Thermodynamic State Variables And Equation Of State

Edited By Vishal kumar | Updated on Jul 02, 2025 06:28 PM IST

Thermodynamic variables such as temperature, pressure, volume and internal energy are what define the state of a thermodynamic system whose importance cannot be overemphasized. However, the quartet is crucial in deploying the attributes of the state. As such, the equation of state seeks to harmonize them all.

This Story also Contains

Definition of Thermodynamic Variables
What are the Extensive and Intensive properties/variables
What is the Equation of state?
What is the Thermodynamic process?
What is the State and Path function?
Solved Examples Based on Thermodynamic Variables and Equation of State
Summary

In this article, we will cover the concept of the 'Thermodynamic variables and equation of state’. This topic falls under the broader category of Thermodynamics, which is a crucial chapter in Class 11 physics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), National Eligibility Entrance Test (NEET), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), two questions have been asked on this concept. And for NEET one question was asked from this concept.

Let's read this entire article to gain an in-depth understanding of thermodynamic variables and the equation of state.

Definition of Thermodynamic Variables

Thermodynamic variables: Any thermodynamic system can be described by specifying some of the variables i.e. its pressure(P), volume(V), temperature(T), internal energy(U) and the number of moles(n). These parameters are called thermodynamic variables.

What are the Extensive and Intensive properties/variables

Intensive properties do not depend on the amount of matter that is present. These are bulk properties. Examples of intensive properties are - Density, Temperature etc. and the Extensive properties are those properties which depend on the amount of matter that is present. Examples of extensive properties are - Volume, Weight etc.

What is the Equation of state?

The relation between the thermodynamic variables (P, V, T) of the system is called the equation of state.

For n moles of an ideal gas, the equation of state is PV = nRT

For n moles of a real gas equation of state is $(P + \frac{an^{2}}{V^{2}})(V- nb) = nRT$

What is the Thermodynamic process?

The process of change of state of a system involves the change of thermodynamic variables such as pressure P, volume V and temperature T of the system. The process is known as the thermodynamic process.

Some important processes are

(i) Isothermal process (ii) Adiabatic process (iii) Isobaric process (iv) Isochoric process

(v) Cyclic and non-cyclic process (vi) Reversible and irreversible process

Later, we will study all these processes one by one in detail.

What is the State and Path function?

State or Point function does not depend on the path followed by the thermodynamic process but it depends on the final and initial position of the process. The Path function depends on the path followed by a thermodynamic process and not on the initial and final states of the system. An example of a point function is Internal energy and an example of a path function is Heat and work.

Solved Examples Based on Thermodynamic Variables and Equation of State

Example 1: One mole of an ideal gas passes through a process where pressure and volume obey the relation $P = P_0 [ 1- \frac{1}{2}\left ( \frac{V_0}{V} \right )^2]$ .Here $P_0$ and $V_0$ are constant. Calculate the change in temperature of gas if its volume changes from $V_0 \: \: to \: \: 2 V_0$ :

1) $\frac{1P_0 V_0 }{2R}$

2) $\frac{5 P_0 V_0 }{4 R}$

3) $\frac{3 P_0 V_0 }{4 R}$

4) $\frac{ P_{0} V_{0}}{4 R}$

Solution:

$PV=nRT$

$\frac{nRT }{V} = P_0 [ 1- \frac{1}{2}\left ( \frac{V_0}{V} \right )^2] \\\\ \\T = \frac{P_0 V }{nR} [ 1- \frac{1}{2}\left ( \frac{V_0}{V} \right )^2] \\\\ T_i = \frac{P_0 V_0 }{R} [ 1- \frac{1}{2}\left ( \frac{V_0^2}{V_0^2} \right )].... at \ V = V_0 \\\\ T_f = \frac{P_0 2V_0 }{R} [ 1- \frac{1}{8}\left ( \frac{V_0^2}{V_0^2} \right )].... \ at \ V = 2 V_0$

$\delta T = T_f - T_i = \frac{7 P_0 V_0 }{4 R } - \frac{P_0 V_0 }{2 R } = \frac{5 P_0 V_0 }{4 R }$

Hence, the answer is the option (2).

Example 2: Which of the following is not a thermodynamic variable?

1) Pressure

2) Temperature

3) Momentum

4) Volume

Solution:

Thermodynamics Variable

Parameters which define the Thermodynamics system.

e.g. pressure, volume, temperature

Hence, the answer is the option 3.

Example 3: Which of the following parameters does not characterize the thermodynamic state of matter?

1) Temperature

2) Pressure

3) Work

4) Volume

Solution:

Equation of state

The relation between the Thermodynamics variable (P, V, T) of the system is called an equation of state.

$PV=nRT$

(P, V, T) represent the Thermodynamic state of matter.

Work does not represent the thermodynamic state of matter.

Hence, the answer is the option (3).

Example 4: Which one of the following statements is false?

1) Work is a state function

2) Temperature is a state function

3) Change in the state is completely defined when the initial and final states are specified.

4) Work appears at the boundary of the system.

Solution:

As we have learnt,
Path functions are properties or quantities whose values depend on the transition of a system from the initial state to the final state.
E.g.: Work, heat, loss of energy due to friction, etc. are some common examples of a path function.
Hence, the answer is the option 1.

Example 5: $\left(\mathrm{P}+\frac{\mathrm{a}}{\mathrm{V}^2}\right)(\mathrm{V}-\mathrm{b})=\mathrm{RT}$ represents the equation of the state of some gases. Where P is the pressure, V is the volume, T is the temperature, and a, b, and R are the constants. The physical quantity, which has a dimensional formula as that of $\frac{b^{2}}{a}$ will be :

1) Compressibility

2) Energy density

3) Modulus of rigidity

4) Bulk modulus

Solution:

$\begin{aligned} & {[\mathrm{b}]=\left[\mathrm{L}^3\right]} \\ & \begin{aligned} {[\mathrm{a}] } & =\left[\mathrm{PV}^2\right] \\ & =\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]\left[\mathrm{L}^6\right] \\ & =\left[\mathrm{ML}^5 \mathrm{~T}^{-2}\right] \end{aligned} \\ & \frac{\left[\mathrm{b}^2\right]}{[\mathrm{a}]}=\frac{\left[\mathrm{L}^6\right]}{\left[\mathrm{ML}^5 \mathrm{~T}^{-2}\right]}=\left[\mathrm{M}^{-1} \mathrm{~L}^1 \mathrm{~T}^2\right] \end{aligned}$

Hence, the answer is the Option (1).

Summary

Temperature, pressure, and volume are examples of thermodynamic variables while internal energy is another important one. The state of a thermodynamic system is described in terms of temperature, pressure, volume, and internal energy, and these properties are called the thermodynamic variables. Formulated as an equation, this equation describes how the system behaves under different circumstances as the arrangement that connects these variables altogether. Such relationships enable us to know how the system will behave under various conditions.

Frequently Asked Questions (FAQs)

1. What are thermodynamic state variables and why are they important?

Thermodynamic state variables are properties that describe the current state of a system, regardless of how it reached that state. They are important because they allow us to characterize a system's condition and predict its behavior. Examples include temperature, pressure, volume, and entropy. These variables are crucial for understanding and analyzing thermodynamic processes and systems.

2. How do state variables help in understanding and applying the Gibbs phase rule?

The Gibbs phase rule (F = C - P + 2) relates the number of degrees of freedom (F) to the number of components (C) and phases (P) in a system. State variables are the quantities that represent these degrees of freedom. The rule helps determine how many state variables need to be specified to fully define a system's state. This is crucial in fields like materials science and chemical engineering for understanding phase equilibria and designing separation processes.

3. How do state variables relate to the concept of chemical potential?

Chemical potential is a state variable that describes how the Gibbs free energy of a system changes with the addition or removal of a particular component. It's a function of other state variables like temperature, pressure, and composition. Chemical potential is crucial for understanding phase equilibria, chemical reactions, and mass transfer processes. It provides a quantitative measure of the tendency of a substance to diffuse, react, or change phase.

4. What is the significance of conjugate pairs of state variables in thermodynamics?

Conjugate pairs are sets of state variables where one is intensive (like pressure) and the other extensive (like volume). The product of a conjugate pair often has units of energy. Examples include temperature-entropy and pressure-volume. These pairs are significant because they appear together in many thermodynamic equations and help describe energy transfer processes. Understanding conjugate pairs is crucial for analyzing work, heat transfer, and other energy exchanges in thermodynamic systems.

5. How do state variables relate to the concept of exergy in thermodynamics?

Exergy is a measure of the maximum useful work that can be extracted from a system as it comes to equilibrium with its surroundings. It's calculated using state variables of both the system and its environment. Understanding how state variables like temperature, pressure, and chemical potential differ between a system and its surroundings is crucial for exergy analysis. This concept is important in energy engineering for assessing the efficiency of processes and identifying opportunities for improvement.

6. What's the difference between extensive and intensive state variables?

Extensive state variables depend on the amount of matter in the system and scale with system size. Examples include volume, mass, and total energy. Intensive state variables are independent of the amount of matter and remain constant when the system is divided. Examples include temperature, pressure, and density. Understanding this distinction is crucial for analyzing thermodynamic systems of different sizes.

7. Why is it important to specify the reference state when discussing some state variables?

Specifying a reference state is crucial for state variables like internal energy, enthalpy, and Gibbs free energy, which don't have absolute values. The choice of reference state affects the numerical values of these variables but not their changes during processes. This concept is important in thermochemistry and in comparing different systems. For example, enthalpies of formation are typically referenced to elements in their standard states, allowing consistent comparisons across different compounds.

8. Why is entropy considered a state variable despite its association with disorder?

Entropy is a state variable because, like other state variables, its value depends only on the current state of the system, not on how it reached that state. Although entropy is often associated with disorder, it's more accurately described as a measure of the number of possible microscopic arrangements of a system. The fact that entropy always increases in isolated systems (the Second Law of Thermodynamics) doesn't change its status as a state variable.

9. What is the significance of reduced state variables in thermodynamics?

Reduced state variables are dimensionless quantities obtained by dividing a state variable by its critical value (e.g., reduced temperature = T/Tc). They are significant because they allow for the comparison of different substances on a common scale. The principle of corresponding states, which states that all fluids behave similarly at the same reduced conditions, is based on these variables. This concept is powerful for predicting properties of substances for which we have limited data.

10. How do state variables relate to the concept of thermodynamic potentials?

Thermodynamic potentials, such as internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy, are state functions expressed in terms of other state variables. Each potential is particularly useful for describing systems under specific constraints. For example, Gibbs free energy is most useful for systems at constant temperature and pressure. Understanding these relationships helps in determining equilibrium conditions and predicting spontaneous processes in various systems.

11. How does the equation of state relate to thermodynamic state variables?

The equation of state is a mathematical relationship between thermodynamic state variables for a given system. It describes how these variables are interconnected and allows us to predict the behavior of the system under different conditions. For example, the ideal gas law (PV = nRT) is an equation of state that relates pressure, volume, temperature, and the amount of gas in a system.

12. How does the ideal gas equation of state differ from real gas equations of state?

The ideal gas equation of state (PV = nRT) assumes that gas molecules have no volume and no intermolecular forces. Real gas equations of state, such as the van der Waals equation, account for molecular volume and intermolecular attractions. These more complex equations provide better accuracy for gases at high pressures or low temperatures where molecular interactions become significant. Understanding these differences is crucial for accurately predicting gas behavior in various conditions.

13. How does the concept of degrees of freedom relate to state variables?

Degrees of freedom in thermodynamics refer to the number of independent state variables needed to fully describe a system's state. The number of degrees of freedom is determined by the phase rule: F = C - P + 2, where F is the number of degrees of freedom, C is the number of components, and P is the number of phases. This concept is crucial for understanding how many variables we need to specify to define a system's state completely.

14. What is the significance of the critical point in relation to state variables?

The critical point is a unique condition where the distinctions between liquid and gas phases disappear. At this point, certain state variables exhibit peculiar behavior. For example, the compressibility becomes infinite, and the difference in density between liquid and gas phases vanishes. Understanding the critical point is crucial for studying supercritical fluids and phase behavior in extreme conditions, with applications ranging from chemical processing to geothermal energy extraction.

15. What role do partial derivatives play in relating thermodynamic state variables?

Partial derivatives are essential tools for relating thermodynamic state variables. They allow us to express how one variable changes with respect to another while holding other variables constant. For example, the heat capacity at constant volume (Cv) is defined as the partial derivative of internal energy with respect to temperature at constant volume. These relationships help us understand the interconnections between state variables and derive important thermodynamic equations.

16. Why can't we use kinetic energy as a state variable in thermodynamics?

Kinetic energy is not a state variable because it depends on the frame of reference and the system's history. State variables must be independent of the path taken to reach the current state. Kinetic energy can change based on the observer's perspective or the system's past motion, making it unsuitable as a state variable in thermodynamics.

17. Can you explain why temperature is considered a state variable?

Temperature is a state variable because it describes the average kinetic energy of particles in a system, regardless of how that energy was acquired. It doesn't depend on the system's history or the path taken to reach its current state. Temperature is an intensive property that remains constant throughout a system at equilibrium, making it an ideal state variable for thermodynamic analysis.

18. How do state variables differ from process variables in thermodynamics?

State variables describe the condition of a system at a specific point, independent of how it reached that state. Examples include temperature and pressure. Process variables, on the other hand, describe changes that occur during a thermodynamic process and depend on the path taken. Examples include heat transferred and work done. Understanding this distinction is crucial for analyzing thermodynamic cycles and processes.

19. Why is it important that state variables are path-independent?

The path-independence of state variables is crucial because it allows us to analyze thermodynamic systems without needing to know their entire history. This property enables us to predict a system's behavior based solely on its current state, simplifying calculations and making it possible to study complex thermodynamic processes. It also ensures that the laws of thermodynamics hold true regardless of how a system reaches a particular state.

20. How does the concept of equilibrium relate to thermodynamic state variables?

Thermodynamic equilibrium occurs when a system's state variables remain constant over time in the absence of external influences. At equilibrium, the system's properties are uniform throughout, and there are no net flows of energy or matter. State variables are particularly useful for describing systems at equilibrium, as they provide a complete description of the system's condition.

21. How do state variables help us understand phase transitions?

State variables are crucial for understanding phase transitions because they allow us to identify and characterize different phases of matter. During a phase transition, certain state variables (like temperature) remain constant while others (like volume or entropy) change abruptly. By monitoring these changes in state variables, we can precisely define phase boundaries and understand the energetics of phase transitions, which is essential in many areas of science and engineering.

22. How do state variables relate to the laws of thermodynamics?

State variables are fundamental to expressing and understanding the laws of thermodynamics. The First Law, relating to energy conservation, is often expressed in terms of changes in internal energy (a state variable). The Second Law introduces entropy (another state variable) and describes its behavior. The Third Law defines the behavior of entropy as temperature approaches absolute zero. By using state variables, these laws can be applied universally to various thermodynamic systems and processes.

23. How do fluctuations in state variables relate to the stability of thermodynamic systems?

Fluctuations in state variables are inherent in all thermodynamic systems due to the microscopic motion of particles. The magnitude of these fluctuations relative to the system's average properties is an indicator of stability. In stable systems, fluctuations are typically small and self-correcting. Near critical points or during phase transitions, fluctuations can become large, indicating instability. Understanding these fluctuations is crucial in statistical thermodynamics and in studying phenomena like critical opalescence.

24. How do state variables behave differently in open vs. closed systems?

In closed systems, the mass remains constant, so extensive state variables like total volume or energy can only change through interactions with the surroundings. In open systems, mass can enter or leave, allowing for changes in extensive variables due to mass transfer. Intensive variables like temperature or pressure behave similarly in both systems. Understanding these differences is crucial when applying thermodynamic principles to real-world systems, such as in chemical engineering or environmental science.

25. What is the Maxwell relation and how does it relate state variables?

Maxwell relations are equations that relate the partial derivatives of thermodynamic quantities. They are derived from the fact that mixed partial derivatives of thermodynamic potentials are equal. For example, (∂T/∂V)S = -(∂P/∂S)V. These relations are powerful tools for deriving equations linking different state variables and for solving problems where direct measurement of certain quantities is difficult. They highlight the interconnected nature of thermodynamic state variables.

26. How do equations of state help in predicting phase behavior?

Equations of state help predict phase behavior by relating state variables like pressure, volume, and temperature. They can be used to construct phase diagrams, which show how a substance behaves under different conditions. For instance, the van der Waals equation can predict the existence of a liquid phase and critical point, which the ideal gas law cannot. This predictive power is crucial in fields like petroleum engineering and chemical processing, where understanding phase behavior is essential.

27. Why is it important to distinguish between state functions and path functions in thermodynamics?

Distinguishing between state functions (which depend only on the current state) and path functions (which depend on the path taken between states) is crucial in thermodynamics. State functions, like internal energy or entropy, allow us to analyze systems without knowing their history. Path functions, like heat or work, are important for understanding specific processes. This distinction is key to applying the First Law of Thermodynamics and calculating changes in system properties correctly.

28. Why is it important to understand the behavior of state variables near phase transitions?

Near phase transitions, many state variables exhibit unusual behavior. For example, heat capacity often spikes, and compressibility can become very large. Understanding these behaviors is crucial for accurately predicting system properties and behavior during processes involving phase changes. This knowledge is applied in diverse fields, from designing efficient heat exchangers to understanding geological processes like volcanic eruptions.

29. How do equations of state account for intermolecular forces?

Real gas equations of state, unlike the ideal gas law, incorporate terms to account for intermolecular forces. For example, the van der Waals equation includes a term (a/V^2) to account for attractive forces and another (b) for the volume occupied by molecules. More sophisticated equations like the Peng-Robinson or Redlich-Kwong equations provide even better approximations for real gas behavior. Understanding these modifications is crucial for accurately predicting gas behavior in high-pressure or low-temperature conditions.

30. What is the significance of partial molar quantities in thermodynamics?

Partial molar quantities describe how an extensive property of a mixture changes when the amount of one component is varied while keeping pressure, temperature, and the amounts of other components constant. They are crucial for understanding the behavior of mixtures and solutions. For example, the partial molar volume explains why the total volume of a mixture might not be the sum of the volumes of its components. This concept is essential in fields like solution thermodynamics and chemical engineering.

31. Why is it important to understand the behavior of state variables in non-equilibrium systems?

While state variables are well-defined for systems at equilibrium, understanding their behavior in non-equilibrium systems is crucial for real-world applications. In non-equilibrium situations, intensive properties like temperature or pressure may vary within the system. Studying how state variables evolve during non-equilibrium processes is essential for understanding phenomena like heat transfer, diffusion, and chemical reactions, which are fundamental to many technological and natural processes.

32. What is the significance of the Joule-Thomson coefficient in relation to state variables?

The Joule-Thomson coefficient describes how the temperature of a gas changes during an isenthalpic expansion (constant enthalpy). It's defined in terms of state variables: (∂T/∂P)H. The sign and magnitude of this coefficient are important for understanding processes like gas liquefaction and throttling. For instance, it explains why some gases cool upon expansion while others heat up. This concept is crucial in refrigeration technology and in the design of gas processing equipment.

33. How do state variables help in understanding and applying the Le Chatelier principle?

Le Chatelier's principle describes how a system at equilibrium responds to changes in state variables like temperature, pressure, or concentration. By understanding how these variables affect the equilibrium, we can predict the direction of shift in response to external changes. This principle is crucial in chemical engineering, particularly in optimizing reaction conditions and in understanding how industrial processes respond to perturbations.

34. What is the relationship between state variables and thermodynamic cycles?

Thermodynamic cycles, such as the Carnot cycle or refrigeration cycle, are described by a series of processes that eventually return the system to its initial state. State variables are crucial for analyzing these cycles because the change in any state variable over a complete cycle must be zero. This property allows us to calculate efficiencies and work outputs of cycles, which is fundamental in the design of heat engines, refrigerators, and power plants.

35. How do state variables behave in systems with long-range interactions, like plasmas or gravitational systems?

In systems with long-range interactions, the behavior of state variables can deviate significantly from that in conventional thermodynamic systems. For example, the concept of extensivity may break down, and the system may exhibit negative heat capacities or other counterintuitive properties. Understanding these behaviors is crucial in fields like astrophysics and plasma physics, where long-range forces dominate system behavior.

36. What is the significance of the triple point in terms of state variables?

The triple point is a unique set of conditions (specific values of state variables like temperature and pressure) where three phases of a substance coexist in equilibrium. It's a fixed point on a substance's phase diagram and is often used as a reference point. For example, the triple point of water is used to define the Kelvin temperature scale. Understanding the triple point is crucial in fields like cryogenics and in calibrating thermometers.

37. How do state variables relate to the concept of fugacity in thermodynamics?

Fugacity is a state variable that replaces pressure in chemical potential expressions for real gases. It's a measure of the tendency of a substance to escape from a phase. Fugacity is related to other state variables through equations of state and is crucial for accurately describing phase equilibria and chemical equilibria in non-ideal systems. This concept is particularly important in chemical engineering and petroleum engineering for modeling complex mixtures and processes.

38. What is the significance of the virial equation of state?

The virial equation of state is a series expansion that describes deviations from