Thermodynamics B/C

Test your knowledge of various Science Olympiad events.
User avatar
WhatScience?
Member
Member
Posts: 394
Joined: July 16th, 2017, 4:03 pm
Division: C
State: NJ

Re: Thermodynamics B/C

Postby WhatScience? » February 8th, 2018, 1:26 pm

I used to feel this way about Physics too, but once you understand the concepts involved formerly impossible questions appear quite simple, and the whole learning process becomes very fulfilling.
This is why I like physics more than bio.

I dropped A and P and Microbes because it was just too much memorization. Thermo at least feels more like a science, where everything has a logical solution.
Did you just...
imply...
that microbio...
isn’t a science...
I did NOT

When I say feels like a science, I mean the logic part.

This is just what I enjoy in a science, not what makes up a science.
"When you clean your room, you are increasing the total chaos of the universe" - Hank Green Crash Course (Entropy)

Events 2018

Thermodynamics, Potions and Poisons, Disease Detectives, Optics, and Towers

UTF-8 U+6211 U+662F
Exalted Member
Exalted Member
Posts: 1491
Joined: January 18th, 2015, 7:42 am
Division: C
State: PA

Re: Thermodynamics B/C

Postby UTF-8 U+6211 U+662F » February 8th, 2018, 4:46 pm

Okay, I'll ask the next one, and if someone else wants to give the rest of the answers for the other question, he/she can?

A mathematician homeowner measures temperatures in a temperature unit he created called S. This system has units on a logarithmic scale, so heating something by 1 degree Celsius would cause the temperature in S to multiply by some constant factor. This homeowner heats his house at a rate of 100 kW when it is 10 S out to maintain an inside temperature of 90 S. If it is 5 S outside, how much heat (in kJ) must he add to his house per minute to maintain an inside temperature of 405 S (assuming the same conditions as before).

User avatar
Justin72835
Member
Member
Posts: 175
Joined: June 25th, 2017, 7:06 am
Division: Grad
State: TX

Re: Thermodynamics B/C

Postby Justin72835 » February 8th, 2018, 5:19 pm

Okay, I'll ask the next one, and if someone else wants to give the rest of the answers for the other question, he/she can?

A mathematician homeowner measures temperatures in a temperature unit he created called S. This system has units on a logarithmic scale, so heating something by 1 degree Celsius would cause the temperature in S to multiply by some constant factor. This homeowner heats his house at a rate of 100 kW when it is 10 S out to maintain an inside temperature of 90 S. If it is 5 S outside, how much heat (in kJ) must he add to his house per minute to maintain an inside temperature of 405 S (assuming the same conditions as before).
Pretty creative question! I'm not 100% sure on this one (my Algebra 2 skills are a bit rusty) but I'll give it a shot.

The rate of heat that must be supplied is proportional to the change in temperature:



Since this equation works only if the scale is linear, we have to modify it a bit:



If we apply this to the second scenario:



Cross-multiplying gives:



Then to get heat transferred per minute, just multiply by 60 and change the units a bit:

"The fault, dear Brutus, is not in our stars,
But in ourselves, that we are underlings."


University of Texas at Austin '23
Seven Lakes High School '19

UTF-8 U+6211 U+662F
Exalted Member
Exalted Member
Posts: 1491
Joined: January 18th, 2015, 7:42 am
Division: C
State: PA

Re: Thermodynamics B/C

Postby UTF-8 U+6211 U+662F » February 8th, 2018, 5:51 pm

Okay, I'll ask the next one, and if someone else wants to give the rest of the answers for the other question, he/she can?

A mathematician homeowner measures temperatures in a temperature unit he created called S. This system has units on a logarithmic scale, so heating something by 1 degree Celsius would cause the temperature in S to multiply by some constant factor. This homeowner heats his house at a rate of 100 kW when it is 10 S out to maintain an inside temperature of 90 S. If it is 5 S outside, how much heat (in kJ) must he add to his house per minute to maintain an inside temperature of 405 S (assuming the same conditions as before).
Pretty creative question! I'm not 100% sure on this one (my Algebra 2 skills are a bit rusty) but I'll give it a shot.

The rate of heat that must be supplied is proportional to the change in temperature:



Since this equation works only if the scale is linear, we have to modify it a bit:



If we apply this to the second scenario:



Cross-multiplying gives:



Then to get heat transferred per minute, just multiply by 60 and change the units a bit:

Overall right, but pay attention to the question details (I bolded the part of the question I'm referring to specifically). Your turn!

User avatar
Justin72835
Member
Member
Posts: 175
Joined: June 25th, 2017, 7:06 am
Division: Grad
State: TX

Re: Thermodynamics B/C

Postby Justin72835 » February 8th, 2018, 6:50 pm

Overall right, but pay attention to the question details (I bolded the part of the question I'm referring to specifically). Your turn!
Oh yeah, looks like I completely missed that part of the question :/.

You have a well-insulated rigid container separated into two equal halves by a small removable membrane of negligible volume. On one side of the membrane, you have 3 moles of a monatomic gas at a temperature of 25K. On the other side, you have a vacuum. If the membrane separating the two halves was quickly removed, allowing the gas to expand into the other half of the container, calculate the change in entropy of the gas.

Note that because the gas expanded by itself (called free expansion), no work was done by the gas. Also, because the container was insulated and the gas expanded in just a fraction of a second, you can assume that no heat was transferred. Lastly, because of the last two conditions, you know that the gas experienced no change in temeprature.
"The fault, dear Brutus, is not in our stars,
But in ourselves, that we are underlings."


University of Texas at Austin '23
Seven Lakes High School '19

UTF-8 U+6211 U+662F
Exalted Member
Exalted Member
Posts: 1491
Joined: January 18th, 2015, 7:42 am
Division: C
State: PA

Re: Thermodynamics B/C

Postby UTF-8 U+6211 U+662F » February 8th, 2018, 7:05 pm

Overall right, but pay attention to the question details (I bolded the part of the question I'm referring to specifically). Your turn!
Oh yeah, looks like I completely missed that part of the question :/.

You have a well-insulated rigid container separated into two equal halves by a small removable membrane of negligible volume. On one side of the membrane, you have 3 moles of a monatomic gas at a temperature of 25K. On the other side, you have a vacuum. If the membrane separating the two halves was quickly removed, allowing the gas to expand into the other half of the container, calculate the change in entropy of the gas.

Note that because the gas expanded by itself (called free expansion), no work was done by the gas. Also, because the container was insulated and the gas expanded in just a fraction of a second, you can assume that no heat was transferred. Lastly, because of the last two conditions, you know that the gas experienced no change in temeprature.
Answer?
S = klnW The number of states doubles for each particle, so overall, there are 2^N times W in the new container. S = kln(2^N * W) = k * (Nln2 + lnW) = kNln2 + klnW Change in S = kNln2 = 3 mol * R * ln 2 = [b]17.29 J/K[/b] As far as I know, the temperature isn't relevant as long as it remains constant.

User avatar
Justin72835
Member
Member
Posts: 175
Joined: June 25th, 2017, 7:06 am
Division: Grad
State: TX

Re: Thermodynamics B/C

Postby Justin72835 » February 8th, 2018, 7:28 pm

Overall right, but pay attention to the question details (I bolded the part of the question I'm referring to specifically). Your turn!
Oh yeah, looks like I completely missed that part of the question :/.

You have a well-insulated rigid container separated into two equal halves by a small removable membrane of negligible volume. On one side of the membrane, you have 3 moles of a monatomic gas at a temperature of 25K. On the other side, you have a vacuum. If the membrane separating the two halves was quickly removed, allowing the gas to expand into the other half of the container, calculate the change in entropy of the gas.

Note that because the gas expanded by itself (called free expansion), no work was done by the gas. Also, because the container was insulated and the gas expanded in just a fraction of a second, you can assume that no heat was transferred. Lastly, because of the last two conditions, you know that the gas experienced no change in temeprature.
Answer?
S = klnW The number of states doubles for each particle, so overall, there are 2^N times W in the new container. S = kln(2^N * W) = k * (Nln2 + lnW) = kNln2 + klnW Change in S = kNln2 = 3 mol * R * ln 2 = [b]17.29 J/K[/b] As far as I know, the temperature isn't relevant as long as it remains constant.
Correct. Dang, that's a really cool way of doing that problem. I was thinking that you could do it this way:



But I think the method you used was much more elegant. Your turn!

Edit: Can you explain how you knew that there were 2^N times W microstates in the new container? Thanks.
"The fault, dear Brutus, is not in our stars,
But in ourselves, that we are underlings."


University of Texas at Austin '23
Seven Lakes High School '19

UTF-8 U+6211 U+662F
Exalted Member
Exalted Member
Posts: 1491
Joined: January 18th, 2015, 7:42 am
Division: C
State: PA

Re: Thermodynamics B/C

Postby UTF-8 U+6211 U+662F » February 9th, 2018, 4:26 pm

Can you explain how you knew that there were 2^N times W microstates in the new container? Thanks.
Each particle has two times the number of places it can be, thus 2 times the number of microstates for each particle, thus 2^N (because of how probability works).

Lord Kelvin collaborates with a Russian scientist but notices there is a discrepancy in their data! Lord Kelvin measured his temperature in Kelvin, and the Russian scientist measured his temperature in Réaumur! How can Lord Kelvin figure out the temperature in Kelvin based on this Russian scientist's measurements?

User avatar
Justin72835
Member
Member
Posts: 175
Joined: June 25th, 2017, 7:06 am
Division: Grad
State: TX

Re: Thermodynamics B/C

Postby Justin72835 » February 10th, 2018, 10:21 am

Can you explain how you knew that there were 2^N times W microstates in the new container? Thanks.
Each particle has two times the number of places it can be, thus 2 times the number of microstates for each particle, thus 2^N (because of how probability works).

Lord Kelvin collaborates with a Russian scientist but notices there is a discrepancy in their data! Lord Kelvin measured his temperature in Kelvin, and the Russian scientist measured his temperature in Réaumur! How can Lord Kelvin figure out the temperature in Kelvin based on this Russian scientist's measurements?
Answer
I'm pretty sure that for this problem you would just use an equation to convert between the two temperatures. For converting from Kelvin to Réaumur: R = (K - 273.15)*0.8 For converting from Réaumur to Kelvin: K = 1.25R + 273.15
"The fault, dear Brutus, is not in our stars,
But in ourselves, that we are underlings."


University of Texas at Austin '23
Seven Lakes High School '19

UTF-8 U+6211 U+662F
Exalted Member
Exalted Member
Posts: 1491
Joined: January 18th, 2015, 7:42 am
Division: C
State: PA

Re: Thermodynamics B/C

Postby UTF-8 U+6211 U+662F » February 10th, 2018, 10:57 am

Answer
I'm pretty sure that for this problem you would just use an equation to convert between the two temperatures. For converting from Kelvin to Réaumur: R = (K - 273.15)*0.8 For converting from Réaumur to Kelvin: K = 1.25R + 273.15
Correct, your turn!

User avatar
Justin72835
Member
Member
Posts: 175
Joined: June 25th, 2017, 7:06 am
Division: Grad
State: TX

Re: Thermodynamics B/C

Postby Justin72835 » February 10th, 2018, 11:17 am

Answer
I'm pretty sure that for this problem you would just use an equation to convert between the two temperatures. For converting from Kelvin to Réaumur: R = (K - 273.15)*0.8 For converting from Réaumur to Kelvin: K = 1.25R + 273.15
Correct, your turn!
You have a heat refrigerator that requires 1340 W of work. You also find that the refrigerator releases heat into a heat sink at a rate of 4250 W. How much heat can the refrigerator extract from a cold reservoir and what is its coefficient of performance?
"The fault, dear Brutus, is not in our stars,
But in ourselves, that we are underlings."


University of Texas at Austin '23
Seven Lakes High School '19

UTF-8 U+6211 U+662F
Exalted Member
Exalted Member
Posts: 1491
Joined: January 18th, 2015, 7:42 am
Division: C
State: PA

Re: Thermodynamics B/C

Postby UTF-8 U+6211 U+662F » February 10th, 2018, 11:30 am

You have a heat refrigerator that requires 1340 W of work. You also find that the refrigerator releases heat into a heat sink at a rate of 4250 W. How much heat can the refrigerator extract from a cold reservoir and what is its coefficient of performance?
Answer
First Law of Thermodynamics: [math]1340 J + Q_c = 4250 J[/math]. [math]Q_c = 2910 J[/math]. [math]COP = \frac{Q_h}{W} = \frac{4250 J}{1340 J} = 3.17[/math]. [b]2910 W[/b] and [b]3.17[/b]

User avatar
Justin72835
Member
Member
Posts: 175
Joined: June 25th, 2017, 7:06 am
Division: Grad
State: TX

Re: Thermodynamics B/C

Postby Justin72835 » February 10th, 2018, 2:09 pm

You have a heat refrigerator that requires 1340 W of work. You also find that the refrigerator releases heat into a heat sink at a rate of 4250 W. How much heat can the refrigerator extract from a cold reservoir and what is its coefficient of performance?
Answer
First Law of Thermodynamics: [math]1340 J + Q_c = 4250 J[/math]. [math]Q_c = 2910 J[/math]. [math]COP = \frac{Q_h}{W} = \frac{4250 J}{1340 J} = 3.17[/math]. [b]2910 W[/b] and [b]3.17[/b]
Overall correct but since we're dealing with a heat refrigerator the formula is actually COP = Qc/W. This is because we are interested in the heat we can extract (Qc) rather than the heat given off at the very end (Qc). Everything is else was good. Your turn!
"The fault, dear Brutus, is not in our stars,
But in ourselves, that we are underlings."


University of Texas at Austin '23
Seven Lakes High School '19

UTF-8 U+6211 U+662F
Exalted Member
Exalted Member
Posts: 1491
Joined: January 18th, 2015, 7:42 am
Division: C
State: PA

Re: Thermodynamics B/C

Postby UTF-8 U+6211 U+662F » February 10th, 2018, 7:19 pm

A 500. g iron rod at 50. degrees Celsius is dropped into a (very big) beaker filled with 2.0 L of water at 20. degrees Celsius. 80.0% of the heat transferred from the iron rod is lost to the surroundings and is not transferred to the water. Find the power in watts of the transfer of energy into the water if the system reaches thermal equilibrium in exactly one minute.

Use the values of c = 0.50 J/(g*K) for iron and c = 4.0 J/(g*K) for water. Use appropriate significant figures.

User avatar
Justin72835
Member
Member
Posts: 175
Joined: June 25th, 2017, 7:06 am
Division: Grad
State: TX

Re: Thermodynamics B/C

Postby Justin72835 » February 10th, 2018, 8:43 pm

A 500. g iron rod at 50. degrees Celsius is dropped into a (very big) beaker filled with 2.0 L of water at 20. degrees Celsius. 80.0% of the heat transferred from the iron rod is lost to the surroundings and is not transferred to the water. Find the power in watts of the transfer of energy into the water if the system reaches thermal equilibrium in exactly one minute.

Use the values of c = 0.50 J/(g*K) for iron and c = 4.0 J/(g*K) for water. Use appropriate significant figures.
Answer
Since the 80% heat lost by the iron rod goes into the surroundings and 20% of it goes into changing the water's temperature, we get the following equation: [math]Q_{iron}=-(Q_{water}+Q_{surroundings})=-5Q_{water}[/math] Now we can plug in all our known values to find the equilibrium temperature, then the heat transferred: [math]m_{iron}c_{iron}(T_f-50)=-5m_{water}c_{water}(T_f-20)[/math] where Tf = 20.1863 degrees Celsius. Plugging this value back into the equation, you find that the Qiron = 7453.42 J. Dividing this value by 60 gives P = 124 W.
"The fault, dear Brutus, is not in our stars,
But in ourselves, that we are underlings."


University of Texas at Austin '23
Seven Lakes High School '19


Return to “2018 Question Marathons”

Who is online

Users browsing this forum: No registered users and 5 guests