{C}\phantom{\rule{0.25em}{0ex}}$ to $\phantom{\rule{0.25em}{0ex}}\text{45}\text{.}0\text{º}\text{C}\text{.}$ (Most of the drop in level is actually due to escaping bubbles of air.)

0.832 mm

(a) The density of water at $0\text{º}\text{C}$ is very nearly $\text{1000}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}$ (it is actually $9\text{99}\text{.}{\text{84 kg/m}}^{3}$ ), whereas the density of ice at $0\text{º}\text{C}$ is $9{\text{17 kg/m}}^{3}$ . 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 $\beta \approx 3\alpha ,$ by calculating the change in volume $\text{Δ}V$ of a cube with sides of length $L\text{.}$

We know how the length changes with temperature: $\text{Δ}L={\mathrm{\alpha L}}_{0}\text{Δ}T$ . Also we know that the volume of a cube is related to its length by $V={L}^{3}$ , so the final volume is then $V={V}_{0}+\text{Δ}V={\left({L}_{0}+\text{Δ}L\right)}^{3}$ . Substituting for $\text{Δ}L$ gives

$V={\left({L}_{0}+{\mathrm{\alpha L}}_{0}\text{Δ}T\right)}^{3}={L}_{0}^{3}{\left(1+\alpha \text{Δ}T\right)}^{3}\text{.}$

Now, because $\alpha \text{Δ}T$ is small, we can use the binomial expansion:

$V\approx {L}_{0}^{3}\left(1+3\alpha \Delta T\right)={L}_{0}^{3}+3\alpha {L}_{0}^{3}\text{Δ}T.$

So writing the length terms in terms of volumes gives $V={V}_{0}+\text{Δ}V\approx {V}_{0}+{3\alpha V}_{0}\text{Δ}T,$ and so

$\text{Δ}V={\mathrm{\beta V}}_{0}\text{Δ}T\approx {3\alpha V}_{0}\text{Δ}T,\phantom{\rule{0.25em}{0ex}}\text{or}\phantom{\rule{0.25em}{0ex}}\beta \approx 3\alpha .$

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
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?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
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
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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How we can toraidal magnetic field
How we can create polaidal magnetic field
4
Because I'm writing a report and I would like to be really precise for the references
where did you find the research and the first image (ECG and Blood pressure synchronized)? Thank you!!