Physics

Chapter 3: The First Law of Thermodynamics

Imron Rosyadi

Introduction to Thermodynamics

Learning Objectives

By the end of this section, you will be able to:

• Define a thermodynamic system, its boundary, and its surroundings.
• Explain the roles of all the components involved in thermodynamics.
• Define thermal equilibrium and thermodynamic temperature.
• Link an equation of state to a system.

Thermodynamic Systems and Surroundings

A thermodynamic system is any part of the universe whose thermodynamic properties are of interest.

• Boundary: The imagined wall separating the system from its surroundings.
• Surroundings (Environment): Everything outside the system that interacts with it.
  • Example: For a car engine, burning gasoline is the system; the piston, exhaust, radiator, and air are the surroundings. The cylinder walls and piston form the boundary.

Figure 3.2: (a) General system and surroundings. (b) Car engine as a system.

Thermodynamics focuses on macroscopic properties rather than individual molecules. This simplification allows us to treat systems as continuous and in equilibrium.

We usually assume uniform properties within the system, simplifying analysis.

Types of Systems

• Isolated System: Completely separated from its environment; no exchange of heat or matter.
  • Idealized concept, often used as a model.
  • Example: Gas in immovable, thermally insulating walls.
• Closed System: Can exchange energy (heat, work) but not matter with its surroundings.
  • Example: A pressure cooker (approximately).
• Open System: Can exchange both energy and matter with its surroundings.
  • Most real-world systems are open.
  • Example: A boiling tea kettle.

Figure 3.3: (a) Boiling tea kettle (open system). (b) Pressure cooker (closed system approximation).

The distinction between system types is crucial for setting up thermodynamic problems and applying the correct laws. For most of this chapter, we’ll consider closed systems unless specified.

Thermal Equilibrium & Zeroth Law

Macroscopic Properties: We focus on average properties (pressure, volume, temperature) rather than individual molecular behaviors. - A system is typically assumed to be in equilibrium for simplified analysis.

Thermal Equilibrium: When two objects (or parts of a system) in thermal contact reach the same temperature, and there is no net heat transfer between them over time.

Zeroth Law of Thermodynamics:

If object 1 is in thermal equilibrium with objects 2 and 3, respectively, then objects 2 and 3 must also be in thermal equilibrium.

\[ \text{If } T_1 = T_2 \text{ and } T_1 = T_3, \text{ then } T_2 = T_3. \]

Note

This law fundamentally defines temperature: objects in thermal equilibrium have the same temperature.

The zeroth law is often overlooked because it seems intuitive, but it’s essential for defining temperature consistently across different systems and for building thermometers. It establishes temperature as a fundamental state variable.

Equation of State

To characterize a thermodynamic system, we use an equation of state:

A relationship between the basic measurable properties of the system (volume, pressure, temperature).

For a closed system, it’s symbolically written as: \[ f(p, V, T) = 0 \]

• \(V\): Volume
• \(p\): Pressure
• \(T\): Temperature

Tip

Example: For an ideal gas, the equation of state is \(pV - nRT = 0\), or \(pV = nRT\).

Variables:

• Extensive Variable: Depends on the amount of matter (e.g., volume, total energy, number of moles).
• Intensive Variable: Independent of the amount of matter (e.g., pressure, temperature).

The equation of state is like a “fingerprint” for a substance, describing how its macroscopic properties are interconnected. While theoretically existing for any system, finding its exact form can be challenging and often requires experimental data. The ideal gas law is a simple, yet powerful, approximation.

Work, Heat, and Internal Energy

Learning Objectives

By the end of this section, you will be able to:

• Describe the work done by a system, heat transfer between objects, and internal energy change of a system.
• Calculate the work, heat transfer, and internal energy change in a simple process.

Work Done by a System

Work is done when a force moves an object through a displacement. In thermodynamics, a system (like a gas) can do work by changing its volume against an external pressure.

Consider a gas in a cylinder with a movable piston:

• If the gas expands, it pushes the piston outwards, doing work.
• If the piston pushes inwards, work is done on the gas (compression).

The infinitesimal work done by the gas is: \[ dW = Fdx = pAdx \] Since \(dV = Adx\) (change in volume), this becomes: \[ dW = p dV \]

Figure 3.4: Work done by an expanding gas.

This definition of work (pdV) is specific to systems that undergo volume changes, like gases. For other systems (e.g., stretched rubber bands), the work term might look different.

The sign convention for work is crucial: positive work is done by the system (e.g., expansion), while negative work is done on the system (e.g., compression).

Calculating Work

For a finite change in volume from \(V_1\) to \(V_2\), the net work is found by integration:

\[ W = \int_{V_1}^{V_2} p dV \]

• This integral is valid for a quasi-static process: A process that occurs slowly enough for the system to remain in thermal equilibrium at each infinitesimal step.
• Graphically, work is the area under the \(pV\) curve on a pressure-volume diagram.
  • Work is positive for expansion (\(V_2 > V_1\)).
  • Work is negative for compression (\(V_2 < V_1\)).

Figure 3.5: Work done by gas expansion as the area under the pV curve.

Quasi-static processes are an idealization that simplifies calculations significantly. In reality, all processes involve some degree of non-equilibrium, but for many practical purposes, the quasi-static approximation is sufficient. Emphasize the graphical interpretation of work as the area under the curve—this is a very useful conceptual tool.

Work is Path Dependent

The work done by a system depends on the specific thermodynamic path taken between initial and final states.

Figure 3.6: Three different paths (ABC, AC, ADC) between states A and C.

• Path AC (Isothermal Expansion): Constant temperature \(T\).
  • For an ideal gas: \(p = \frac{nRT}{V}\)
  • \(W = \int_{V_1}^{V_2} \frac{nRT}{V} dV = nRT \ln \left(\frac{V_2}{V_1}\right)\)
• Path ABC: Isobaric expansion (A to B) then isochoric cooling (B to C).
  • From A to B (\(p_1\) constant): \(W_{AB} = p_1(V_2 - V_1)\)
  • From B to C (\(V_2\) constant): \(W_{BC} = 0\)
  • Total work: \(W_{ABC} = p_1(V_2 - V_1)\)

Notice \(W_{AC} \neq W_{ABC}\), showing path dependence.

This path dependence is a key concept in thermodynamics, distinguishing work from state functions like internal energy. Students should understand why the area under the curve changes for different paths.

You can draw the different areas for each path on the board to visually reinforce the concept.

Internal Energy (\(E_{int}\))

The internal energy of a thermodynamic system is the sum of the mechanical energies of all its constituent molecules or entities.

\[ E_{int} = \sum_i (\overline{K_i} + \overline{U_i}) \]

• \(\overline{K_i}\): Average kinetic energy of molecule \(i\) (translational, rotational, vibrational).
• \(\overline{U_i}\): Average potential energy due to interactions between molecules.

For an ideal monatomic gas:

• Only translational kinetic energy: \(\overline{K_i} = \frac{1}{2}m_i \overline{v_i^2}\).
• No interatomic interactions: \(\overline{U_i} = 0\).
• From kinetic theory, \(\frac{1}{2}m_i \overline{v_i^2} = \frac{3}{2} k_B T\).

So, for \(n\) moles of an ideal monatomic gas: \[ E_{int} = n N_A \left(\frac{3}{2} k_B T\right) = \frac{3}{2} nRT \]

Important

The internal energy of an ideal monatomic gas depends only on temperature (\(T\)), not on pressure or volume.

It’s important to differentiate between the macroscopic internal energy of the system and the microscopic energies of its constituent particles. The internal energy is a state function, meaning its value depends only on the current state of the system, not on how it got there. This contrasts with work and heat, which are path-dependent. For real gases or polyatomic gases, internal energy also includes rotational and vibrational modes, making it more complex, but the principle remains.

First Law of Thermodynamics

Learning Objectives

By the end of this section, you will be able to:

• State the first law of thermodynamics and explain how it is applied.
• Explain how heat transfer, work done, and internal energy change are related in any thermodynamic process.

The First Law of Thermodynamics

The First Law of Thermodynamics is a statement of energy conservation for thermodynamic systems. It relates changes in internal energy to heat transfer and work done.

If \(Q\) is the heat exchanged between a system and its environment, and \(W\) is the work done by or on the system, the change in internal energy (\(\Delta E_{int}\)) of the system is:

\[ \Delta E_{int} = Q - W \]

Sign Conventions:

Quantity	Description	Sign
\(Q > 0\)	Heat added to system	Positive
\(Q < 0\)	Heat removed from system	Negative
\(W > 0\)	Work done by system (expansion)	Positive
\(W < 0\)	Work done on system (compression)	Negative

The first law is arguably the most important in thermodynamics. It quantifies how energy flows into and out of a system and how it is stored.

Emphasize that the sign conventions are arbitrary but must be consistent. This particular convention is common in physics. In chemistry, work done on the system is often positive.

Internal Energy as a State Function

While \(Q\) and \(W\) are path-dependent, their difference (\(\Delta E_{int} = Q - W\)) is path-independent.

Figure 3.7: Different paths between states A and B.

• For any path from state A to state B, \(\Delta E_{int}\) is the same.
• State function: A property whose value depends only on the current state of the system, not on the path taken to reach that state (e.g., \(E_{int}\), pressure, volume, temperature).

Differential form of the First Law:

\[ dE_{int} = dQ - dW \] - For infinitesimal changes, where \(dE_{int}\) is the infinitesimal change in internal energy, \(dQ\) is infinitesimal heat exchanged, and \(dW\) is infinitesimal work done.

This concept of state function is fundamental. It means we don’t need to know the entire history of a system to determine its internal energy. This simplifies calculations greatly, especially for complex processes.

This property allows us to define internal energy as a property of the system itself.

Example: Changes of State and the First Law

A system moves from state A to state B, receiving 400 J of heat and doing 100 J of work.

1. Change in internal energy (A to B):

   \(\Delta E_{int, AB} = Q_{AB} - W_{AB} = 400 \text{ J} - 100 \text{ J} = 300 \text{ J}\).

2. Change in internal energy (B to A):

   For a closed path, \(\Delta E_{int} = 0\). So, \(\Delta E_{int, AB} + \Delta E_{int, BA} = 0\).

   \(\Delta E_{int, BA} = - \Delta E_{int, AB} = -300 \text{ J}\).

3. Heat absorbed along a different path (A to B):

   If \(W'_{AB} = -400 \text{ J}\) (work done on the system) along a different path from A to B, \(\Delta E_{int, AB} = Q'_{AB} - W'_{AB}\)

   \(300 \text{ J} = Q'_{AB} - (-400 \text{ J})\)

   \(300 \text{ J} = Q'_{AB} + 400 \text{ J}\)

   \(Q'_{AB} = -100 \text{ J}\). (Heat is removed from the system).

Note

Internal energy change is independent of the path. Heat and work depend on the path.

Walk through each step of the example clearly, reinforcing the sign conventions and the state function concept. Part (c) is particularly good for illustrating that different paths can involve different amounts of heat and work, even if the change in internal energy is the same.

Thermodynamic Processes

Learning Objectives

By the end of this section, you will be able to:

• Define a thermodynamic process.
• Distinguish between quasi-static and non-quasi-static processes.
• Calculate physical quantities, such as the heat transferred, work done, and internal energy change for isothermal, adiabatic, and cyclical thermodynamic processes.

What is a Thermodynamic Process?

A thermodynamic process is the manner in which a system’s state changes from an initial state to a final state.

• Characterized by changes in thermodynamic variables (e.g., pressure, volume, temperature, moles for an ideal gas).

Quasi-static Process: - An idealized process where the change occurs infinitesimally slowly. - The system remains in thermodynamic equilibrium at every instant. - Macroscopic properties (P, V, T) are well-defined throughout the process. - Can be represented by a well-defined path on a state diagram (e.g., \(pV\) diagram).

Non-quasi-static Process: - Occurs rapidly, involving non-equilibrium intermediate states. - Macroscopic properties may not be well-defined during the process. - Cannot be represented by a defined path on a state diagram.

Figure 3.8: Quasi-static vs. non-quasi-static processes.

Emphasize that quasi-static processes are theoretical idealizations but are extremely useful for analysis. Real processes are always non-quasi-static to some degree.

The ability to plot a path on a pV diagram is a hallmark of a quasi-static process.

Types of Quasi-Static Processes

Several important types of quasi-static processes are defined by keeping one thermodynamic variable constant:

1. Isothermal Process:
   • Constant temperature (\(T = \text{constant}\)).
   • System is in thermal equilibrium with a large heat bath.
   • For an ideal gas, \(pV = \text{constant}\) (hyperbolic curve on \(pV\) diagram). Figure 3.10: Isothermal expansion on a pV diagram.
2. Isobaric Process:
   • Constant pressure (\(p = \text{constant}\)).
   • Work done: \(W = p(V_2 - V_1)\).
3. Isochoric Process:
   • Constant volume (\(V = \text{constant}\)).
   • Work done: \(W = \int p dV = 0\).
4. Adiabatic Process:
   • No heat transfer (\(Q = 0\)).
   • System is thermally insulated from its environment.
   • Expansion leads to cooling; compression leads to heating. Figure 3.11: Adiabatic expansion.

For each process, briefly describe how it’s achieved experimentally and what its implications are for the first law. For example, for isochoric, \(W=0 \Rightarrow \Delta E_{int} = Q\). For adiabatic, \(Q=0 \Rightarrow \Delta E_{int} = -W\). The pV diagram representation for each type of process is important for conceptual understanding.

Cyclic Processes

A cyclic process is one where the system returns to its initial state at the end of the process.

• Since internal energy is a state function, its change over a complete cycle is zero: \[ \Delta E_{int, \text{cycle}} = 0 \]
• Applying the First Law of Thermodynamics: \[ Q_{\text{cycle}} - W_{\text{cycle}} = \Delta E_{int, \text{cycle}} = 0 \] Therefore: \[ Q_{\text{cycle}} = W_{\text{cycle}} \quad (\text{for a cyclic process}) \] This means the net heat added to the system equals the net work done by the system over a complete cycle.

Note

Reversible Process: A process that can be made to retrace its path by differential changes in the environment. All reversible processes must be quasi-static. (Not all quasi-static processes are reversible, e.g., if friction is involved).

Cyclic processes are fundamental to understanding engines and refrigerators. They demonstrate how energy can be continuously converted between heat and work.

The concept of reversibility is closely linked to the second law of thermodynamics, which will be covered later, but it’s good to introduce it here.

Heat Capacities of an Ideal Gas

Learning Objectives

By the end of this section, you will be able to:

• Define heat capacity of an ideal gas for a specific process.
• Calculate the specific heat of an ideal gas for either an isobaric or isochoric process.
• Explain the difference between the heat capacities of an ideal gas and a real gas.
• Estimate the change in specific heat of a gas over temperature ranges.

Molar Heat Capacity at Constant Volume (\(C_V\))

Consider an ideal gas in a fixed-volume vessel (isochoric process).

• Since volume is constant, \(dV=0\), so work done \(dW = p dV = 0\).
• From the First Law: \(dE_{int} = dQ - dW = dQ\).
• The heat added at constant volume is related to the temperature change by: \[ dQ = nC_V dT \] where \(n\) is the number of moles and \(C_V\) is the molar heat capacity at constant volume.

Therefore, for an ideal gas, the change in internal energy for any process is: \[ dE_{int} = nC_V dT \] This is because internal energy of an ideal gas depends only on temperature.

It is crucial to emphasize that while \(C_V\) is measured at constant volume, the relationship \(dE_{int} = nC_V dT\) holds for any process in an ideal gas, because \(E_{int}\) is solely a function of temperature for an ideal gas. This is a common point of confusion.

Molar Heat Capacity at Constant Pressure (\(C_p\))

Consider an ideal gas in a vessel with a movable piston, under constant external pressure (isobaric process).

• As heat is added, the gas expands, doing work \(dW = p dV\).
• The heat added at constant pressure is: \[ dQ = nC_p dT \] where \(C_p\) is the molar heat capacity at constant pressure.

From the First Law (\(dE_{int} = dQ - dW\)):

\[ dE_{int} = nC_p dT - p dV \]

For an ideal gas, \(pV = nRT\). Differentiating with constant \(p\):

\(p dV = nR dT\).

Substitute this into the First Law:

\[ dE_{int} = nC_p dT - nR dT = n(C_p - R) dT \]

Since \(dE_{int}\) for an ideal gas is always \(nC_V dT\):

\[ nC_V dT = n(C_p - R) dT \]

This leads to the important relationship:

\[ C_p = C_V + R \] or \[ C_p - C_V = R \]

Explain why \(C_p\) is greater than \(C_V\): at constant pressure, some of the added heat goes into doing work of expansion, in addition to increasing the internal energy. At constant volume, all the added heat goes into increasing internal energy. This relationship is a direct consequence of the ideal gas law and the first law of thermodynamics.

Heat Capacities and Degrees of Freedom

From the kinetic theory of gases, we found the molar heat capacity at constant volume to be: \[ C_V = \frac{d}{2}R \] where \(d\) is the number of degrees of freedom of a molecule.

Using \(C_p = C_V + R\): \[ C_p = \frac{d}{2}R + R = \left(\frac{d}{2} + 1\right)R \]

Gas Type	Degrees of Freedom (\(d\))	\(C_V\)	\(C_p\)	\(C_p - C_V\)
Monatomic	3 (translational)	\(\frac{3}{2}R\)	\(\frac{5}{2}R\)	\(R\)
Diatomic	5 (3 trans + 2 rot)	\(\frac{5}{2}R\)	\(\frac{7}{2}R\)	\(R\)
Polyatomic	6 (3 trans + 3 rot)	\(3R\) (approx)	\(4R\) (approx)	\(R\)

Note

Real Gases: Heat capacities of real gases are often slightly higher due to vibrational motion, especially at higher temperatures, but \(C_p - C_V\) remains close to \(R\).

Connect this back to the kinetic theory. Explain degrees of freedom: independent ways a molecule can store energy (translational, rotational, vibrational). Mention that the values in the table are approximations, particularly for polyatomic gases, as vibrational modes can be “activated” at higher temperatures, increasing \(d\).

Adiabatic Processes for an Ideal Gas

Learning Objectives

By the end of this section, you will be able to:

• Define adiabatic expansion of an ideal gas.
• Demonstrate the qualitative difference between adiabatic and isothermal expansions.

Adiabatic Process (\(Q=0\))

In an adiabatic process, no heat is allowed to enter or leave the system (\(Q=0\)). - This occurs when the system is perfectly insulated or the process is very fast.

Consequences of Adiabatic Processes: - Compression: Work is done on the gas (\(W < 0\)), so \(\Delta E_{int} = -W > 0\). Internal energy increases, leading to a rise in temperature. - Expansion: Gas does work on the surroundings (\(W > 0\)), so \(\Delta E_{int} = -W < 0\). Internal energy decreases, leading to a drop in temperature.

Free Expansion of an Ideal Gas:

• Gas expands into a vacuum (against \(p=0\)).
• \(W = \int p dV = 0\) (no work done).
• If insulated (\(Q=0\)), then \(\Delta E_{int} = Q - W = 0\).
• For an ideal gas, \(\Delta E_{int} = 0 \Rightarrow \Delta T = 0\). So, temperature does not change during free expansion of an ideal gas.

Figure 3.13: Free expansion of a gas.

Highlight that adiabatic processes are critical in many engineering applications, like internal combustion engines (compression stroke) and refrigeration cycles.

The free expansion example is a good counter-intuitive result that shows the power of the first law for ideal gases.

Quasi-static Adiabatic Process for an Ideal Gas

For a quasi-static adiabatic process of an ideal gas, the following relationships hold:

\[ pV^\gamma = \text{constant} \] where \(\gamma\) is the adiabatic index or ratio of molar heat capacities: \[ \gamma = \frac{C_p}{C_V} \] Since \(C_p = C_V + R\), we know \(\gamma > 1\).

Other forms of the adiabatic condition: \[ TV^{\gamma-1} = \text{constant} \] \[ T^\gamma p^{1-\gamma} = \text{constant} \]

Tip

Example: For a monatomic ideal gas, \(C_V = \frac{3}{2}R\), \(C_p = \frac{5}{2}R\), so \(\gamma = \frac{5/2 R}{3/2 R} = \frac{5}{3} \approx 1.67\).

For a diatomic ideal gas, \(C_V = \frac{5}{2}R\), \(C_p = \frac{7}{2}R\), so \(\gamma = \frac{7/2 R}{5/2 R} = \frac{7}{5} = 1.40\).

The derivation of \(pV^\gamma = \text{constant}\) is given in the text, but the focus here should be on understanding and applying the result.

Explain what gamma represents: it’s a measure of how “stiff” the gas is with respect to temperature changes during volume changes.

Adiabatic vs. Isothermal Expansion

Comparing paths on a \(pV\) diagram:

Figure 3.15: Adiabatic vs. isothermal expansions.

• Isothermal curve (\(T = \text{constant}\)):
  • \(p = \frac{nRT}{V} \Rightarrow p \propto \frac{1}{V}\)
  • Slope: \(\frac{dp}{dV} = -\frac{p}{V}\)
• Adiabatic curve (\(pV^\gamma = \text{constant}\)):
  • \(p = \frac{\text{constant}}{V^\gamma} \Rightarrow p \propto \frac{1}{V^\gamma}\)
  • Slope: \(\frac{dp}{dV} = -\gamma \frac{p}{V}\)

Since \(\gamma > 1\), the adiabatic curve is steeper than the isothermal curve at any given point. - This means for a given volume change, the pressure drops more steeply in an adiabatic expansion (due to temperature decrease) than in an isothermal expansion (where temperature is kept constant).

This visual comparison is very important for understanding the qualitative differences between these two fundamental processes. Emphasize that the greater steepness of the adiabatic curve is directly related to the temperature drop during expansion (or rise during compression) because no heat is exchanged to maintain constant temperature.

Key Takeaways

1. Thermodynamic Systems: Defined by their boundaries and interaction with surroundings (isolated, closed, open).
2. State Functions: Internal energy (\(E_{int}\)), pressure (\(p\)), volume (\(V\)), and temperature (\(T\)) depend only on the system’s current state, not the path.
3. Zeroth Law: Defines thermal equilibrium and temperature; if \(T_1=T_2\) and \(T_1=T_3\), then \(T_2=T_3\).
4. Work and Heat: Both are path-dependent ways of transferring energy into or out of a system. Work done by a gas: \(W = \int p dV\).
5. First Law of Thermodynamics: Energy conservation: \(\Delta E_{int} = Q - W\).
6. Quasi-static Processes: Idealized slow processes where the system is always in equilibrium.
7. Special Processes for Ideal Gases:
   • Isothermal (\(T=\) constant): \(\Delta E_{int} = 0\), so \(Q=W\). \(W = nRT \ln(V_2/V_1)\).
   • Isochoric (\(V=\) constant): \(W=0\), so \(\Delta E_{int} = Q = nC_V \Delta T\).
   • Isobaric (\(p=\) constant): \(W=p \Delta V\). \(\Delta E_{int} = Q-W = nC_p \Delta T - p \Delta V\).
   • Adiabatic (\(Q=0\)): \(\Delta E_{int} = -W\). \(pV^\gamma = \text{constant}\) where \(\gamma = C_p/C_V\).
8. Heat Capacities: \(C_p = C_V + R\) for ideal gases, due to work done during constant pressure expansion. \(C_V = \frac{d}{2}R\), where \(d\) is degrees of freedom.
9. Cyclic Process: \(\Delta E_{int} = 0\), so \(Q_{\text{cycle}} = W_{\text{cycle}}\).

Key Equations

Equation	Description
\(dW = p dV\)	Infinitesimal work done by a gas
\(W = \int_{V_1}^{V_2} p dV\)	Work done by a gas for a finite change in volume
\(E_{int} = \frac{3}{2} nRT\)	Internal energy of an ideal monatomic gas
\(\Delta E_{int} = Q - W\)	First Law of Thermodynamics
\(W = nRT \ln \left(\frac{V_2}{V_1}\right)\)	Work done in isothermal process (ideal gas)
\(dE_{int} = nC_V dT\)	Change in internal energy (ideal gas, any process)
\(C_p = C_V + R\)	Relation between molar heat capacities (ideal gas)
\(pV^\gamma = \text{constant}\)	Condition for quasi-static adiabatic process (ideal gas)
\(\gamma = \frac{C_p}{C_V}\)	Adiabatic index

Key Terms

Term	Definition
Thermodynamic System	Anything whose thermodynamic properties are of interest.
Surroundings	Everything outside the system that interacts with it.
Boundary	Imagined wall separating the system from its surroundings.
Isolated System	A system completely separated from its environment (no heat or matter exchange).
Closed System	A system that can exchange energy (heat, work) but not matter with its surroundings.
Open System	A system that can exchange both energy and matter with its surroundings.
Thermal Equilibrium	State where objects in thermal contact have the same temperature and no net heat transfer.
Zeroth Law of Thermodynamics	If objects A and B are each in thermal equilibrium with object C, then A and B are in thermal equilibrium with each other.
Equation of State	A mathematical relationship between a system’s measurable properties (e.g., \(f(p, V, T) = 0\)).
Extensive Variable	A property that depends on the amount of matter in the system (e.g., volume).
Intensive Variable	A property that is independent of the amount of matter in the system (e.g., temperature).
Internal Energy (\(E_{int}\))	The sum of the mechanical energies of all molecules in a system.
First Law of Thermodynamics	States that the change in internal energy of a system equals the heat added to the system minus the work done by the system (\(\Delta E_{int} = Q - W\)).
State Function	A property whose value depends only on the current state of the system, not on the path taken to reach that state.
Thermodynamic Process	The manner in which a system’s state changes from an initial to a final state.
Quasi-static Process	An idealized process that occurs infinitesimally slowly, maintaining thermal equilibrium at each step.
Isothermal Process	A process in which the temperature of the system remains constant (\(\Delta T = 0\)).
Isobaric Process	A process in which the pressure of the system remains constant (\(\Delta p = 0\)).
Isochoric Process	A process in which the volume of the system remains constant (\(\Delta V = 0\)).
Adiabatic Process	A process in which no heat is exchanged between the system and its surroundings (\(Q = 0\)).
Cyclic Process	A process in which the system returns to its initial state (\(\Delta E_{int} = 0\)).
Molar Heat Capacity at Constant Volume (\(C_V\))	Heat required to raise the temperature of 1 mole of gas by 1 K at constant volume.
Molar Heat Capacity at Constant Pressure (\(C_p\))	Heat required to raise the temperature of 1 mole of gas by 1 K at constant pressure.
Adiabatic Index (\(\gamma\))	The ratio of molar heat capacities, \(\gamma = C_p/C_V\).