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### How can internal energy be expressed as a function of any two of $p, v, T$?

In the book of Irey, Theormodynamics, the author states that (while talking about single phase substances) For a simple compressible media, we may choose our measurable independent variables any ...
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### Does the total energy in a volume (gas) increase linearly with increases in pressure?

If a volume has a certain pressure, then double that pressure at another time, is the total energy stored doubled? I'm sure this has to do with thermodynamics and if I remember correctly there is a ...
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### Why is there Net Work Done on a Pressure/Volume Cycle?

The area enclosed by the rectangle is said to be the Net Work Done. My question is why would there be any net workdone if it returns to its initial position (W). Shouldn't it act like a force-...
### $p$-$V$ diagram and internal energy
I recently learned that if the initial state is equal to the final state in the $p$-$V$ diagram then the internal energy is equal to zero so that means $Q=W$. My question is how can you know that only ...
I'm supposed to derive the following thermodynamic identity: $$- \left( \frac{\partial U}{\partial V} \right)_{T,N} + T \left( \frac{\partial P}{\partial T} \right)_{V,N} = P$$ ...by starting with ...