Before introducing the temperature scale now known as the Kelvin scale, Kelvin s

Question

Before introducing the temperature scale now known as the Kelvin scale, Kelvin suggested
a logarithmic scale in which the QH/QL = exp(theta_H)/exp(theta_L) where theta_H and theta_L denote,
respectively, the temperatures of the hot and cold reservoirs on this scale.

a) Show that theta = ln(T)+C where C is a constant
b) On the Kelvin scales, temperatures can range between 0 and infinity. Determine the
corresponding range of values for ??
c) Obtain an expression for the thermal efficiency of any system undergoing a reversible
power cycle while operating between reservoirs at temperatures ?H and ?L on the
logarithmic scale

Explanation / Answer

(i)   Consider QL be at temperature T in kelvin scale and in the log scale; QH be at a temperature T+dT in kelvin scale and +d in log scale.

QH/QL = e(+d)- = ed in the log scale

   =(T+dT)/T=1+(dT/T) in the kelvin scale.

ed =1+(dT/T)             -(1)

expansion of ex: ex =1+[x/1!] +[x2/2!]+...

ed =1+ [d/1!] +[d2/2!]+...=1+ d   [neglecting higher powers, as d is very small]

in (1), 1+d=1+dT/T

=>d=dT/T

=>= ln(T) +c

(ii)   As T tends to 0, tends to -

as T tends to , tends to +

thus, ranges from - to +

(iii)for a reversible cycle, efficiency=1-(QL/QH)

   given, QH/QL=eH/eL =eH-L

efficiency=1-eL-H

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