# Volume expansion and pressure change

fish
A brass cube, 10 cm on a side, is heated with a temperature change of 200 deg. C. By what percentage does its volume change?

Vo = 10cm = 1000cm^3 = .001m^3
T = 200 C
brass = 19x10^-6 coefficient of linear expansion

V=Vo(1+3 x brass x T)
or
change in V = Vo(3 x brass x T)
= .001m^3(3 x 19x10^-6 x 200 C)
= .0000114 = 1.14x10^-5

% change = change in V/Vo
= .0000114/.001
= .0114 = 1.1%

answer in the book is 1.1x10^-3 %

What I am doing wrong here?

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A radial tire is inflated to a gauge pressure of 35 lb/in^2 at 60 deg. F.
If the temperature increases to 100 deg F while the volume of the
tire remains constant, what is the tire's new pressure?

T1=60 F = 289 K
T2=100 F = 311 K
V1=V2
P1= 35 lb/in^2
Find P2

P1/T1=P2/T2
P2=P1T2/T1

P2 = (35 lb/in^2 x 311 K)/289 K
P2 = 37.7 lb/in^2

Also tried this:

P=pa + pg
14.7 (atm) + 35 lb/in^2
= 49.7 lb/in^2 = absolute pressure

P2 = (49.7 lb/in^2 x 311 K)/289 K
P2 = 53.5 lb/in^2

answer in the book is 39 lb/in^2

any help would be appreciated

gnome
In the first one:

You wrote it correctly here:
V=Vo(1+3 x brass x T)
but when you put in the numbers you lost the "1 +":
= .001m^3(3 x 19x10^-6 x 200 C)
Fix that & you'll get the correct answer.

[Edited to delete erroneous comment about question 2.]

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Homework Helper
Gold Member
Brass? You're fine, the answer key is in error --- lotsa luck with your teacher on that detail.

Tire pressure? The T conversion is fine, the answer key is correct, and you have omitted ONE step in one of the two calculations you did --- the other needs more than one correction.

Need a hint beyond that?

gnome
For the brass:

No, the answer key is correct.

this line:
= .001m^3(3 x 19x10^-6 x 200 C)

should be

= .001m^3(1 + 3 x 19x10^-6 x 200 C)

Fix that, & you will get the same answer that the book gives.

For the tire:
Sorry about my erroneous comment, which I'll now delete to avoid confusion. (As Bystander said, you've got it, if you just do one more step.)

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Homework Helper
Gold Member
A 19 ppm/K LINEAR expansion coefficient integrated over 200K produces a 10 ppm change in VOLUME? Not on this planet, not in this solar system, ....

You are seeing a very common phenomenon occurring in answer keys --- student help working the problems and not being proofread by textbook authors.

Ambitwistor
Originally posted by gnome
For the brass:

No, the answer key is correct.

No, it isn't.

this line:
= .001m^3(3 x 19x10^-6 x 200 C)

should be

= .001m^3(1 + 3 x 19x10^-6 x 200 C)

Fine. Then you get V = 0.0010114 m3. If you calculate the percent change in V, that's

&Delta;V/V0 x 100% = (V-V0)/V0 x 100% = 1.14%

as fish obtained.

(Actually, you can get it quicker by just noting that the coefficient of volume expansion &beta; is defined by &Delta;V/V0 = &beta;&Delta;T; if you use the fact that &beta; = 3&alpha; where &alpha; is the linear coefficient fish calls "brass", you immediately obtain &Delta;V/V = 3&alpha;&Delta;T = 0.0114.)

gnome
Arggghh!!

Sorry, I read it too quickly & didn't realized he was shortcutting to just the change in volume when he wrote ".001m^3(3 x 19x10^-6 x 200 C)"

Beyond that, I'll just plead insanity, or something....(& get back to my own homework)

Ambitwistor
I bet the solutions manual mistakenly listed the change in volume (nonsensically multiplied by 100), instead of the fractional change in volume.

fish
For the tire pressure,
P2 = 53.5 lb/in^2 - 14.7 lb/in^2 = 38.8 = 39 lb/in^2

In the preface of the book, it says each solution in the solutions manual has been checked for accuracy by a minimum of 5 instructors. I've found other errors in the answer keys and also errors in the example problems of the main textbook. What Bystander said was pretty interesting...not in this solar system

thanks for the help guys