What is the correct solution for the temperature expansion problem?

In summary, the conversation discussed finding the temperature change needed to raise the pressure in a steel container filled with alcohol by 6 bar. It involved using the compressibility and temperature coefficients of expansion for alcohol and steel in a calculation, which resulted in a solution of 0.97K. However, the answer in the book was listed as 9.7K, indicating a possible typo in the back of the book.
  • #1
bolzano95
89
7

Homework Statement


A steel container is filled with alcohol. For how much kelvins we have to raise the temperature for the pressure to be raised by 6 bar?
Compressibility of alcohol is
[tex]
\kappa=1.15 \cdot 10^{-4} bar^{-1}
[/tex]
temperature coefficient of volume expansion of alcohol is
[tex]
\beta_a= 7.5 \cdot 10^{-4} K^{-1}
[/tex]
temperature coefficient of length expansion of steel is
[tex]
\alpha_s= 1.2 \cdot 10^{-5} K^{-1}
[/tex]

Homework Equations


Look at attempt at solution.

The Attempt at a Solution


I solved this problem like this:
[tex]
\frac{\Delta V}{V}=(\beta_a-3\alpha)\Delta T\\
\frac{\Delta V}{V}=\kappa\Delta p\\
\frac{\kappa\Delta p}{\beta_a-3\alpha}=\Delta T= 0.97K\\
[/tex]
But the solution is wrong. The correct answer is 9.7K. I'm interested where I made a mistake.
 
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  • #2
It looks like your answer is off by a factor of 10. This may be due to a typo or an error in the conversion of compressibility units to bar-1.
 
  • #3
I checked and get 0.97K. Do you think the solving process is correct?
 
  • #4
The method is correct. If you are certain that the input numbers are also correct, then the answer in the back of the book may be a typo.
 

1. What is temperature expansion?

Temperature expansion is the phenomenon where a material expands or contracts when its temperature changes. This is due to the movement of molecules within the material, causing it to take up more or less space.

2. Why does temperature expansion occur?

Temperature expansion occurs because of the kinetic energy of molecules. As the temperature increases, the molecules have more energy and move further apart, causing the material to expand. Similarly, as the temperature decreases, the molecules lose energy and move closer together, causing the material to contract.

3. How does temperature affect the expansion of different materials?

Temperature affects the expansion of different materials differently. Some materials, like metals, expand more with an increase in temperature, while others, like water, have a unique behavior where they expand when their temperature decreases. This is known as thermal expansion and contraction.

4. What are the applications of temperature expansion in everyday life?

Temperature expansion has various applications in everyday life. It is used in the design of bridges, roads, and buildings, where materials need to be able to expand and contract with changes in temperature without causing structural damage. It is also used in thermometers, thermostats, and other temperature-sensitive devices.

5. How is temperature expansion measured and calculated?

Temperature expansion is measured using a device called a dilatometer, which measures the change in length or volume of a material with a change in temperature. The coefficient of thermal expansion (CTE) is used to calculate the amount of expansion or contraction that will occur in a material with a change in temperature. CTE is a measure of how much a material will expand or contract per unit of temperature change.

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