# Homework Help: (Thermo) Energy as Taylor expansion

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1. Sep 16, 2015

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