How is the slope of the temperature graph determined?

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Homework Help Overview

The discussion revolves around determining the slope of a temperature graph related to thermal equilibrium between two blocks with equal mass but different initial temperatures. The problem involves analyzing the relationship between the final temperature and the initial temperatures of the blocks, specifically focusing on the specific heats of the blocks.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the derivation of the slope of the temperature graph, questioning how it relates to the specific heats of the blocks. There is an exploration of the expression for final temperature in terms of initial temperatures and specific heats.

Discussion Status

Some participants have expressed understanding of the mathematical expression provided, indicating a productive exchange of ideas. However, there remains a focus on clarifying the reasoning behind the slope's derivation, suggesting ongoing exploration of the topic.

Contextual Notes

The problem is framed within a series of experiments with specific constraints, such as the thermal insulation of the container and the equal mass of the blocks. The discussion reflects on the assumptions related to specific heats and their implications for the temperature graph.

ShizukaSm
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In a series of experiments, block B is to be placed in a thermally insulated container with
block A, which has the same mass as blockB. In each experiment, block B is initially at a certain temperature TB, but temperature TA of block A is changed from experiment to experiment. Let Tf
represent the final temperature of the two blocks when they reach thermal equilibrium in any of the experiments. The graph(attatched) gives temperature Tf versus the initial temperature TA for a range of possible values of TA, from TA = 0 K to TA = 500 K. The vertical axis scale is set by Tfs= 400 K. What are:
(a)temperature TB.
(b) the ratio cB/cA of the specific heats of the blocks?
sfa.JPG


Ok so, I was able to solve this problem, however, my book answer used a method that I did not understand:
Slope.JPG


How can he infer that the slope is equal to \frac{c_A}{c_A+c_B}? Where did that come from?
 
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ShizukaSm said:
In a series of experiments, block B is to be placed in a thermally insulated container with
block A, which has the same mass as blockB. In each experiment, block B is initially at a certain temperature TB, but temperature TA of block A is changed from experiment to experiment. Let Tf
represent the final temperature of the two blocks when they reach thermal equilibrium in any of the experiments. The graph(attatched) gives temperature Tf versus the initial temperature TA for a range of possible values of TA, from TA = 0 K to TA = 500 K. The vertical axis scale is set by Tfs= 400 K. What are:
(a)temperature TB.
(b) the ratio cB/cA of the specific heats of the blocks?
View attachment 60477

Ok so, I was able to solve this problem, however, my book answer used a method that I did not understand:
View attachment 60478

How can he infer that the slope is equal to \frac{c_A}{c_A+c_B}? Where did that come from?
Rewriting the expression as ##T_f=\left(\frac{c_A}{c_A+c_B}\right) T_A+\left(\frac{c_B}{c_A+c_B}\right) T_B## might help. :wink:
 
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Mandelbroth said:
Rewriting the expression as ##T_f=\left(\frac{c_A}{c_A+c_B}\right) T_A+\left(\frac{c_B}{c_A+c_B}\right) T_B## might help. :wink:

Oh, yes it does! Thanks a lot:smile:
 
ShizukaSm said:
Oh, yes it does! Thanks a lot:smile:
You're most certainly welcome. Good luck with the physics. :wink:
 

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