Question

A piston-cylinder device initially contains \(15 \mathrm{ft}^{3}\) of helium gas at 25 psia and \(70^{\circ} \mathrm{F}\). Helium is now compressed in a polytropic process \(\left(P V^{n}=\text { constant }\right)\) to 70 psia and \(300^{\circ} \mathrm{F}\). Determine \((a)\) the entropy change of helium, \((b)\) the entropy change of the surroundings, and (c) whether this process is reversible, irreversible, or impossible. Assume the surroundings are at \(70^{\circ} \mathrm{F}\).

Short answer

Question: Determine if the given polytropic process of helium gas inside a piston-cylinder device is reversible, irreversible, or impossible based on the entropy change. The initial and final states are provided, and the process follows a polytropic equation \(PV^n=\text{constant}\). Solution: After converting the temperatures to the absolute scale (Rankine), applying the ideal gas law to find the mass of helium, calculating the polytropic exponent n, and finding the entropy changes of the gas and surroundings, we determine the total entropy change. Based on this value, we can conclude if the process is reversible (total entropy change = 0), irreversible (total entropy change > 0), or impossible (total entropy change < 0).

Step-by-step solution

Converting temperature to absolute temperature scale

The first step is to convert the given initial and final temperatures in Fahrenheit to the absolute temperature scale (Rankine): Initial Temperature \(T_1 = 70^\circ \mathrm{F} + 459.67 = 529.67\,\mathrm{R}\) Final Temperature \(T_2 = 300^\circ \mathrm{F} + 459.67 = 759.67\,\mathrm{R}\)

Applying the ideal gas law

Now we will apply the ideal gas law, \(PV = mRT\), to find the mass of the helium gas: \(P_1 V_1 = m R T_1\) where, \(P_1 = 25\,\text{psia}\), \(V_1 = 15\,\text{ft}^3\), \(R = 1545/({29\text{lb}_m}/{\text{lb}_m\text{R}})\) is the specific gas constant for helium, and \(T_1 = 529.67\,\mathrm{R}\). Solving the equation for \(m\), we have the mass of the helium gas.

Determining the polytropic exponent n

Using the polytropic process equation, \(P_1 V_1^n = P_2 V_2^n\), we can find the exponent n. Given \(P_2 = 70\, \text{psia}\) and \(T_2 = 759.67\,\mathrm{R}\), we can find \(V_2\) using the ideal gas law: \(P_2 V_2 = m R T_2\) Solving for \(V_2\) and plugging the values in the polytropic process equation, we can find the value of n.

Finding the entropy change of the helium gas

The entropy change for an ideal gas undergoing a polytropic process is given by: \(\Delta s_{\text{gas}} = m C_v \ln\frac{T_2}{T_1} - n m R \ln\frac{P_2}{P_1}\) where \(C_v\) is the specific heat at constant volume for helium. Using the values of \(T_1\), \(T_2\), \(P_1\), \(P_2\), and \(m\) calculated before and n, we can calculate the entropy change of the gas.

Finding the entropy change of the surroundings

The heat transfer in an internally reversible polytropic process is: \(q_{rev}=-\frac{n}{n-1}mR(T_1-T_2)\) Now we can find the entropy change of the surroundings by considering the transfer of heat into it: \(\Delta s_{\text{surroundings}} = \frac{q_{rev}}{T_{\text{surroundings}}}\) Given \(T_{\text{surroundings}}= 70^\circ \mathrm{F} + 459.67 = 529.67\,\mathrm{R}\) and using the calculated value of \(q_{rev}\), we can find the entropy change of the surroundings.

Determine whether the process is reversible, irreversible, or impossible

Now we can determine the nature of the process based on the total entropy change: \(\Delta s_{\text{total}} = \Delta s_{\text{gas}} + \Delta s_{\text{surroundings}}\) If \(\Delta s_{\text{total}} = 0\), then the process is reversible. If \(\Delta s_{\text{total}} > 0\), then the process is irreversible. If \(\Delta s_{\text{total}} < 0\), then the process is impossible. Using the calculated values of \(\Delta s_{\text{gas}}\) and \(\Delta s_{\text{surroundings}}\), we can determine the nature of the process.

Key concepts

Piston-Cylinder Device

The piston-cylinder device is a fundamental component often used to illustrate thermodynamic processes in physics and engineering. This apparatus consists of a cylinder that is sealed by a moveable piston. Inside, it can contain various gases or fluids. The piston is designed to move freely within the cylinder, allowing for changes in volume as force is applied or as the internal state of the gas alters. This change in volume directly impacts parameters such as pressure and temperature within the gas, based on the principles of thermodynamics.

Understanding how the piston-cylinder operates is crucial when analyzing thermodynamic cycles, such as those found in engines or refrigeration systems. Processes like expansion, compression, heat transfer, and work done can all be modeled using this device. In the example problem provided, helium gas is compressed, which would mean that the piston is pushing down, decreasing the volume and increasing pressure and temperature, leading to questions about entropy change during this polytropic process.

Polytropic Process

A polytropic process is a thermodynamic process that follows the relation \(PV^n = \text{constant}\), where \(P\) represents pressure, \(V\) is the volume, and \(n\) is the polytropic exponent. Depending on the value of \(n\), this process can represent a variety of different physical scenarios. For example, when \(n=0\), it corresponds to an isobaric process (constant pressure), \(n=1\) to an isothermal process (constant temperature), and \(n=\infty\) to an isochoric process (constant volume). Understanding the characteristics of a polytropic process can help predict how a system behaves when pressure, volume, and temperature change.

In the context of the exercise, the helium gas undergoing a polytropic compression means that the relation between the initial and final states can be used to find the polytropic exponent \(n\). This is important, as it influences how we understand the system's behavior and calculate the entropy change. The value of \(n\) determines how energy is transferred within the system, and considering entropy, which is a measure of disorder or randomness, we can determine how ordered or disordered the system becomes during this specific process.

Ideal Gas Law

The ideal gas law, represented as \(PV = mRT\), is pivotal in understanding the behavior of gases under varying conditions of pressure (\(P\)), volume (\(V\)), and temperature (\(T\)). Here, \(m\) denotes the gas mass, and \(R\) is the specific gas constant unique to each gas. The ideal gas law combines Boyle's Law, Charles's Law, and Avogadro's Law and assumes that the molecules of the gas are point particles that interact with each other only through elastic collisions.

The law is extremely useful when dealing with problems like the exercise provided, as it allows us to compute the mass of the gas when the pressure, volume, and temperature are known. When combined with the concept of a polytropic process, the ideal gas law provides a foundation for calculating changes in the system, such as entropy change. However, it is crucial to note that the ideal gas law is an approximation. It assumes no intermolecular forces and that the volume occupied by the gas molecules themselves is negligible, which holds true under normal conditions but not for all gases at all temperatures and pressures.