A gaseous sample consisting of \(1.00 \mathrm{~mol}\) molecules is described by the equation of state \(p \mathrm{~V}_{\mathrm{m}}=\mathrm{RT}(1+B p)\). Initially at \(373 \mathrm{~K}\), it undergoes JouleThomson expansion from \(100 \mathrm{~atm}\) to \(1.00 \mathrm{~atm}\). Given that \(C_{p, \mathrm{~m}}=R, \mu=0.21\) \(\mathrm{K} \mathrm{atm}^{-1}, B=-0.525(\mathrm{~K} / T) \mathrm{atm}^{-1}\), and that these are constant over the temperature range involved, calculate \(\Delta T\) and \(\Delta S\) for the gas.

The equation of state is a mathematical model that describes the behavior of a given quantity of matter under various conditions of temperature, pressure, and volume.
For an ideal gas, the simplest equation of state is the Ideal Gas Law, given by PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the universal gas constant, and T is temperature in Kelvin.
In our exercise, the equation of state is given as a modified version for a real gas:

Modified Equation of State

For real gases, deviations from ideal behavior are common, and these deviations are accounted for by implementing additional terms in the equation of state. In the exercise, this real gas equation incorporates a second virial coefficient, B, which accounts for the interactions between gas molecules and the volume they occupy:
\[ pV_m=RT(1+Bp) \]This modified equation suggests that as pressure (p) changes, these interactions play a significant role. The term Bp reflects the non-ideality of the gas and is also temperature-dependent, indicating that as temperature changes, so too does the behavior of the gas.Understanding this modified equation is crucial for accurately predicting the change in temperature and entropy during a process such as Joule-Thomson expansion. By using this equation, we can account for the real behavior of the gas, making our predictions more realistic than if we were to assume ideal gas behavior.

Entropy is a property of a thermodynamic system that measures the degree of disorder or randomness. Entropy change, denoted by \( \Delta S \), during a process quantifies the change in this disorder, and it's a core concept in understanding thermodynamic processes. In the context of thermodynamics, entropy is often associated with the second law, which states that the total entropy of an isolated system can never decrease over time.

Entropy Change in Joule-Thomson Expansion

During a Joule-Thomson expansion, a gas undergoes an isenthalpic process, meaning there is no change in enthalpy (\( H \)). In our exercise, the Joule-Thomson expansion is associated with a cooling effect. The process is adiabatic (no heat exchange with the surroundings), and the change in entropy is determined using the specific heat at constant pressure, \( C_{p,\,m} \).The entropy change formula, derived for an ideal gas, is also applicable to real gases for small temperature changes, and it is given by:
\[ \Delta S = C_{p,\,m} \ln\left(\frac{T_{\text{final}}}{T_{\text{initial}}}\right) \]By using this relation, we can quantify how the discrete properties of the gas change in response to the process. Since entropy also relates to heat transfer and irreversibility, this calculation is pivotal for understanding the intrinsic properties of the thermodynamic cycle being analyzed.

Ideal gases are hypothetical gases that follow the ideal gas law perfectly at all pressures, temperatures, and volumes. This means they obey the equation PV = nRT without any deviations. Ideal gas properties make it straightforward to predict gas behavior since these gases do not show interactions between their particles and occupy no volume, two assumptions which are not true for real gases.

Deviation from Ideal Gas Behavior

While ideal gases are a useful conceptual tool in thermodynamics, real gases only behave ideally under certain conditions, usually at high temperatures and low pressures where the interactions between molecules are negligible compared to the kinetic energy of the molecules.In the exercise, the properties of the real gas are such that the second virial coefficient comes into play to correct for these non-ideal behaviors, indicating a deviation from what would be predicted by the Ideal Gas Law alone. Understanding the ideal gas properties helps students establish a base reference point from which to compare the more complex behavior of real gases, such as the one in the given problem.