Question

A piston-cylinder device, with a set of stops on for the top, initially contains 3 kg of air at \(200 \mathrm{kPa}\) and \(27^{\circ} \mathrm{C}\). Heat is now transferred to the air, and the piston rises until it hits the stops, at which point the volume is twice the initial volume. More heat is transferred until the pressure inside the cylinder also doubles. Determine the work done and the amount of heat transfer for this process. Also, show the process on a \(P\) -v diagram.

Short answer

Based on the given information, determine the work done and the amount of heat transferred during the two-stage process, and represent the process on a \(P\)-\(v\) diagram. In the first stage, the piston rises until it hits the stops, and the volume becomes twice the initial volume. During the second stage, more heat is transferred until the pressure inside the cylinder doubles. To find the work done and the heat transferred during the two-stage process, follow these steps: 1. Calculate the initial volume using the ideal gas law. 2. Calculate the volume, temperature, and pressure for each stage. 3. Determine the work done during each stage. 4. Calculate the heat transfer for each stage. 5. Represent the process on a \(P\)-\(v\) diagram. Using the provided information, calculate the values of the work done and heat transferred, and draw a \(P\)-\(v\) diagram for the process.

Step-by-step solution

Find the initial volume

Using the initial state of the air, we can determine the volume of the air by applying the ideal gas law: $$PV = mRT$$ Where \(P\) is pressure, \(V\) is volume, \(m\) is mass, \(R\) is the specific gas constant for air, and \(T\) is temperature. We can rearrange the equation to solve for the initial volume \(V_1\): $$V_1 = \frac{mRT_1}{P_1}$$ Using given values \(P_1= 200 \,\mathrm{kPa}\), \(T_1 = 27^{\circ}\mathrm{C} = 300\,\mathrm{K}\), and \(m = 3\,\mathrm{kg}\), and specific gas constant for air \(R = 287\mathrm{\frac{J}{kg \cdot K}}\), we can calculate \(V_1\).

Calculate volume, temperature, and pressure for each stage

For stage 1, we have: $$V_2 = 2V_1$$ Next, we determine the final temperature after the first stage \(T_2\) by using the ideal gas law: $$ T_2 = \frac{P_1 V_2}{mR} $$ For stage 2, the pressure inside the cylinder doubles: $$P_3 = 2P_1$$ Thus, we can find the final temperature after the second stage \(T_3\): $$ T_3 = \frac{P_3 V_2}{mR} $$

Determine the work done during each stage

For stage 1, since the process is isobaric (constant pressure): $$W_{1\to2}=P_1\left(V_2 - V_1\right)$$ For stage 2, since the process is isochoric (constant volume), there will be no work done: $$W_{2\to3}=0$$ Total work done \(W = W_{1\to2} + W_{2\to3}\).

Calculate the heat transfer for each stage

Applying the first law of thermodynamics for each stage, we arrive at: Stage 1: $$Q_{1\to2} = \Delta U_{1\to2} + W_{1\to2}$$ $$Q_{1\to2}=nc_v\left(T_2-T_1\right) + W_{1\to2}$$ Stage 2: $$Q_{2\to3} = \Delta U_{2\to3} + W_{2\to3}$$ $$Q_{2\to3}=nc_v\left(T_3-T_2\right)$$ Total heat transfer \(Q = Q_{1\to2} + Q_{2\to3}\).

Represent the process on a P-v diagram

To create a \(P\)-\(v\) diagram, you will plot the following points: 1. Initial state \((P_1, V_1)\) 2. After stage 1 \((P_1, V_2)\) 3. After stage 2 \((P_3, V_2)\) Draw a horizontal line from the initial state to the point after stage 1, and then draw a vertical line from the point after stage 1 to the point after stage 2.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in thermodynamics that relates the pressure, volume, and temperature of an ideal gas. It is given by the formula
\( PV = nRT \)
where
\( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles of gas, \( R \) is the universal gas constant, and \( T \) is the absolute temperature measured in Kelvin.

Application in Piston-Cylinder Devices

In the context of the piston-cylinder device, this law helps us establish the relationship between the gas's state variables when minimal heat loss and no phase change are assumed. This is crucial in determining the initial and final conditions of processes involving these devices, just as in the given exercise where the initial volume of air in the cylinder was found using this principle. Understanding the ideal gas law is essential for grasping how changes in temperature, volume, and pressure affect ideal gases, facilitating the calculation of work done and heat transfer in thermodynamic processes.

Work Done in Thermodynamic Processes

In thermodynamics, work done refers to the energy transferred when a force acts upon a system to change its volume. It is a path-dependent variable meaning that the work done on or by the system depends on the actual process followed during the expansion or compression of the gas.

Isobaric and Isochoric Processes

Two types of processes often encountered in thermodynamics are isobaric (constant pressure) and isochoric (constant volume) processes. In an isobaric process, the work done by the expanding gas can be calculated by the formula:
\( W = P \times \bigtriangleup V \)
where \( P \) is the pressure and \( \bigtriangleup V \) is the change in volume. On the other hand, in an isochoric process, the volume of the system does not change, meaning no work is done, regardless of any pressure change. Understanding these concepts helps students solve problems where they need to calculate the net work done during the various stages of a thermodynamic cycle.

Heat Transfer in Thermodynamics

Heat transfer in thermodynamics refers to the energy transferred due to the temperature difference between a system and its surroundings. It's a central concept because it helps us understand how energy is absorbed or released during thermodynamic processes.

First Law of Thermodynamics

The first law of thermodynamics, also known as the law of energy conservation, is key to calculating heat transfer. It states that the change in the internal energy of a system is equal to the heat added to the system minus the work done by the system on its surroundings. This can be expressed as:
\( Q = \bigtriangleup U + W \)
where \( Q \) is heat transfer, \( \bigtriangleup U \) is the change in internal energy, and \( W \) is work done. In the given exercise, we apply this law to determine the amount of heat transferred during both stages of the thermodynamic process. By knowing the work done and the change in internal energy, which is derived from the heat capacity at constant volume \( c_v \) and the temperature change, we can calculate the total heat transfer required for the process.

P-v Diagram Representations

P-v diagrams, or pressure-volume diagrams, are visual representations of the changes in pressure and volume within thermodynamic processes. They are valuable tools for engineers and scientists as they illustrate the relationship between these two state variables during physical transformations.

Visualizing Thermodynamic Processes

In these diagrams, the x-axis typically represents the volume \( V \) while the y-axis represents the pressure \( P \). Different processes are shown as lines or curves connecting initial and final states. For isobaric processes, the line is horizontal, indicating constant pressure; for isochoric processes, it is vertical, signifying constant volume. The area under a line in a P-v diagram corresponds to the work done by or on the system during that particular process. By graphically summarizing the problem, P-v diagrams allow for a clearer understanding of the whole process, which is particularly helpful for students solving thermodynamics problems and visualizing the problem, as seen in the provided textbook exercise where we chart the two-stage heat transfer process.