For the isothermal expansion of a gas into a vacuum, \(\Delta E=0, q=0,\) and \(w=0 .\) (a) Is this a spontaneous process? (b) Explain why no work is done by the system during this process. (c) What is the "driving force" for the expansion of the gas: enthalpy or entropy?

To determine if a process is spontaneous, we need to consider the change in Gibbs free energy, which is given by the following equation: \(\Delta G = \Delta H - T \Delta S\) where \(\Delta G\) is the change in Gibbs free energy, \(\Delta H\) is the change in enthalpy, \(T\) is the temperature, and \(\Delta S\) is the change in entropy. A process is considered spontaneous if \(\Delta G < 0\). In the given problem, since the process is isothermal, we can re-write the equation as: \(\Delta G = q + w - T \Delta S\) We are given that \(q = 0\) and \(w = 0\), so the equation simplifies to: \(\Delta G = - T \Delta S\) If \(\Delta S > 0\), the process will be spontaneous since \(\Delta G\) will be less than zero. An expansion of a gas results in an increase in entropy because the particles of the gas have more space to move around, which means \(\Delta S > 0\). Thus, the process is spontaneous.

During the isothermal expansion of a gas into a vacuum, the gas expands freely with no pressure exerted externally (since it is in a vacuum). Work is defined as the energy transferred due to the force acting on an object over a certain distance, and in this case, the force would be exerted by the pressure of the gas. The formula for work done in a thermodynamic system involving a gas is given by: \(w = - \int_{V_1}^{V_2} P_{ext} dV\) where \(w\) is the work done by the gas, \(V_1\) and \(V_2\) are the initial and final volumes of the gas, and \(P_{ext}\) is the external pressure. In this case, since the gas is expanding into a vacuum, \(P_{ext} = 0\), and hence, the work done is equal to \(0\).

In order to determine the driving force for the expansion of the gas, we need to consider whether the change in enthalpy (\(\Delta H\)) or the change in entropy (\(\Delta S\)) is the dominant factor in the Gibbs free energy equation (\(\Delta G = \Delta H - T \Delta S\)). Since the process is isothermal, \(\Delta E = 0\), and the internal energy of a gas depends only on its temperature, the enthalpy of the gas also remains constant during the expansion process because \(\Delta H = \Delta E + P_\Delta V\). Therefore, \(\Delta H = 0\). On the other hand, as mentioned earlier, the entropy of the gas increases during expansion, which means \(\Delta S > 0\). Since \(\Delta H = 0\) and \(\Delta S > 0\), the driving force for the gas expansion is the entropy change. The expansion of the gas results in an increase in disorder or randomness in the system, which is consistent with an increase in entropy.