Question

A gas initially at \(p_{1}=1\) bar and occupying a volume of 1 liter is compressed within a piston-cylinder assembly to a final pressure \(p_{2}=4\) bar. (a) If the relationship between pressure and volume during the compression is \(p V=\) constant, determine the volume, in liters, at a pressure of 3 bar. Also plot the overall process on a graph of pressure versus volume. (b) Repeat for a linear pressure-volume relationship between the same end states.

Short answer

For \(pV = \text{constant}\), volume at 3 bar is 0.333 liters; for linear relationship, volume at 3 bar is 0.5 liters.

Step-by-step solution

Understand the initial conditions

The initial pressure is given by \(p_{1} = 1\) bar and the initial volume \(V_{1} = 1\) liter.

Understand the final conditions

The final pressure is given by \(p_{2} = 4\) bar, and we need to determine the volume at \(p = 3\) bar.

Use the given relationship \(pV = \text{constant}\)

Since \(pV\) is constant, we have \(p_{1}V_{1} = p_{2}V_{2}\). Therefore, \(1 \times 1 = 4 \times V_{2}\). Solving for \(V_{2}\): \[ V_{2} = \frac{1}{4} = 0.25 \text{ liters} \].

Calculate the volume at 3 bar

Using the relation \(pV = \text{constant}\), we need to find the volume \(V\) when \(p = 3\) bar: \[ 1 \times 1 = 3 \times V \]. Solving for \(V\): \[ V = \frac{1}{3} = 0.333 \text{ liters} \]

Plot the process on a P-V graph

On the P-V graph, plot points (1 liter, 1 bar), (0.333 liters, 3 bar), and (0.25 liters, 4 bar). Connect the points to show the relationship \(pV = \text{constant}\).

For a linear P-V relationship

The linear relationship can be expressed as \(p = aV + b\). Using the initial and final conditions, \(1 = a \times 1 + b\) and \(4 = a \times 0.25 + b\). Solve these equations to find \(a\) and \(b\).

Solve for a and b

From \(1 = a + b\) and \(4 = 0.25a + b\), subtract the first equation from the second: \[ 4 - 1 = 0.25a - a \] to get \[ 3 = -0.75a \] therefore \[ a = -4 \]. Substitute \(a\) back to find \(b\): \[ 1 = -4 + b \] thus \[ b = 5 \].

Find the volume at 3 bar using the linear relation

If \(p = 3\) bar, solve for \(V\): \[ 3 = -4V + 5 \] which gives \[ 4V = 5 - 3 \] so \[ V = 0.5 \text{ liters} \].

Plot the linear process on a P-V graph

On the P-V graph, plot points (1 liter, 1 bar) and (0.25 liters, 4 bar). Because the relationship is linear, draw a straight line between these points.

Key concepts

Pressure-Volume Relationship

In thermodynamics, the pressure-volume (P-V) relationship is crucial for understanding how gases behave when confined within a system. When we say that the relationship between pressure and volume is such that the product remains constant, we are referring to Boyle's Law. This law is mathematically represented as:
\( pV = C \)
where p is pressure, V is volume, and C is a constant. In our exercise, we initially have a pressure of 1 bar and a volume of 1 liter. When the gas is compressed, the pressure increases, and the volume decreases. By understanding this relationship, we can calculate the new volume when the pressure changes, as shown in the exercise steps.
Another type of relationship in our example is a linear P-V relationship represented as:
\( p = aV + b \)
Here, the relationship between pressure and volume changes linearly from one state to another. To find the constants a and b, we use given boundary conditions and solve the resulting equations. This helps in determining intermediate states during the compression.

Piston-Cylinder Assembly

A piston-cylinder assembly is a basic yet fundamental apparatus in thermodynamics. It consists of a cylindrical chamber fitted with a movable piston. This setup helps in studying the behavior of gases under varying pressure and volume. When the gas is compressed or expanded within the cylinder, the piston's position changes to accommodate the new volume of gas.
For instance, in our exercise, we compress the gas inside the cylinder. Initially, the gas is at 1 bar pressure and occupies 1 liter volume. As the gas is compressed to a higher pressure of 3 or 4 bars, the volume decreases proportionally, following the specified P-V relationship.
One can visualize this easily: imagine a syringe (without a needle) filled with air and the plunger that can move freely. As you push the plunger in, the volume of air decreases, but the pressure increases. This is essentially what happens in a piston-cylinder assembly. It provides an excellent way to apply theoretical relationships practically.

P-V Graph

A P-V graph, or Pressure-Volume graph, is a valuable tool in thermodynamics to visualize changes in a system. It plots pressure (P) on the Y-axis and volume (V) on the X-axis. By plotting different states of a system on this graph, one can better understand the process and relationships involved.
In our exercise, we plot the initial state (1 liter, 1 bar) and final states (0.25 liters, 4 bars) and (0.333 liters, 3 bars) on the P-V graph. Since these points represent the relationship pV = constant, they form a hyperbolic curve on the graph. For the linear relationship, the points are connected by a straight line.
Visualizing the changes on a P-V graph helps clarify how pressure and volume are interdependent. When dealing with complex processes, this simple graphical representation can provide insights that would be harder to grasp through equations alone. It helps to easily compare different types of processes like isothermal (constant temperature), isobaric (constant pressure), or adiabatic (no heat exchange), based on their unique curves on the P-V graph.