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In thermodynamics, we study the energy interactions that occur in a system. For the separation of natural gas into \(\text{CH}_4\) (methane) and \(\text{C}_2\text{H}_6\) (ethane), we harness principles of thermodynamics to understand and calculate the energy changes involved. Specifically, we focus on variables like temperature, pressure, and volume, which are paramount in determining energy changes. By analyzing these transformations, we can be more efficient in designing processes like gas separation. The principles of thermodynamics help us ensure that we are moving energy in the most efficient way possible, minimizing wasted energy and allowing us to calculate the minimum work required for the process.

The ideal gas law is a cornerstone in the realm of thermodynamics. It links pressure (\text{\text{P}}), volume (\text{\text{V}}), temperature (\text{\text{T}}), and the amount of gas (\text{\text{n}}) using the equation: \[ PV = nRT \] where \(R\) is the universal gas constant (\text{0.0821 L·atm/(K·mol)}). In this particular problem, the natural gas mixture at the inlet has a volumetric flow rate of \text{100 m³/s} at \text{20°C} and \text{1 atm}. By converting the temperature to Kelvin (\text{273 K + 20°C}), and plugging in the values on the right side of the ideal gas equation, we can solve for the molar flow rate (\text{N_in}). This step is critical as it allows us to understand the amount of gas we are handling in the system, setting up for the calculation of downstream component flows.

Entropy, denoted as \( S \), brings the concept of disorder to thermodynamics. In gas separation processes, we encounter entropy change due to mixing and separating gases. For ideal gases undergoing isothermal processes (constant temperature), the entropy change can be given by the formula: \[ \text{ΔS} = nR \text{ln}\frac{V_2}{V_1} \]For our problem, since we are dealing with molar flow rates, this translates to: \[ \text{ΔS}_{\text{component}} = n_{component} R \text{ln} \frac{n_{component}}{n_{\text{in}}} \] We need to calculate the entropy change separately for \text{CH}_4 and \text{C}_2\text{H}_6. Summing these gives the total entropy change of the system. Understanding entropy change helps quantify the energy dispersal in the separation process, impacting the theoretical work required.

Molar flow rate is a key aspect when dealing with chemical processes like gas separation. It refers to the quantity of a component flowing per unit time, usually expressed in \text{mol/s}. In this context, we need to determine the molar flow rate of the incoming natural gas mixture. Utilizing the ideal gas law, we found \text{n}_{\text{in}}, the total molar flow rate of the mixture. The device then separates this stream into \text{CH}_4 and \text{C}_2\text{H}_6} based on their mole fractions (\text{y}). We calculate the molar flow rates of individual components as: \[n_{\text{C}_2\text{H}_6} = y \times n_{\text{in}} \] \[n_{\text{CH}_4} = (1 - y) \times n_{\text{in}} \]This information is pivotal as it feeds into further calculations including entropy change and theoretical work.

Theoretical work is a measure of the minimum energy required to achieve a process, calculated under ideal conditions with no losses. For the given problem, the work input needed for the gas separation device can be determined using the total entropy change calculated earlier. The relationship between work and entropy is given by: \[ \text{W}_{\text{min}} = -T \text{ΔS}_{\text{total}} \] Where \text{T} is the temperature in Kelvin. This gives a measure of the minimal energy input to drive the separation process at a steady state. By varying the mole fraction \text{y} from \text{0.05} to \text{0.50} and calculating \text{W}_{\text{min}} for each \text{y}, we can understand how the minimum theoretical work changes with the composition of the natural gas.