(a) Calculate the work done upon expansion of 1 mol of gas quasi-statically and isothermally from volume \(v_{i}\) to a volume \(v_{f}\), when the equation of state is $$ \left(P+\frac{a}{v^{2}}\right)(v-b)=R T $$ where \(a\) and \(b\) are the van der Waals constants. (b) If \(a=1.4 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{4} / \mathrm{mol}\) and \(b=3.2 \times 10^{-5} \mathrm{~m}^{3} / \mathrm{mol}\), how much work is done when the gas expands from a volume of 10 liters to a volume of \(22.4\) liters at \(20^{\circ} \mathrm{C} ?\)

Isothermal expansion is a process where a gas expands at a constant temperature. Because the temperature remains constant, the internal energy of the gas does not change. This process requires an exchange of heat with the surroundings to maintain the constant temperature. In the context of the van der Waals gas, the isothermal expansion is described by the equation of state: \ \ \(\left(P + \frac{a}{v^{2}}\right)(v-b) = R T \) \ \ During isothermal expansion, the pressure of the gas changes as the volume changes, but the temperature remains fixed. This means that any work done by (or on) the gas as it expands is counteracted by heat exchange with the surroundings to maintain the constant temperature. The work done during an isothermal expansion can be expressed using integral calculus, which allows us to take into account the changing pressure.

A quasi-static process is a process that happens infinitely slowly, so that the system remains in equilibrium at all stages. In reality, no process can be truly quasi-static, but it is a useful idealization for simplifying the analysis of thermodynamic processes. \ \ For a gas undergoing an isothermal expansion quasi-statically, the process is slow enough that we can assume every infinitesimal stage is an equilibrium state. This allows us to use the equation of state at every point in the process to calculate the work done: \ \ \(\text{W} = \int_{v_i}^{v_f} P \mathrm{d}v \) \ \ This integral gives us the total work done in a quasi-static process by integrating the infinitesimal work done at each stage over the entire volume change.

Work done by a gas during expansion or compression can be calculated using the pressure-volume relationship. For an isothermal and quasi-static process, work can be determined by integrating pressure over volume: \ \ \(\text{W} = \int_{v_i}^{v_f} P \mathrm{d}v \) \ \ For a gas following the van der Waals equation, we can express pressure (\text{P}) in terms of volume (\text{v}) and integrate accordingly. \ \ The solution involves substituting the pressure expression from the van der Waals equation into the integral and separating it into parts that can be evaluated independently. The complex integral is split into two simpler integrals: one involving the ideal gas constant (R \(\text{T} \mathrm{d}v / (v - b) \)) and another involving the attraction constant (\text{a} \(\text{d}v / v^2) \).

Thermodynamic integrals are crucial for calculating work done in processes involving gases. The integrals we encounter depend on the form of the equation of state governing the behavior of the gas. \ \ For the van der Waals gas, the integral for work done during an isothermal expansion is: \ \ \(\text{W} = \int_{v_i}^{v_f} \left(\frac{R T}{v - b} - \frac{a}{v^2}\right) \mathrm{d}v \) \ \ Breaking this into two parts, we have: \ \ \(\text{W} = R T \int_{v_i}^{v_f} \frac{\text{d}v}{v-b} - a \int_{v_i}^{v_f} \frac{\text{d}v}{v^2} \) \ \ For each integral, use standard calculus results: \ \ \(\text{W}_1 = R T \ln \left(\frac{v_f - b}{v_i - b}\right) \) \ \ \(\text{W}_2 = a \left( \frac{1}{v_i} - \frac{1}{v_f} \right) \) \ \ These integrals, once summed, give the total work done.

The van der Waals equation introduces two new constants, `a` and `b`, to the ideal gas law to account for the behavior of real gases: \ \ \(\left(P + \frac{a}{v^{2}}\right)(v-b) = R T \) \ \ - `a` adjusts for the intermolecular forces present in real gases. It represents the attraction between gas molecules, which reduces the pressure compared to an ideal gas. \ \ - `b` accounts for the finite volume occupied by gas molecules. It corrects the volume term in the ideal gas equation by subtracting the volume excluded by the gas molecules themselves. \ \ These constants vary for different gases and are determined experimentally. In the context of the problem, they are crucial for accurately calculating the work done by the gas during expansion.