Thermodynamic Processes for Ideal Gases

Hello! Welcome to your fifth lesson in the thermodynamics module.

In our last lesson, we learned how to apply the First Law of Thermodynamics, $\Delta U=Q-W$, to analyze energy changes in closed systems. We saw how to use property tables for substances like refrigerants to find the change in internal energy.

Today, we'll build directly on that foundation. The learning objective is to analyze common thermodynamic processes for ideal gases: isobaric, isochoric, isothermal, and adiabatic.

By focusing on ideal gases, we can move from relying on tables to using direct, powerful formulas. This formula-based approach is not only faster but also deepens our understanding of the physics. Mastering these four processes is crucial, as they form the fundamental "building blocks" of the thermodynamic cycles that power aircraft engines, such as the Brayton cycle for jet engines.

1. Overview of the Four Key Processes

Let's begin with a concise summary of the four processes. Each is defined by a single property that remains constant. This constraint has a unique impact on the energy balance described by the First Law.

To give these abstract processes a more physical meaning, it's helpful to visualize them. The classic example of a gas trapped in a piston-cylinder device is perfect for this.

2. The Formulas for Ideal Gas Processes

Now we arrive at the core of today's lesson: the specific formulas for calculating work ($W$), heat transfer ($Q$), and internal energy change ($\Delta U$) for an ideal gas undergoing these processes.

Specific Heats: $c_v$ and $c_p$

Before we detail each process, we must introduce two critical properties of an ideal gas: specific heats.

• Specific heat at constant volume ($c_v$): The energy required to raise the temperature of a unit mass of a substance by one degree while its volume is held constant.
• Specific heat at constant pressure ($c_p$): The energy required to raise the temperature of a unit mass of a substance by one degree while its pressure is held constant.

For an ideal gas, the change in specific internal energy ($\Delta u$) and specific enthalpy ($\Delta h$) depend only on temperature change:
$\Delta u=c_v\Delta T=c_v(T_2-T_1)$
$\Delta h=c_p\Delta T=c_p(T_2-T_1)$
These two equations are fundamental and hold true for an ideal gas regardless of the process it undergoes.

Analysis of Each Process

Let's break down each of the four processes.

1. Isochoric (Constant Volume) Process

• Condition: $V=\text{constant}$
• Work ($W$): Since $\Delta V=0$, the boundary work is zero.
  $W=0$
• First Law: $\Delta U=Q-W⟹\Delta U=Q$
• Energy Changes: Any heat added goes directly into increasing the internal energy.
  $\Delta U=m{c}_v\Delta T$
  $Q=\Delta U=m{c}_v\Delta T$

2. Isobaric (Constant Pressure) Process

• Condition: $P=\text{constant}$
• Work ($W$): The gas expands or contracts against a constant pressure.
  $W=P(V_2-V_1)$
  Using the Ideal Gas Law ($PV=mRT$), this can also be written as:
  $W=mR(T_2-T_1)$
• First Law: $\Delta U=Q-W$
• Energy Changes:
  $\Delta U=m{c}_v\Delta T$
  For a constant pressure process, the heat transfer is equal to the change in enthalpy:
  $Q=\Delta H=m{c}_p\Delta T$
  This is because some of the heat energy must also do the work of expansion, which is accounted for in the definition of enthalpy ($H=U+PV$).

The following video provides an excellent, detailed walkthrough of solving an isobaric process problem, including a clear explanation of specific heats.

3. Isothermal (Constant Temperature) Process

• Condition: $T=\text{constant}$
• Internal Energy ($\Delta U$): Since $T$ is constant, $\Delta T=0$. For an ideal gas, this means there is no change in internal energy.
  $\Delta U=m{c}_v\Delta T=0$
• First Law: $\Delta U=Q-W⟹0=Q-W⟹Q=W$
• Energy Changes: All heat added to the system is immediately converted into work done by the system.
  $W={\int }_1^{2}PdV$
  Using $P=mRT/V$, this integrates to:
  $Q=W=mRT\ln \left(\frac{V_2}{V_1}\right)=mRT\ln \left(\frac{P_1}{P_2}\right)$

4. Adiabatic (No Heat Transfer) Process

• Condition: $Q=0$ (a perfectly insulated or very rapid process).
• First Law: $\Delta U=Q-W⟹\Delta U=-W$
• Energy Changes: Any work done by the system comes at the expense of its own internal energy.
  $\Delta U=m{c}_v(T_2-T_1)$
  $W=-\Delta U=-m{c}_v(T_2-T_1)=m{c}_v(T_1-T_2)$
• Process Relations: Unlike the other processes, here P, V, and T all change. They are related by the following equations, where $\gamma =c_p/c_v$ is the specific heat ratio (approx. 1.4 for air):
  $P{V}^{\gamma }=\text{constant}$
  $T{V}^{\gamma -1}=\text{constant}$

Summary

The four processes we analyzed are the theoretical cornerstones for understanding real-world engines. This image provides a concise summary of the key relationships for each.

Key Takeaways:

• For an ideal gas, changes in internal energy depend only on temperature change: $\Delta U=m{c}_v\Delta T$.
• Isochoric (constant V): No work is done ($W=0$), so all heat transfer changes the internal energy ($Q=\Delta U$).
• Isobaric (constant P): Work is done ($W=P\Delta V$), and heat transfer equals the change in enthalpy ($Q=\Delta H=m{c}_p\Delta T$).
• Isothermal (constant T): Internal energy does not change ($\Delta U=0$), so all heat transfer is converted to work ($Q=W$).
• Adiabatic ($Q=0$): No heat is transferred, so work is done at the expense of internal energy ($W=-\Delta U$).

Preview of the Next Lesson:
We have now thoroughly analyzed processes in closed systems where the mass is fixed. However, many aerospace components—like jet engine compressors, combustors, and turbines—are open systems, where mass flows through them. In the next lesson, we will adapt the First Law of Thermodynamics to create the Steady-Flow Energy Equation (SFEE). This will allow us to analyze the energy transfers in these critical open-system devices.