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Isochoric Process

Updated Jun 13, 2026

An Isochoric Process is a thermodynamic process during which the volume of a closed system remains constant. Because the volume does not change, the system performs no boundary work on its surroundings, and no work is performed upon the system by the surroundings. In such a process, any heat energy transferred to the system results directly in a change in the internal energy and, consequently, a change in temperature and pressure.

On This Page

  1. Short Definition
  2. Detailed Explanation
  3. Key Formula / Syntax / Principle
  4. Important Characteristics
  5. Practical Example
  6. Common Confusions or Misconceptions
  7. Related Terms
  8. Why It Matters

Short Definition

An isochoric process, also known as a constant-volume process, an isovolumetric process, or an isometric process, is a state change in which the volume (##V##) of the system remains fixed (##dV = 0##). In the context of the first law of thermodynamics, this implies that the work done (##W##) by the system is zero, making the change in internal energy (##\Delta U##) equal to the heat energy (##Q##) added to or removed from the system.

Detailed Explanation

In thermodynamics, an isochoric process describes the behavior of a gas or fluid confined within a rigid, non-deformable container. When heat is added to such a system, the molecules gain kinetic energy. Since the walls of the container cannot move, the gas cannot expand to relieve the increasing pressure. Therefore, the temperature and pressure rise in direct proportion to the heat input.

The term "isochoric" is derived from the Greek words *isos*, meaning "equal," and *chora*, meaning "space" or "volume." This process is one of the fundamental "polytropic" processes used to model the behavior of engines, refrigerators, and various industrial systems.

The First Law of Thermodynamics in Isochoric Conditions

The first law of thermodynamics is expressed as:

### \Delta U = Q - W ###

Where:

For any process involving a change in volume, work is defined as:

### W = \int_{V_1}^{V_2} P \, dV ###

In an isochoric process, the volume is constant (##V_1 = V_2##), meaning ##dV = 0##. Consequently, the integral evaluates to zero:

### W = 0 ###

Substituting this back into the first law equation:

### \Delta U = Q ###

This relationship demonstrates that in a constant-volume process, all heat transferred into the system increases the internal energy, which is typically manifested as a rise in temperature.

Ideal Gas Behavior (Gay-Lussac's Law)

For an ideal gas undergoing an isochoric process, the relationship between pressure and temperature is governed by the Ideal Gas Law:

### PV = nRT ###

Since ##n##, ##R##, and ##V## are constant, the ratio of pressure to temperature remains constant:

### \dfrac{P}{T} = \text{constant} \quad \text{or} \quad \dfrac{P_1}{T_1} = \dfrac{P_2}{T_2} ###

This is known as Gay-Lussac's Law. It implies that if you double the absolute temperature (in Kelvin) of a gas in a rigid container, the pressure will also double.

Key Formula / Syntax / Principle

The following formulas are essential for calculating the properties of an isochoric process:

### W = 0 ###
### Q = n C_v \Delta T ###

Where ##n## is the number of moles, ##C_v## is the molar specific heat at constant volume, and ##\Delta T## is the change in temperature (##T_2 - T_1##).

### \Delta U = n C_v \Delta T ###
### \Delta S = n C_v \ln\left(\dfrac{T_2}{T_1}\right) ###
### \dfrac{P_1}{T_1} = \dfrac{P_2}{T_2} ###
Table 1: Summary of Isochoric Process Variables
Variable Status Physical Significance
Volume (##V##) Constant (##dV=0##) The boundaries of the system do not move.
Work (##W##) Zero No energy is transferred via mechanical displacement.
Heat (##Q##) Equal to ##\Delta U## All thermal energy added goes into internal energy.

Important Characteristics

The isochoric process possesses several distinct characteristics that differentiate it from isobaric (constant pressure), isothermal (constant temperature), and adiabatic (no heat transfer) processes:

  • Vertical Line on P-V Diagram: On a Pressure-Volume (P-V) graph, an isochoric process is represented by a perfectly vertical line. This is because the volume (x-axis) does not change while the pressure (y-axis) increases or decreases.
  • No Boundary Work: Since work is the area under the curve on a P-V diagram, and a vertical line has no area beneath it, the work done is zero.
  • Specific Heat Capacity: The heat required to raise the temperature of a gas at constant volume is lower than the heat required at constant pressure (##C_v < C_p##). This is because, at constant volume, no energy is "wasted" on performing expansion work.
  • Temperature and Pressure Correlation: There is a direct linear relationship between absolute temperature and pressure. As the kinetic energy of the molecules increases, they strike the container walls more frequently and with greater force.

Practical Example

A common real-world approximation of an isochoric process is the heating of a gas inside a rigid metal cylinder, such as a fire extinguisher or a scuba tank, left in the sun.

Imagine a sealed steel tank containing an ideal gas at an initial pressure of ##200 \, \text{kPa}## and a temperature of ##300 \, \text{K}##. If the tank is heated until the temperature reaches ##600 \, \text{K}##, we can determine the final pressure.

Using Gay-Lussac's Law:

### P_2 = P_1 \times \left(\dfrac{T_2}{T_1}\right) ###
### P_2 = 200 \, \text{kPa} \times \left(\dfrac{600 \, \text{K}}{300 \, \text{K}}\right) ###
### P_2 = 400 \, \text{kPa} ###

In this example, the volume remained constant because the steel tank is rigid. The doubling of the temperature resulted in a doubling of the pressure. No mechanical work was done, but the internal energy of the gas increased significantly.

Another example is the Otto Cycle (the theoretical cycle for spark-ignition internal combustion engines). The heat addition (combustion) and heat rejection phases are modeled as isochoric processes because the piston is assumed to be momentarily stationary at the Top Dead Center (TDC) or Bottom Dead Center (BDC) during those moments.

Common Confusions or Misconceptions

  • "No Work" vs. "No Energy Change": A common mistake is assuming that because no work is done, no energy change occurs. In reality, while ##W = 0##, the heat transfer ##Q## can be very high, leading to massive changes in internal energy and temperature.
  • Constant Volume vs. Constant Mass: While many isochoric processes involve a closed system with constant mass, the defining feature is the constant volume. A system could theoretically have mass entering or leaving, but if the volume is fixed, the boundary work remains zero. However, in standard thermodynamics, we usually assume a closed system.
  • Real vs. Ideal Gases: For ideal gases, ##P/T## is perfectly constant. For real gases at very high pressures or low temperatures, intermolecular forces and the volume of the molecules themselves cause deviations from this linear relationship.
Table 2: Comparison with Other Thermodynamic Processes
Process Type Constant Parameter Work Done (##W##)
Isochoric Volume (##V##) ##0##
Isobaric Pressure (##P##) ##P(V_2 - V_1)##
Isothermal Temperature (##T##) ##nRT \ln(V_2/V_1)##
  • Isobaric Process: A process occurring at constant pressure.
  • Isothermal Process: A process occurring at constant temperature.
  • Adiabatic Process: A process where no heat or mass is transferred between the system and its surroundings.
  • Internal Energy (##U##): The total energy contained within a system, including kinetic and potential energy at the molecular level.
  • Specific Heat Capacity (##C_v##): The amount of heat energy required to raise the temperature of a unit mass of a substance by one degree at constant volume.

Why It Matters

Understanding the isochoric process is vital for engineering and safety. Many industrial systems involve high-pressure gases stored in rigid containers. If these containers are exposed to heat (such as during a fire), the pressure increases isochorically. Engineers must calculate these pressure rises to design "burst disks" or relief valves that prevent catastrophic tank failure.

In thermodynamics, the isochoric process serves as a baseline for measuring the specific heat of substances. By measuring how much heat is required to raise the temperature of a substance without allowing it to expand, scientists can determine the substance's internal energy characteristics. Furthermore, it is a key component in the theoretical analysis of heat engines, helping engineers optimize the efficiency of power cycles by understanding where energy is added and where it is lost.

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