a. Expand the differential off in terms of partial derivatives with respect to the natural variables of f b. Using the result from a. and applying the 1st Law of Thermodynamics, what are af da I and af of la? др c. From the equality of mixed partial derivatives, show that ƏT i la - Əs θα T
a. Expand the differential off in terms of partial derivatives with respect to the natural variables of f b. Using the result from a. and applying the 1st Law of Thermodynamics, what are af da I and af of la? др c. From the equality of mixed partial derivatives, show that ƏT i la - Əs θα T
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![**Educational Content on Helmholtz Free Energy**
3. **Specific Helmholtz Free Energy**
The specific Helmholtz free energy \( f \) is related to the specific internal energy \( u \) as:
\[
f(T, \alpha) = u - Ts
\]
where the "natural variables" of \( f \) are temperature and specific volume.
**a.** Expand the differential of \( f \) in terms of partial derivatives with respect to the natural variables of \( f \).
**b.** Using the result from a. and applying the 1st Law of Thermodynamics, determine the following partial derivatives:
\[
\left( \frac{\partial f}{\partial \alpha} \right)_T \quad \text{and} \quad \left( \frac{\partial f}{\partial T} \right)_{\alpha}
\]
**c.** From the equality of mixed partial derivatives, show that:
\[
\left( \frac{\partial p}{\partial T} \right)_{\alpha} = \left( \frac{\partial s}{\partial \alpha} \right)_T
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Transcribed Image Text:**Educational Content on Helmholtz Free Energy**
3. **Specific Helmholtz Free Energy**
The specific Helmholtz free energy \( f \) is related to the specific internal energy \( u \) as:
\[
f(T, \alpha) = u - Ts
\]
where the "natural variables" of \( f \) are temperature and specific volume.
**a.** Expand the differential of \( f \) in terms of partial derivatives with respect to the natural variables of \( f \).
**b.** Using the result from a. and applying the 1st Law of Thermodynamics, determine the following partial derivatives:
\[
\left( \frac{\partial f}{\partial \alpha} \right)_T \quad \text{and} \quad \left( \frac{\partial f}{\partial T} \right)_{\alpha}
\]
**c.** From the equality of mixed partial derivatives, show that:
\[
\left( \frac{\partial p}{\partial T} \right)_{\alpha} = \left( \frac{\partial s}{\partial \alpha} \right)_T
\]
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