Answer to Question #267898 in Molecular Physics | Thermodynamics for phys

Question

Carnot cycle. Consider a Carnot engine with a monoatomic ideal gas as an operating substance (i.e. the substance that undergoes the cyclic process to convert heat into work). One property of the ideal gas is that its internal energy is related only to its temperature, s.t. U=32nRTU=32nRT. The cyclic process is Show that you will get:

η=1− τc/τh

by getting the ratio of the total work done over the total amount of heat entering the system.

Answer 1

$U =\dfrac32 nRT\\ \quad\\ U\ \alpha\ T$

This means that the internal energy of the monoatomic gas is related only to its temperature

Efficiency (η) = $\dfrac{∆U}{U_h}$ = $\dfrac{U_h-U_c}{U_h}$

but internal energy is dependent and relates with internal energy,

$\therefore\ η= \dfrac{T_h-T_c}{T_h} = \dfrac{T_h}{T_h} - \dfrac{T_c}{T_h}\\ = 1-\dfrac{T_c}{T_h}$