# (Thermo) Energy as Taylor expansion

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1. Sep 16, 2015

### RadiumBlue

1. The problem statement, all variables and given/known data
I've attached a screenshot of the problem, which will probably provide much better context than my retelling. I'm having problems with parts f and g. The most relevant piece of information is:

"To get used to the process of Taylor expansions in two variables, first we will let $V$ be a constant. For the following three functions compute $\Delta E$ in terms of $\Delta T$ with $V$ constant.

1:$E = \alpha V T^{17}$ where $\alpha$ is a constant. "
...
etc.

2. Relevant equations
Taylor series
$\Delta E = \frac{dE}{dT} \Delta T$

3. The attempt at a solution
My problem with this question is I'm not quite sure what it's asking/what answer it wants. Does it want just the first two terms of the taylor expansion for each equation, using V as a constant?

I solved part e this way:

Taylor expansion of E with respect to T:
$E(T) = E(T_i) + \frac{dE}{dT} (T-T_i) ...$

Using only the linear term as the problem states, and subtracting E(T_i)

$E(T) - E(T_i) = \frac{dE}{dT} (T-T_i)$

Substituting

$\Delta E = \frac{dE}{dT} \Delta T$

I don't know how to proceed for part F. Would it be this for the first equation?

$\alpha V T^{17} = \alpha V (T_i)^{17} + 17\alpha V T^{16} (T-(T_i))$

$\alpha V T^{17} - \alpha V (T_i)^{17} = 17\alpha V T^{16} (\Delta T)$

and so forth? Or is it more complicated than that?

#### Attached Files:

• ###### thermo1.png
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2. Sep 21, 2015

### Staff: Admin

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

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