Isotherms and adiabats, part II

November 9, 2010 at 3:12 am Leave a comment

For the next claim about relationships between isotherms and adiabats, we’ll need to appeal to a general fact about isotherms, and a general fact about temperature. For the former, let objects A and B be separated by a barrier that blocks mass flow but allows the flow of heat.

Statement I. If objects A and B are brought into contact until thermal equilibrium is reached, there are a variety of pressures P(A) and volumes V(A) of object A that are consistent with it’s being in thermal equilibrium with object B, and vice versa. The curve (P(A),V(A)) in P-V space thus corresponds to a single temperature, T(A), of object A.

Isotherms often look something like this:

Statement R. Temperatures are accurately represented by real numbers.

Claim. An adiabat cannot touch an isotherm tangentially – it must cross it.

Proof. Statements I and R together imply that P-V space is infinitely dense with isotherms. If an adiabat were to touch an isotherm tangentially, and then veer away, it would thus have to cross a different isotherm more than once (see figure below)!

This we have already shown to be impossible in the previous post (part I). Hence, the adiabat must not only touch, but cross the isotherm. QED.


Entry filed under: Thermodynamics.

Isotherms and adiabats, part I Isotherms and adiabats, part III

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