Question

Suppose that the entropy of a certain substance (not an Einstein solid) is given by S = a(E)¹/², where a is a constant. What is the energy E as a function of the temperature T? (Use the following as necessary: an and T.)

Answer 1

The energy E as a function of the temperature
`\( T \) is given by \( E = a^2 T^2 \).`

Step-by-step explanation:

To derive the relationship between energy E and temperature T from the entropy expression
`\( S = a(E)^(1/2) \)`, we start by squaring both sides to eliminate the square root:
`\( S^2 = aE \)`. Next, differentiate both sides with respect to temperature T :

`\[ 2S (dS)/(dT) = a (dE)/(dT) \]`

Now, substitute
`\( (dS)/(dT) = (a)/(2√(E)) (dE)/(dT) \) and \( (1)/(T) = (dT)/(dE) \):`

`\[ 2S \left((a)/(2√(E)) (dE)/(dT)\right) = a (dE)/(dT) \]`

Simplify and integrate:

`\[ S √(E) = (a)/(2) \ln\left((E)/(E_0)\right) \]`

Solve for
`\( E \) in terms of \( T \)` to obtain the final result:

`\[ E = a^2 T^2 \]`

This demonstrates that the energy of the substance is proportional to the square of the temperature.