A sketch shows a convex curve for together with many of its tangent lines

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Thermodynamic potentials and the formal structure of thermodynamics · Thermodynamic potentials and the formal structure of thermodynamics

Thermal Physics, Ch Thermodynamic potentials, Legendre transforms and Maxwell relations

Problem

A sketch shows a convex curve

y(x)

for

x > 0

together with many of its tangent lines. Measuring the slope

s

and the intercept on the

y

axis of each tangent drawn, every member of the family is found to obey

y = sx - \tfrac34 s^{4/3}, \qquad s > 0,

so that the sketch gives the intercept as a function of the slope. (

x

y

and

s

are pure numbers here.) ((a)) Write down

\phi(s)

, the Legendre transform of

y

, reading it off the family of tangents. ((b)) Recover

x

as a function of

s

from

\phi

. ((c)) Hence find the curve

y(x)

itself. ((d)) Give the coordinates

(x,y)

of the point at which the tangent of slope

s = 8.00

touches the curve.

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