Suppose that subsequent to this heating, 1.90 104 J of thermal energy is removed

Question

Suppose that subsequent to this heating, 1.90 104 J of thermal energy is removed from the gas isothermally. Find the final volume in terms of the initial volume of the example, V0. (Hint: Follow the same steps as in the example, but in reverse. Use the values in the Practice It section. Also note that the initial volume in this exercise is 1.25V0.)

Practice IT section (A balloon contains 5.90 moles of a monatomic ideal gas. As energy is added to the system by heat (say, by absorption from the Sun), the volume increases by 25% at a constant temperature of 34.0°C.)

Explanation / Answer

I think you used the following formula for isothermal work done on an ideal gas:
W = - nRTln( V_final/V_initial )
Since the internal energy of an ideal gas depends only on temperature, its change in internal energy while undergoing an isothermal process equals zero. So work done on the gas and heat absorbed
in such a process have same magnitude but opposite sign:
Q = -W = nRTln( V_final/V_initial )

To find the volume after the isothermal compression just solve the second formula for V_final:
Q = -W = nRTln( V_final/V_initial )
<=>
Q/(nRT) = ln( V_final/V_initial )
=>
V_final = V_initial e^( Q/(nRT) )

with
V_initial = 1.25V
Q = - 1.2×10 J (negative because heat is removed)
n = 4.40 mol
R = 8.3145 Jmol¹K¹
T = 301K
i get the following result:
V_final = 1.25 V e^(-1.08975) = 0.42V

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