3. A possible equation of state for a gas takes the form P.V=R·T·e VRT in which a and R are constants. P, V, T are the pressure, volume, and temperature of the gas, respectively. Show that (3)T() P()v = -1.
3. A possible equation of state for a gas takes the form P.V=R·T·e VRT in which a and R are constants. P, V, T are the pressure, volume, and temperature of the gas, respectively. Show that (3)T() P()v = -1.
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![**Problem 3:**
A possible equation of state for a gas takes the form:
\[ P \cdot V = R \cdot T \cdot e^{-\frac{\alpha}{VRT}} \]
where \(\alpha\) and \(R\) are constants. \(P, V, T\) are the pressure, volume, and temperature of the gas, respectively.
Show that:
\[
\left( \frac{\partial P}{\partial V} \right)_T \left( \frac{\partial V}{\partial T} \right)_P \left( \frac{\partial T}{\partial P} \right)_V = -1.
\]
**Hint:** Take the natural log of this equation before you take the partial derivatives.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F775c65b2-d298-4974-84c2-1b9ec352df93%2F4fdba982-72f7-4db5-8b13-0c930b120ac8%2F7vyonvc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 3:**
A possible equation of state for a gas takes the form:
\[ P \cdot V = R \cdot T \cdot e^{-\frac{\alpha}{VRT}} \]
where \(\alpha\) and \(R\) are constants. \(P, V, T\) are the pressure, volume, and temperature of the gas, respectively.
Show that:
\[
\left( \frac{\partial P}{\partial V} \right)_T \left( \frac{\partial V}{\partial T} \right)_P \left( \frac{\partial T}{\partial P} \right)_V = -1.
\]
**Hint:** Take the natural log of this equation before you take the partial derivatives.
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