Notes : Thermodynamic State Variables and Equation of State

Thermodynamic State Variables and Equation of State

Every equilibrium state of a thermodynamic system is completely described by a set of macroscopic quantities known as Thermodynamic State Variables or State Functions.

What are Thermodynamic State Variables?

State variables are measurable physical quantities that define the state of a thermodynamic system when it is in thermodynamic equilibrium.

For a gas, the common state variables are:

Pressure ($P$)
Volume ($V$)
Temperature ($T$)
Mass ($M$)
Number of moles ($\mu$)

These variables completely describe the equilibrium state of the system.

Condition for State Variables

State variables can only be assigned meaningful values when the system is in a state of thermodynamic equilibrium.

In equilibrium:

Temperature is uniform throughout the system.
Pressure is uniform throughout the system.
No net macroscopic changes occur with time.

Non-Equilibrium States

When a system changes rapidly, pressure and temperature may vary from one region to another. In such situations, a single value of pressure or temperature cannot be assigned to the entire system.

1. Free Expansion of Gas

When a partition separating a gas from a vacuum is suddenly removed, the gas expands rapidly.

Pressure is not uniform.
Different regions have different densities.
The system is not in equilibrium.

Therefore, state variables cannot properly describe the system during the expansion process.

2. Explosive Chemical Reactions

In explosive reactions such as ignition of petrol vapour by a spark:

Temperature changes violently.
Pressure changes rapidly.
Properties are highly non-uniform.

Hence, thermodynamic state variables lose their meaning during the reaction.

Only after the system settles into equilibrium can pressure and temperature again be assigned unique values.

Equation of State

The thermodynamic state variables are not completely independent. They are connected through a mathematical relation called the Equation of State.

For an ideal gas containing a fixed number of moles:

$$PV=\mu RT$$

where:

$P$ = Pressure
$V$ = Volume
$\mu$= Number of moles
$R$ = Universal Gas Constant
$T$ = Absolute Temperature

This equation shows that if any two independent variables are known, the third variable can be determined.

Isotherm

An Isotherm is a curve on a Pressure-Volume ($P-V$) graph representing a constant temperature process.

$$PV=\text{constant}$$

Classification of State Variables

1. Extensive Variables

Extensive variables depend on the size or amount of matter in the system.

Examples:

Volume ($V$)
Mass ($M$)
Internal Energy ($U$)
Heat Supplied ($Q$)
Number of Moles ($\mu$)

If the system is divided into two equal parts, each part contains half the value of every extensive variable.

$X_{\text{half}}=\frac{X}{2}$

2. Intensive Variables

Intensive variables do not depend on the size of the system.

Examples:

Pressure ($P$)
Temperature ($T$)
Density ($\rho$)

When the system is divided into two equal parts, intensive variables remain unchanged.

$$Y_{\text{half}}=Y$$

Divide-in-Half Test

If the quantity becomes half after division, it is extensive.
If the quantity remains unchanged, it is intensive.

Consistency of Thermodynamic Equations

The classification of variables helps verify whether a thermodynamic equation is physically meaningful.

Consider the First Law of Thermodynamics:

$$\Delta Q=\Delta U+P\Delta V$$

Nature of each term:

$\Delta Q$ → Extensive
$\Delta U$ → Extensive
$P$ → Intensive
$\Delta V$ → Extensive

The product $P\Delta V$ is extensive.

Since every term in the equation is extensive, the equation is thermodynamically consistent.

Important Formulae

Ideal Gas Equation

$$PV=\mu RT$$

Isothermal Process

$$PV=\text{constant}$$

First Law of Thermodynamics

$$\Delta Q=\Delta U+P\Delta V$$

Density Formula

$$\rho=\frac{M}{V}$$

Key Points

State variables describe only equilibrium states.
Pressure, volume and temperature are important state variables.
State variables are related through an equation of state.
For an ideal gas, $PV=\mu RT$.
Extensive variables depend on system size.
Intensive variables do not depend on system size.
Isotherms are constant-temperature curves.
The First Law of Thermodynamics contains only extensive terms.

Frequently Asked Questions (FAQ)

What are thermodynamic state variables?

State variables are macroscopic quantities such as pressure, volume and temperature that completely describe an equilibrium state of a thermodynamic system.

What is the equation of state?

The mathematical relation among thermodynamic state variables is called the equation of state.

What is the equation of state for an ideal gas?

$PV=\mu RT$

What is the difference between extensive and intensive variables?

Extensive variables depend on system size, whereas intensive variables are independent of system size.

Is pressure an intensive variable?

Yes, pressure remains unchanged when a system is divided into smaller parts.

Why are state variables defined only in equilibrium?

Because pressure and temperature must have uniform values throughout the system.

What is an isotherm?

An isotherm is a curve on a P–V graph representing a constant-temperature process.

Can heat and work be state variables?

No. Heat and work are path functions and not state variables.

Multiple Choice Questions (MCQs)

1. Which of the following is a state variable?

A) Heat
B) Work
C) Temperature
D) Friction

Answer: C) Temperature

2. State variables are defined only for:

A) Non-equilibrium states
B) Equilibrium states
C) Chemical reactions
D) Explosions

Answer: B) Equilibrium states

3. The equation of state for an ideal gas is:

A) $PV=RT$
B) $PV=\mu RT$
C) $P=VRT$
D) $V=\mu R$

Answer: B) $PV=\mu RT$

4. Which quantity is extensive?

A) Temperature
B) Pressure
C) Density
D) Volume

Answer: D) Volume

5. Which quantity is intensive?

A) Internal Energy
B) Volume
C) Pressure
D) Mass

Answer: C) Pressure

6. Density is a:

A) Extensive Variable
B) Intensive Variable
C) State Equation
D) Thermodynamic Process

Answer: B) Intensive Variable

7. An isotherm represents:

A) Constant Pressure
B) Constant Volume
C) Constant Temperature
D) Constant Density

Answer: C) Constant Temperature

8. In free expansion of gas:

A) Pressure remains uniform
B) Temperature remains uniform
C) System is not in equilibrium
D) State variables remain well-defined

Answer: C) System is not in equilibrium

9. The product $P\Delta V$ is:

A) Intensive
B) Extensive
C) Dimensionless
D) State Variable

Answer: B) Extensive

10. Which law contains the equation $\Delta Q=\Delta U+P\Delta V$?

A) Zeroth Law
B) Boyle's Law
C) First Law of Thermodynamics
D) Charles's Law

Answer: C) First Law of Thermodynamics

Quiz

Question 1

Which of the following is an intensive variable?

A) Volume
B) Mass
C) Temperature
D) Internal Energy

Answer: C) Temperature

Question 2

The equation $PV=\mu RT$ is known as:

A) Boyle's Law
B) Charles's Law
C) Ideal Gas Equation
D) Newton's Law

Answer: C) Ideal Gas Equation

Question 3

Which quantity remains unchanged when a system is divided into two equal parts?

A) Volume
B) Mass
C) Internal Energy
D) Pressure

Answer: D) Pressure

Question 4

An isotherm represents a process at: