Temperature graph slope

In summary, the experiment involves placing block B and block A in a thermally insulated container and changing the temperature of block A in each experiment. The final temperature of the two blocks, Tf, is represented on a graph with a scale of 400 K. The questions are asking for the temperature of block B, TB, and the ratio of the specific heats of the blocks, cB/cA. The slope of the graph is equal to cA/(cA+cB), which can be used to solve for the unknown values.
  • #1
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 [itex]\frac{c_A}{c_A+c_B}[/itex]? Where did that come from?
 
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  • #2
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 [itex]\frac{c_A}{c_A+c_B}[/itex]? 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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  • #3
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:
 
  • #4
ShizukaSm said:
Oh, yes it does! Thanks a lot:smile:
You're most certainly welcome. Good luck with the physics. :wink:
 
  • #5


The slope of a graph represents the rate of change between two variables. In this case, the slope of the graph represents the change in temperature (Tf) with respect to the change in initial temperature (TA). We can use this information to find the ratio of the specific heats of the two blocks.

The specific heat of a substance is the amount of heat required to raise the temperature of one unit mass of the substance by one degree. In this case, we have two blocks (A and B) with the same mass, so we can assume that they have the same specific heat (cA = cB).

Now, in each experiment, block B is initially at a certain temperature TB, but the temperature of block A (TA) is changed. This means that the change in temperature for block A (ΔTA) is equal to the change in temperature for block B (ΔTB).

Using the slope of the graph, we can write the following equation:

Slope = ΔTf/ΔTA = ΔTf/(ΔTA + ΔTB)

Since we know that ΔTA = ΔTB, we can rewrite the equation as:

Slope = ΔTf/2ΔTA

Now, we also know that the final temperature (Tf) is equal to the average of the initial temperatures of block A and B (TA and TB), so we can write:

Tf = (TA + TB)/2

Substituting this into the previous equation, we get:

Slope = (Tf - TB)/ΔTA

Finally, we know that the vertical axis scale is set by Tf = 400 K, so we can write:

Slope = (400 K - TB)/ΔTA

Rearranging this equation, we get:

TB = 400 K - Slope x ΔTA

Now, we can see that the slope is equal to the ratio of the specific heats of the blocks, since the specific heats are the only variables in the equation. Therefore, we can write:

TB = 400 K - (cB/cA) x ΔTA

This is how we can infer that the slope is equal to cB/cA. I hope this helps!
 

Related to Temperature graph slope

1. What is the meaning of the slope on a temperature graph?

The slope on a temperature graph represents the rate of change of temperature over time. A steeper slope indicates a larger change in temperature, while a flatter slope indicates a smaller change.

2. How is the slope of a temperature graph calculated?

The slope of a temperature graph is calculated by dividing the change in temperature by the change in time. This is represented by the equation slope = (change in temperature) / (change in time).

3. What does a positive slope on a temperature graph indicate?

A positive slope on a temperature graph indicates that the temperature is increasing over time. This could mean that the temperature is rising due to external factors such as heat sources or weather patterns.

4. What does a negative slope on a temperature graph indicate?

A negative slope on a temperature graph indicates that the temperature is decreasing over time. This could mean that the temperature is dropping due to external factors such as cooling systems or weather patterns.

5. How can the slope of a temperature graph be used to make predictions?

The slope of a temperature graph can be used to make predictions about future temperature changes. By analyzing the slope, one can determine if the temperature is likely to continue increasing, decreasing, or remain constant in the future. This information can be useful for planning and decision making.

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