Cyclic Thermodynamic Processes: Videos & Practice Problems

Cyclic thermodynamic processes involve a system returning to its initial state after a series of steps. The total work done in such a cycle equals the area enclosed within the PV diagram loop, calculated using the equation for work in isobaric processes. Additionally, the change in internal energy over a complete cycle is zero, indicating that the total heat transfer equals the total work done. This highlights the conservation of energy principle in thermodynamics, emphasizing the relationship between heat, work, and internal energy.

Properties of Cyclic Thermodynamic Processes

Cyclic thermodynamic processes are essential concepts in thermodynamics, where a system, such as an ideal gas, undergoes a series of steps and returns to its initial state. This cyclical nature allows for the analysis of work done and heat transfer throughout the process. Understanding these processes involves recognizing two key properties: the relationship between work and the area enclosed in the PV diagram, and the behavior of internal energy over a complete cycle.

To calculate the total work done during a cyclic process, one must sum the work done in each individual step. In a PV diagram, the work done can be represented as the area enclosed by the path taken. For example, if the process includes isobaric (constant pressure) and isovolumetric (constant volume) steps, the work done during isovolumetric steps is zero, simplifying the calculation. The work done during isobaric processes can be calculated using the formula:

W = P \Delta V

where \(W\) is the work done, \(P\) is the pressure, and \(\Delta V\) is the change in volume. The total work done over the cycle can be determined by calculating the area of the rectangle formed in the PV diagram, which also equals the total work done. This leads to the important conclusion that the total work done in a cyclic process is equal to the area enclosed within the loop on the PV diagram.

Another significant aspect of cyclic processes is the change in internal energy (\(\Delta E\)). The internal energy of a system depends solely on its state, not on the path taken to reach that state. For a complete cycle, the change in internal energy is given by:

\Delta E = Q - W

However, since the system returns to its initial state, the overall change in internal energy for the cycle is zero:

Q = W

In summary, cyclic processes reveal that the work done is represented by the area enclosed in the PV diagram, and the internal energy change over a complete cycle is zero, leading to the conclusion that the heat transferred equals the work done. Understanding these principles is crucial for analyzing thermodynamic systems and their behaviors.

Signs in a Three-Step Cyclic Process

In thermodynamics, understanding the relationships between heat (q), work (w), and internal energy (ΔE_internal) is crucial, especially when analyzing processes depicted in a Pressure-Volume (PV) diagram. The first law of thermodynamics states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system:

$$\Delta E_{internal} = q - w$$

In a cyclic process, the system returns to its initial state, which implies that the total change in internal energy over one complete cycle is zero:

$$\Delta E_{cycle} = \Delta E_{AB} + \Delta E_{BC} + \Delta E_{CA} = 0$$

For the first segment of the cycle, from point A to B, which is an isobaric process (constant pressure), the internal energy increases. This indicates that heat is added to the system and work is done by the system. Therefore, both q and w are positive, leading to a positive change in internal energy:

1. **From A to B:**
- ΔE_internal: Positive
- q: Positive
- w: Positive

In the second segment, from B to C, the process is isovolumetric (constant volume). Here, no work is done (w = 0), and since heat is added, the internal energy also increases. Thus, both q and ΔE_internal are positive, while w remains zero:

2. **From B to C:**
- ΔE_internal: Positive
- q: Positive
- w: 0

For the final segment, from C back to A, the process is not isobaric or isovolumetric, but rather a cyclic process. Since the total change in internal energy for the cycle must equal zero, and the previous two segments contributed positively to the internal energy, the change in internal energy from C to A must be negative. Consequently, the work done in this segment is also negative, as it corresponds to moving left on the PV diagram, indicating that work is done on the system. This leads to a negative heat transfer as well:

3. **From C to A:**
- ΔE_internal: Negative
- q: Negative
- w: Negative

In summary, analyzing the signs of q, w, and ΔE_internal in a cyclic process involves applying the first law of thermodynamics and understanding the characteristics of each segment of the cycle. This conceptual approach allows for a logical determination of the relationships between these thermodynamic variables without the need for numerical calculations.

An ideal gas is taken through the four processes shown below. The changes in internal energy for three of these processes are as follows: ΔE_AB = +82 J; ΔE_BC = +15 J; ΔE_DA = –56 J. Find the change in internal energy for the process from C to D.

• A. - 153 J
• B. 41 J
• C. - 41 J
• D. Cannot determine

Answer: (C)

Heat Added Over Complete Cycle

In a cyclic thermodynamic process, the total amount of heat added over the complete cycle can be determined using the principles of thermodynamics. For a gas undergoing such a process, the key concept is that the change in internal energy is zero when the system returns to its initial state. This means that the total heat added, denoted as \( Q_{\text{total}} \), is equal to the total work done, \( W_{\text{total}} \), during the cycle, as stated by the first law of thermodynamics:

\( \Delta U = Q - W \)

Since \( \Delta U = 0 \) for a complete cycle, we can simplify this to:

\( Q_{\text{total}} = W_{\text{total}} \)

The work done in a cyclic process can be visualized as the area enclosed by the path on a pressure-volume (PV) diagram. To calculate this area, one can use geometric formulas depending on the shape of the enclosed area. For example, if the area is a triangle, the work done can be calculated using the formula for the area of a triangle:

\( W = \frac{1}{2} \times \text{base} \times \text{height} \)

In this scenario, if the base of the triangle is 3 units and the height is 30 units, the work done over the cycle would be:

\( W = \frac{1}{2} \times 3 \times 30 = 45 \, \text{joules} \)

However, it is important to consider the direction of the cycle. If the cycle is traversed in a counterclockwise direction, the work done is considered negative. Therefore, in this case, the total work done and the total heat added over the cycle would be:

\( Q_{\text{total}} = -45 \, \text{joules} \)

This indicates that more heat is removed from the system than is added, resulting in a net negative heat transfer for the cycle.

Here’s what students ask on this topic:

What is a cyclic thermodynamic process?

A cyclic thermodynamic process is one in which a system, such as an ideal gas, undergoes a series of steps and returns to its initial state. This means that the state variables (pressure, volume, and temperature) are the same at the beginning and end of the cycle. In a PV diagram, this process forms a closed loop. The total work done in the cycle is equal to the area enclosed by this loop. Additionally, the change in internal energy over a complete cycle is zero, indicating that the total heat transfer equals the total work done.

How do you calculate the work done in a cyclic process?

The work done in a cyclic process can be calculated by finding the area enclosed by the loop in a PV diagram. For isobaric processes, the work done is given by the equation:

$W=P∆V$

where $P$ is the pressure and $∆V$ is the change in volume. The total work done is the sum of the work done in each individual process within the cycle. For a rectangular loop, the area (and thus the work done) is simply the product of the base and height of the rectangle.

Why is the change in internal energy zero in a cyclic process?

The change in internal energy ($\mathrm{∆E}$) over a complete cycle is zero because internal energy is a state function. This means it depends only on the state of the system, not on the path taken to reach that state. Since the system returns to its initial state in a cyclic process, the initial and final internal energies are the same, resulting in $\mathrm{∆E}=0$. Consequently, the total heat transfer (Q) equals the total work done (W) over the cycle, as given by the first law of thermodynamics: $\mathrm{∆E}=Q-W$.

What is the significance of the direction of the loop in a cyclic process?

The direction of the loop in a cyclic process indicates the sign of the work done. If the loop runs clockwise in a PV diagram, the work done by the system is positive. Conversely, if the loop runs counterclockwise, the work done by the system is negative. This is because the area under the curve represents the work done, and the direction determines whether the work is done on or by the system. Understanding this helps in analyzing energy transfer and efficiency in thermodynamic cycles.

How is the total heat transfer related to the work done in a cyclic process?

In a cyclic process, the total heat transfer (Q) is equal to the total work done (W) by the system. This relationship arises from the fact that the change in internal energy ($\mathrm{∆E}$) over a complete cycle is zero. According to the first law of thermodynamics:

$\mathrm{∆E}=Q-W$

Since $\mathrm{∆E}=0$ for a cyclic process, it follows that $Q=W$. This highlights the conservation of energy principle, emphasizing that the energy added as heat is entirely converted to work over the cycle.