We will prove the equivalence of the ideal gas temperature scale theta and the thermodynamic scale t by performing a carnot cycle on an ideal gas. for such a gas pv = nk theta and the internal energy e is a function of theta only. you may wish to proceed as follows: calculate the heat exchanges q_h and q_c as a function of theta_h, theta_c, and volume expansion factors (ratio between volumes at the initial and final stage of the process). calculate the volume expansion factor in an adiabatic process as a function of theta. show that q_h/q_c = theta_h/theta_c. starting from de = tds - pdv show that the equation of state pv = nkt in fact implies that e can only depend on t. what is the most general equation of state consistent with an internal energy that depends only on temperature ?