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A 3.50 mole sample of \(\mathrm{N}_{2}\) in a state defined by \(T_{i}=\) 250\. \(\mathrm{K}\) and \(V_{i}=3.25 \mathrm{L}\) undergoes an isothermal reversible expansion until \(V_{f}=35.5 \mathrm{L}\) Calculate \(w,\) assuming (a) that the gas is described by the ideal gas law, and (b) that the gas is described by the van der Waals equation of state. What is the percent error in using the ideal gas law instead of the van der Waals equation? The van der Waals parameters for \(\mathrm{N}_{2}\) are listed in Table 7.4

Consider the isothermal expansion of 2.35 mol of an ideal gas at \(415 \mathrm{K}\) from an initial pressure of 18.0 bar to a final pressure of 1.75 bar. Describe the process that will result in the greatest amount of work being done by the system with \(P_{\text {external }} \geq 1.75\) bar, and calculate \(w\). Describe the process that will result in the least amount of work being done by the system with \(P_{\text {external}} \geq 1.75\) bar, and calculate \(w .\) What is the least amount of work done without restrictions on the external pressure?

Calculate \(\Delta H\) and \(\Delta U\) for the transformation of \(2.50 \mathrm{mol}\) of an ideal gas from \(19.0^{\circ} \mathrm{C}\) and 1.00 atm to \(550 .^{\circ} \mathrm{C}\) and \(19.5 \mathrm{atm}\) if \(C_{P, m}=20.9+0.042 \frac{T}{\mathrm{K}}\) in units of \(\mathrm{J} \mathrm{K}^{-1} \mathrm{mol}^{-1}\)

An ideal gas undergoes a single-stage expansion against a constant external pressure \(P_{\text {external}}=P_{f}\) at constant temperature from \(T, P_{i}, V_{i},\) to \(T, P_{f}, V_{f}\) a. What is the largest mass \(m\) that can be lifted through the height \(h\) in this expansion? b. The system is restored to its initial state in a single-state compression. What is the smallest mass \(m^{\prime}\) that must fall through the height \(h\) to restore the system to its initial state? c. If \(h=15.5 \mathrm{cm}, P_{i}=1.75 \times 10^{6} \mathrm{Pa}, P_{f}=1.25 \times 10^{6} \mathrm{Pa}\) \(T=280 . \mathrm{K},\) and \(n=2.25 \mathrm{mol},\) calculate the values of the masses in parts (a) and (b).