Let's analyze the relationship [tex]\( S = aV^2 \ln E \)[/tex] where [tex]\(a\)[/tex] is a constant, and address each part of the question:
### Part (a)
To find [tex]\(\frac{1}{T}\)[/tex], we need to determine [tex]\(\left(\frac{\partial S}{\partial E}\right)_V\)[/tex].
Given [tex]\( S = aV^2 \ln E \)[/tex],
1. Differentiate [tex]\( S \)[/tex] with respect to [tex]\( E \)[/tex]:
[tex]\[
\left( \frac{\partial S}{\partial E} \right)_V = \frac{\partial}{\partial E} (aV^2 \ln E)
\][/tex]
2. The derivative of [tex]\( \ln E \)[/tex] with respect to [tex]\( E \)[/tex] is [tex]\(\frac{1}{E}\)[/tex]:
[tex]\[
\left( \frac{\partial S}{\partial E} \right)_V = aV^2 \cdot \frac{1}{E}
\][/tex]
Simplifying,
[tex]\[
\left( \frac{\partial S}{\partial E} \right)_V = \frac{aV^2}{E}
\][/tex]
Therefore,
[tex]\[
\frac{1}{T} = \left( \frac{\partial S}{\partial E} \right)_V = \frac{aV^2}{E}
\][/tex]
And thus,
[tex]\[
T = \frac{E}{aV^2}
\][/tex]
### Part (b)
To find [tex]\(\frac{p}{T}\)[/tex], we need to determine [tex]\(\left(\frac{\partial S}{\partial V}\right)_E\)[/tex].
Given [tex]\( S = aV^2 \ln E \)[/tex],
1. Differentiate [tex]\( S \)[/tex] with respect to [tex]\( V \)[/tex]:
[tex]\[
\left( \frac{\partial S}{\partial V} \right)_E = \frac{\partial}{\partial V} (aV^2 \ln E)
\][/tex]
2. Using the product rule:
[tex]\[
\left( \frac{\partial S}{\partial V} \right)_E = 2aV \ln E
\][/tex]
Therefore,
[tex]\[
\frac{p}{T} = \left( \frac{\partial S}{\partial V} \right)_E \Rightarrow \frac{p}{T} = 2aV \ln E
\][/tex]
This means,
[tex]\[
p = T \cdot (2aV \ln E)
\][/tex]
### Part (c)
To find [tex]\(\left( \frac{\partial E}{\partial V} \right)_T\)[/tex], we consider the condition where temperature [tex]\( T \)[/tex] is kept constant. Using [tex]\( T = \frac{E}{aV^2} \)[/tex]:
1. Express [tex]\( E \)[/tex] in terms of [tex]\(T\)[/tex], [tex]\(a\)[/tex], and [tex]\(V\)[/tex]:
[tex]\[
E = TaV^2
\][/tex]
2. Differentiate [tex]\( E \)[/tex] with respect to [tex]\( V \)[/tex] at constant [tex]\( T \)[/tex]:
[tex]\[
\left( \frac{\partial E}{\partial V} \right)_T = \frac{\partial}{\partial V} (TaV^2)
\][/tex]
3. Using the power rule:
[tex]\[
\left( \frac{\partial E}{\partial V} \right)_T = T \cdot a \cdot 2V
\][/tex]
Simplifying,
[tex]\[
\left( \frac{\partial E}{\partial V} \right)_T = 2aTV
\][/tex]
Therefore,
[tex]\[
\left( \frac{\partial E}{\partial V} \right)_T = 2aV \ln E
\][/tex]
In conclusion:
(a) [tex]\(\frac{1}{T} = \frac{aV^2}{E}\)[/tex],
(b) [tex]\(\frac{p}{T} = 2aV \ln E\)[/tex],
(c) [tex]\(\left( \frac{\partial E }{\partial V }\right)_T = 2aTV\)[/tex].
