Question

A gas is compressed from an initial volume of \(0.42 \mathrm{m}^{3}\) to a final volume of \(0.12 \mathrm{m}^{3} .\) During the quasi-equilibrium process, the pressure changes with volume according to the relation \(P=a V+b,\) where \(a=-1200 \mathrm{kPa} / \mathrm{m}^{3}\) and \(b=600 \mathrm{kPa}\) Calculate the work done during this process ( \(a\) ) by plotting the process on a \(P\) - \(V\) diagram and finding the area under the process curve and ( \(b\) ) by performing the necessary integrations.

Short answer

Answer: The work done on the gas during the quasi-equilibrium compression process is -83.16 kJ.

Step-by-step solution

Identify the given values

The initial volume of the gas is \(V_i=0.42 \mathrm{m}^3\), and the final volume is \(V_f=0.12\mathrm{m}^3\). The pressure-volume relationship is given by \(P(V)=aV+b=-1200V\ \mathrm{kPa} + 600\ \mathrm{kPa}\). We will use these values in our calculations for both methods.

Plot the process on a \(P-V\) diagram and find the area under the curve (Method A)

First, we will plot the pressure-volume relationship \(P(V)\) on a \(P-V\) diagram. Since it is a linear function, we only need to find the initial and final points to draw a straight line representing the process. At the initial volume \(V_i=0.42\ \mathrm{m}^3\), the pressure is: \(P_i = -1200(0.42)+600=-504\ \mathrm{kPa}+600=96\ \mathrm{kPa}\). At the final volume \(V_f=0.12\ \mathrm{m}^3\), the pressure is: \(P_f = -1200(0.12)+600=-144\ \mathrm{kPa}+600=456\ \mathrm{kPa}\). The area under the curve is a trapezoid with base \(V_f - V_i\) and heights \(P_i\) and \(P_f\). The area of a trapezoid is given by \(\frac{1}{2}(b_1+b_2)h\), where \(b_1\) and \(b_2\) are the base lengths and \(h\) is the height. The area under the process curve (work done) is: \(W_A = \frac{1}{2}(P_i+P_f)(V_f - V_i)\). Plug in the values and calculate the work done: \(W_A = \frac{1}{2}(96\ \mathrm{kPa} + 456\ \mathrm{kPa})(0.12\ \mathrm{m}^3 - 0.42\ \mathrm{m}^3) = \frac{1}{2}(552\ \mathrm{kPa})(-0.3\ \mathrm{m}^3) = -83.16\ \mathrm{kJ}\).

Perform the integration (Method B)

Now we will calculate the work done using the integration method. Recall the formula for work done: \(W = -\int_{V_i}^{V_f} P(V)dV\). We have the relationship \(P(V) = -1200V + 600\). So, the work done is: \(W_B = -\int_{0.42}^{0.12} (-1200V + 600)dV\). Integrate the function and substitute the limits of integration: \(W_B = -[-600V^2 + 600V]_{0.42}^{0.12} = -\left((-600(0.12)^2 + 600(0.12)) - (-600(0.42)^2 + 600(0.42))\right) = -83.16\ \mathrm{kJ}\). We find that both methods yield the same result for the work done, \(W=\ -83.16\ \mathrm{kJ}\). This is the work done on the gas during the quasi-equilibrium process.

Key concepts

Quasi-Equilibrium Process

The concept of a quasi-equilibrium process is central to understanding thermodynamic systems. This idealized process occurs so slowly that each intermediate step is effectively in equilibrium with its surroundings. This allows the system to adjust its properties gradually without experiencing rapid or uneven changes. During a quasi-equilibrium process, parameters like pressure and volume change at a rate where the system remains in a sequence of equilibrium states.

For students, this concept explains why certain thermodynamic calculations are simplified, as it is assumed that the system passes through a series of infinitesimally small equilibrium steps, thus allowing properties such as pressure and volume to be well-defined at every stage of the process. This assumption is crucial because it permits the use of mathematical techniques like integration to calculate work done in a process, offering a precise approach to solving complex problems.

Pressure-Volume Relationship

The pressure-volume (P-V) relationship, also known as an equation of state, is an equation that relates the pressure of a system to its volume and temperature. In the context of thermodynamics, it is indispensable for describing the state of a gas during a process. For many systems, especially gases, the ideal gas law provides a good approximation of this relationship. However, real gases often require more complex equations that can include additional constants or terms, as seen in the provided exercise.

Understanding the specific pressure-volume relationship allows students to predict how a system will respond to changes in its state, such as during compression or expansion. Beyond its theoretical importance, this relationship is practical for calculating work, predicting heat transfer, and determining system efficiency in engineering applications.

P-V Diagram

A Pressure-Volume (P-V) diagram is a graphical representation that shows the change in pressure with respect to volume for a thermodynamic process or cycle. Each point on the diagram represents a state of the system with a unique pressure and volume, where the curve graphically represents a process path. The P-V diagram is instrumental in visualizing the work done by or on a system during a thermodynamic process.

The area under the curve of a P-V diagram corresponds to the work done, as it represents the integral of pressure with respect to volume. For students, learning to interpret and draw P-V diagrams is a fundamental skill in thermodynamics. It gives tangible insight into processes such as isothermal, isobaric, isochoric, and adiabatic processes, each of which has a characteristic shape on the diagram. Thus, mastering P-V diagrams enables students to grasp more complex concepts like engine cycles and refrigeration cycles.

Integration Method in Thermodynamics

In thermodynamics, the integration method is a mathematical technique used to calculate work, heat, and other properties of a system. Particularly for work, when a force acts through a distance, or a pressure acts to change the volume of a system, the integration method allows us to calculate the work done during a quasi-equilibrium process by integrating the force over the distance, or in the case of gases, integrating the pressure with respect to volume.

Students often encounter integration when calculating the area under a P-V curve, which corresponds to the work done on or by the system during a process. It's essential to understand that this method assumes a quasi-equilibrium process to be valid. The accuracy and applicability of the integration depend on how closely the real process resembles a quasi-equilibrium process. By learning the integration technique, students are better equipped to solve complex problems and understand the nuances of various thermodynamic processes.