Thermodynamic State Variables And Equation Of State

Edited By Vishal kumar | Updated on Sep 09, 2024 11:07 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.

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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.