Isochoric Process: Definition, Formula and Examples

Isochoric Process

Edited By Vishal kumar | Updated on Jul 02, 2025 06:29 PM IST

An isochoric process involves a scenario in thermodynamics where the volume of a system remains constant while its pressure and temperature may change. This concept is key in understanding how certain systems, like a heated can of gas that doesn't expand, behave under specific conditions. For students studying for board exams and competitive exams like JEE and NEET, grasping the isochoric process is essential. This article simplifies the concept and includes a solved example to show how it applies practically, helping you understand and remember this important thermodynamic principle.

This Story also Contains

What is the Isochoric Process?
Key points in the Isochoric Process
Solved Examples Based on the Isochoric Process
Summary

Isochoric Process

What is the Isochoric Process?

A Thermodynamic process in which volume remains constant is known as the Isochoric Process.

In this process, P and T change keeping P constant. So Gay-Lussac’s law is obeyed in this process.

Key points in the Isochoric Process

- Its Equation of state is given as $\frac{P}{T}=$ constant
$
\text { or } \frac{P_1}{T_1}=\frac{P_2}{T_2}=\text { constant }
$
- P-V Indicator diagram for an isobaric process

Its PV graph has slope $=$ infinity (i.e $\frac{d P}{d V}=\infty$,

The above Graph represents an Isochoric increase in pressure at volume V.

The P-V diagram for this process is a line is parallel to the pressure line.

Specific heat of gas during the Isochoric process is given by

$C_V=\frac{f}{2} R$

The bulk modulus of elasticity during the Isochoric process is given by

$K=\frac{\Delta P}{-\Delta V / V}=\frac{\Delta P}{0}=\infty$

Work done in the Isochoric process-

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$\begin{aligned} & \Delta W=P \Delta V \\ & \text { and as } \Delta V=0 \\ & \text { So } \Delta W=0\end{aligned}$

Internal energy in the Isochoric process

$\Delta U=n C_V \Delta T=n \frac{R}{(\gamma-1)} \Delta T$

Heat in the Isochoric process

From FLTD $\Delta Q=\Delta U+\Delta W$
But $\Delta W=0$
$
\text { So } \Delta Q=\Delta U=n C_V \Delta T=n \frac{R}{(\gamma-1)} \Delta T=\frac{P_f V_f-P_i V_i}{\gamma-1}
$

Example of Isochoric process

Heating of water in a pressure cooker (Valve closed)

Solved Examples Based on the Isochoric Process

Example 1: In an isochoric process, if $T_1=27^{\circ} \mathrm{C}$ and $T_2=127^{\circ} \mathrm{C}$ . Then $P_1 / P_2$ will be equal to:

1) 9/59

2) 2/3

3) 3/4

4) None of these

Solution:

Isochoric Process

When a Thermodynamic Process undergoes a physical change in such a way that its volume remains constant.

$\begin{aligned} & V=\text { constant } \\ & \therefore \frac{P_1}{T_1}=\frac{P_2}{T_2}\end{aligned}$

In the isochoric process, volume remains constant

At constant volume

$\begin{aligned} P & \propto T \\ \Rightarrow & \frac{P_1}{P_2}=\frac{T_1}{T_2}=\frac{273+27}{273+127}=\frac{300}{400}=\frac{3}{4}\end{aligned}$

Hence, the answer is the option 3.

Example 2: For a thermodynamic process $\mathrm{dP} / \mathrm{dV}=\infty$, it shows that the process is
1) Isothermal
2) Isochoric
3) Adiabatic
4) Isobaric

Solution:

Slope in P-V diagram for isochoric process

$
\frac{d P}{d V}=\infty
$

Since the line is parallel to the pressure line.
For isochoric process
$\mathrm{P}-\mathrm{V}$ diagram is
The slope of the P-V diagram
$\mathrm{dP} / \mathrm{dV}$
as $\mathrm{dV}=0$
so $\mathrm{dP} / \mathrm{dV}=\infty$

Hence, the answer is the option 2.

Example 3: A monatomic gas is going under an isochoric process, its specific heat will be

1) 5/2 R

2) 7/2 R

3) 5/3 R

4) 3/2 R

Solution:

For isochoric process

$
C_V=f / 2 R
$
$\mathrm{f}=3$ for monoatomic gas
$
\mathrm{Cv}=3 / 2 \mathrm{R}
$

Hence, the answer is the option 4.

Example 4: A cylinder with a fixed capacity of 67.2 lit. contains helium gas at STP. The amount of heat ( in J) needed to raise the temperature of the gas by:
$20^{\circ} \mathrm{C}$ is : [ Given that $R=8.31 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ ]

1) 748

2) 374

3) 350

4) 700

Solution:

Specific heat for isochoric process

$
\begin{aligned}
& C_V=\frac{f}{2} R \\
& C_p-C_v=R
\end{aligned}
$
wherein
$f_{\text {is the degree of freedom }}$
1 mole at STP $\rightarrow 22.4$ it
So, 67.2 It $\rightarrow \frac{67.2}{22.4}$ mole
and the cylinder has with fixed capacity
So volume is constant means the isochoric process
$
\begin{aligned}
& Q=n C_V \Delta t=n\left(\frac{3 R}{2}\right) \Delta t \\
&=\left(\frac{67.2}{22.4}\right) \times \frac{3}{2} \times 8.314 \times 20 \\
& \Delta Q=748 . J
\end{aligned}
$

Hence, the answer is the option (1).

Example 5: A thermodynamic process undergoes a physical change in such a way that its volume remains constant, then the bulk modulus of elasticity for this process is

1) Zero

2) Infinite

3) can't say anything

4) one

Solution:

Bulk modulus of elasticity:
$
K=\frac{\Delta P}{-(\Delta V / V)}
$

For isochoric process
$
\begin{aligned}
& \Delta V=0 \\
& \text { So } \mathrm{K}=\frac{\Delta P}{0}=\infty
\end{aligned}
$

Hence, the answer is the option 2.

Summary

An isochoric process is a thermodynamic process that refers to the volume of a system remaining the same. However, the pressure and temperature of the system can change. However, no work is done by or on the system as the volume does not change. The system gets energy due to either addition or removal of it, and it appears as changes in internal energy and pressure. The temperature is directly proportional to the pressure for an ideal gas in an isochoric process.

Frequently Asked Questions (FAQs)

1. Why is there no work done in an isochoric process?

No work is done in an isochoric process because work in thermodynamics is defined as pressure times the change in volume (W = P∆V). Since the volume remains constant (∆V = 0), the work done is zero, regardless of any pressure changes.

2. How does the internal energy change in an isochoric process?

In an isochoric process, any change in internal energy is solely due to heat transfer. Since no work is done, the change in internal energy (∆U) is equal to the heat added to or removed from the system (Q). This is expressed as ∆U = Q for an isochoric process.

3. What happens to pressure and temperature during an isochoric heating process?

During an isochoric heating process, both pressure and temperature increase. As heat is added to the system at constant volume, the particles gain kinetic energy, increasing their speed and collision frequency with the container walls, resulting in higher pressure and temperature.

4. What is the relationship between heat capacity at constant volume (Cv) and an isochoric process?

The heat capacity at constant volume (Cv) is directly applicable to isochoric processes. It represents the amount of heat required to raise the temperature of a substance by one degree Celsius while keeping its volume constant. In an isochoric process, Q = nCv∆T, where Q is heat added, n is the number of moles, and ∆T is the temperature change.

5. How does an isochoric process differ from an isobaric process?

An isochoric process maintains constant volume, while an isobaric process maintains constant pressure. In an isochoric process, no work is done, and all heat transfer contributes to changing internal energy. In contrast, an isobaric process involves work due to volume changes, and heat transfer affects both internal energy and work done.

6. Can you give an example of an isochoric process in everyday life?

A common example of an isochoric process is heating food in a sealed, rigid container (like a pressure cooker before the pressure starts to build). As the food is heated, the temperature and pressure inside the container increase, but the volume remains constant due to the rigid walls.

7. How does the ideal gas law apply to an isochoric process?

For an ideal gas undergoing an isochoric process, the ideal gas law (PV = nRT) simplifies to P1/T1 = P2/T2, where subscripts 1 and 2 represent initial and final states. This is because volume (V) and the number of moles (n) remain constant, so pressure and temperature are directly proportional.

8. How does the equation of state for an ideal gas change during an isochoric process?

For an ideal gas undergoing an isochoric process, the equation of state P1V = nRT1 becomes P2V = nRT2, where subscripts 1 and 2 represent initial and final states. Since volume (V) and the number of moles (n) are constant, this simplifies to P1/T1 = P2/T2.

9. How does the First Law of Thermodynamics simplify for an isochoric process?

For an isochoric process, the First Law of Thermodynamics (∆U = Q - W) simplifies to ∆U = Q. This is because the work done (W) is zero in a constant volume process, so all heat added or removed directly changes the internal energy of the system.

10. What is the relationship between isochoric processes and internal energy in monatomic gases?

For monatomic ideal gases, the change in internal energy during an isochoric process is directly proportional to the change in temperature. Specifically, ∆U = (3/2)nR∆T, where n is the number of moles, R is the gas constant, and ∆T is the change in temperature.

11. What is an isochoric process?

An isochoric process, also known as an isovolumetric or constant-volume process, is a thermodynamic process in which the volume of a system remains constant while other properties like pressure and temperature may change. The key feature is that no work is done by or on the system through expansion or compression.

12. How does an isochoric process appear on a P-V diagram?

On a Pressure-Volume (P-V) diagram, an isochoric process appears as a vertical line. This is because the volume remains constant while the pressure changes, resulting in a straight line parallel to the pressure axis.

13. What is the significance of the isochoric process in the Carnot cycle?

The Carnot cycle, which represents the most efficient heat engine possible, does not include isochoric processes. It consists of two isothermal and two adiabatic processes. However, understanding isochoric processes is crucial for comparing the Carnot cycle to real heat engines, which often include processes closer to isochoric conditions.

14. Can an isochoric process be reversible?

Yes, an isochoric process can be reversible if it occurs infinitely slowly and without friction, allowing the system to remain in thermal equilibrium throughout the process. In practice, most real isochoric processes are irreversible due to finite rates of heat transfer and internal friction.

15. What happens to the average kinetic energy of gas molecules during an isochoric heating process?

During an isochoric heating process, the average kinetic energy of gas molecules increases. This is directly related to the increase in temperature, as temperature is a measure of the average kinetic energy of particles in a system.

16. How does the concept of degrees of freedom relate to isochoric processes in gases?

The concept of degrees of freedom is important in understanding the behavior of gases during isochoric processes. For an ideal monatomic gas, there are three translational degrees of freedom, which contribute to the heat capacity at constant volume (Cv). This relationship is expressed as Cv = (3/2)R for monatomic gases, where R is the gas constant.

17. How does the heat capacity ratio (γ) affect isochoric processes in gases?

The heat capacity ratio (γ), defined as Cp/Cv, doesn't directly affect isochoric processes since these occur at constant volume. However, γ is important for comparing isochoric processes to other types of processes and for understanding how a gas will behave when transitioning between different types of processes.

18. How does the concept of enthalpy relate to isochoric processes?

In an isochoric process, the change in enthalpy (∆H) is equal to the change in internal energy (∆U). This is because enthalpy is defined as H = U + PV, and in an isochoric process, the change in PV is zero (∆(PV) = 0) since volume is constant.

19. What is the relationship between isochoric processes and the equipartition theorem?

The equipartition theorem is particularly relevant to isochoric processes as it relates to the internal energy of a system. For an ideal gas in an isochoric process, each degree of freedom contributes (1/2)kT to the average energy per molecule, where k is Boltzmann's constant and T is temperature.

20. How do real gases deviate from ideal behavior during isochoric processes?

Real gases deviate from ideal behavior in isochoric processes, especially at high pressures or low temperatures. These deviations are due to intermolecular forces and the finite size of molecules, which are neglected in the ideal gas model. The van der Waals equation or other real gas equations of state can be used to account for these deviations.

21. Can you explain how the concept of free energy relates to isochoric processes?

In an isochoric process, the change in Helmholtz free energy (∆F) is particularly relevant. Since volume is constant, ∆F = ∆U - T∆S, where U is internal energy, T is temperature, and S is entropy. This relationship is useful for determining the spontaneity and equilibrium conditions of isochoric processes.

22. Can you describe how the concept of exergy applies to isochoric processes?

Exergy, which represents the maximum useful work possible in bringing a system to equilibrium with its environment, changes during an isochoric process. In an isochoric heating process, exergy increases as the temperature difference from the surroundings increases, indicating a greater potential for useful work.

23. Can you explain how the concept of fugacity relates to isochoric processes in real gases?

Fugacity, which is a measure of the tendency of a substance to escape from a phase, can change during an isochoric process in real gases.

24. How is entropy affected in an isochoric process?

In an isochoric process, entropy changes solely due to heat transfer. The change in entropy (∆S) for an isochoric process is given by ∆S = nCv ln(T2/T1), where T2 and T1 are the final and initial temperatures, respectively. Entropy increases if heat is added to the system and decreases if heat is removed.

25. How does an isochoric process affect the density of a gas?

In an isochoric process, the density of a gas remains constant. This is because density is defined as mass per unit volume, and in an isochoric process, both mass and volume remain unchanged. However, the pressure and temperature of the gas may change.

26. Can an isochoric process occur in liquids and solids?

Yes, isochoric processes can occur in liquids and solids, although they are less common than in gases. For instance, heating a liquid in a completely filled, rigid container is an isochoric process. In solids, certain phase transitions can occur at constant volume.

27. What is the significance of the isochoric process in the Otto cycle?

The Otto cycle, which models the operation of a spark-ignition internal combustion engine, includes two isochoric processes. These represent the combustion of fuel-air mixture (heat addition) and the exhaust of combustion products (heat rejection) in the engine cylinder when the piston is at its extreme positions.

28. How does an isochoric process differ from an isothermal process?

An isochoric process maintains constant volume, while an isothermal process maintains constant temperature. In an isochoric process, temperature typically changes, and no work is done. In an isothermal process, volume and pressure can change, and work may be done, but the temperature remains constant.

29. Can you explain the concept of isochoric cooling?

Isochoric cooling is a process where heat is removed from a system while its volume remains constant. This results in a decrease in both temperature and pressure of the system. An example is a sealed container of gas in a refrigerator, where the gas cools without changing its volume.

30. How does the work done in an isochoric process compare to that in an isobaric process?

In an isochoric process, no work is done because the volume remains constant (W = P∆V = 0). In contrast, work is done in an isobaric process because the volume changes while pressure remains constant (W = P∆V ≠ 0). This fundamental difference affects how energy is transferred in these processes.

31. What is the role of isochoric processes in thermodynamic cycles?

Isochoric processes play important roles in various thermodynamic cycles, particularly those modeling internal combustion engines. For example, in the Otto cycle (modeling gasoline engines) and the Diesel cycle (modeling diesel engines), isochoric processes represent rapid combustion and heat rejection phases where volume change is negligible.

32. Can an isochoric process be adiabatic?

Yes, an isochoric process can be adiabatic if it occurs in a perfectly insulated container. In this case, no heat is exchanged with the surroundings (Q = 0), and since no work is done in an isochoric process (W = 0), the internal energy of the system remains constant (∆U = 0).

33. What is the significance of isochoric processes in statistical mechanics?

In statistical mechanics, isochoric processes are important for understanding the behavior of particles in a fixed volume. The partition function for an ideal gas in an isochoric process helps in deriving thermodynamic properties and understanding the statistical behavior of gas molecules at constant volume.

34. How does an isochoric process affect the mean free path of gas molecules?

In an isochoric process, the mean free path of gas molecules changes inversely with pressure. If the system is heated isochorically, the increased molecular speed leads to more frequent collisions, decreasing the mean free path. Conversely, isochoric cooling increases the mean free path.

35. Can you explain how isochoric processes relate to the concept of internal energy in thermodynamics?

Isochoric processes provide a direct way to observe changes in internal energy. Since no work is done (∆W = 0), any heat added or removed directly changes the internal energy (∆U = Q). This makes isochoric processes valuable for studying and measuring internal energy changes in thermodynamic systems.

36. How does the Maxwell-Boltzmann distribution change during an isochoric process?

During an isochoric heating process, the Maxwell-Boltzmann distribution of molecular speeds shifts towards higher velocities and broadens. This reflects the increase in average kinetic energy and the wider range of molecular speeds at higher temperatures, all while maintaining a constant volume.

37. Can you describe how an isochoric process would appear on a T-S (Temperature-Entropy) diagram?

On a Temperature-Entropy (T-S) diagram, an isochoric process appears as a curved line. For an ideal gas, this curve follows the equation S2 - S1 = nCv ln(T2/T1), where S is entropy, T is temperature, n is the number of moles, and Cv is the specific heat capacity at constant volume.

38. How does the concept of thermal equilibrium apply to isochoric processes?

In an isochoric process, thermal equilibrium is reached when the system's temperature becomes uniform throughout and equal to its surroundings (if allowed to equilibrate). The process of reaching equilibrium in a closed, constant-volume system involves internal energy redistribution without volume change.

39. What is the significance of isochoric processes in understanding phase transitions?

Isochoric processes are important in studying certain phase transitions, particularly those that occur at constant volume. For example, the transition between different solid phases of a material under high pressure can often be approximated as an isochoric process.

40. How does an isochoric process affect the compressibility of a gas?

The isothermal compressibility of a gas is not directly applicable during an isochoric process since the volume doesn't change. However, the process can affect the gas's potential compressibility. As pressure increases in an isochoric heating process, the gas becomes less compressible if volume were allowed to change.

41. How do isochoric processes relate to the concept of thermodynamic potentials?

Isochoric processes are directly related to the Helmholtz free energy (F), one of the fundamental thermodynamic potentials. F is particularly useful for systems at constant volume, as its natural variables are temperature and volume. The change in F during an isochoric process gives the maximum work available from the system.

42. What is the significance of isochoric processes in understanding heat engines?

While ideal heat engines like the Carnot cycle don't include isochoric processes, real heat engines often approximate isochoric conditions during rapid heating or cooling phases. Understanding isochoric processes helps in analyzing the efficiency and performance of real heat engines, particularly in internal combustion engines.

43. How does an isochoric process affect the chemical potential of a substance?

In an isochoric process, changes in chemical potential are primarily driven by temperature changes. The chemical potential, which represents the change in Gibbs free energy with respect to the number of particles, can change significantly in isochoric processes, affecting the system's equilibrium conditions.

44. How do isochoric processes relate to the concept of thermodynamic cycles?

Isochoric processes are components of several important thermodynamic cycles, particularly those modeling internal combustion engines. They represent phases where rapid heating or cooling occurs without significant volume change, such as the combustion and exhaust strokes in an Otto cycle.

45. What is the relationship between isochoric processes and the speed of sound in a gas?

The speed of sound in a gas is related to its compressibility, which changes during an isochoric process. As temperature increases in an isochoric heating process, the speed of sound in the gas increases, following the relationship c = √(γRT/M), where c is the speed of sound, γ is the heat capacity ratio, R is the gas constant, T is temperature, and M is the molar mass.

46. How does an isochoric process affect the Joule-Thomson coefficient of a gas?

The Joule-Thomson coefficient, which describes the temperature change of a gas during an isenthalpic expansion, is not directly applicable during an isochoric process. However, the isochoric process can change the gas's state, affecting its Joule-Thomson coefficient for subsequent expansions.