## Isochoric Process – Isometric Process

An** isochoric process** is a thermodynamic process, in which the **volume** of the closed system **remains constant** (V = const). It describes the behavior of gas inside the container, that cannot be deformed. Since the volume remains constant, the heat transfer into or out of the system does not the p∆V work, but only changes the** internal energy** (the temperature) of the system.

For an ideal gas and a** polytropic process**, the case** n ➝ ∞**

*corresponds to an*

**isochoric (constant-volume) process**. In contrast to adiabatic process , in which

*n =**and*

*a system exchanges no heat with its surroundings (Q = 0; W*

*≠0*

*)*

*, in an isochoric process there is a change in the internal energy (due to ∆T≠0) and therefore*

*ΔU*

*≠*

*0 (for ideal gases) and (Q*

*≠*

*0; W*

*=0*

*).*

In engineering of **internal combustion engines**, isochoric processes are very important for their thermodynamic cycles (Otto and Diesel cycle), therefore the study of this process is crucial for automotive engineering.

## Isochoric Process and the First Law

The classical form of the first law of thermodynamics is the following equation:

**dU = dQ – dW**

In this equation dW is equal to **dW = pdV** and is known as the boundary work. Then:

**dU = dQ – pdV**

In isochoric process and the ideal gas, all of heat added to the system will be used to increase the internal energy.

**Isochoric process (pdV = 0):**

**dU = dQ ***(for ideal gas)*

## Isochoric Process – Ideal Gas Equation

See also: What is an Ideal Gas

Let assume an** isochoric heat addition** in an ideal gas. In an ideal gas, molecules have no volume and do not interact. According to the ideal gas law, pressure varies linearly with temperature and quantity, and inversely with volume.

*pV = nRT*

where:

*p*is the absolute pressure of the gas*n*is the amount of substance*T*is the absolute temperature*V*is the volume*R*is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant,

In this equation the symbol R is a constant called the **universal gas constant** that has the same value for all gases—namely, R = 8.31 J/mol K.

The isochoric process can be expressed with the ideal gas law as:

or

On a **p-V diagram**, the process occurs along a horizontal line that has the equation V = constant.

**Pressure-volume work** by the closed system is defined as:

Since the process is **isochoric**, ** dV = 0**, the

**pressure-volume work is equal to zero**. According to the ideal gas model, the internal energy can be calculated by:

*∆U = m c*_{v }*∆T*

where the property **c**_{v}** (J/mol K) **is referred to as **specific heat** (or **heat capacity**) at a constant volume because under certain special conditions (constant volume) it relates the temperature change of a system to the amount of energy added by heat transfer.

Since there is no work done by or on the system, the **first law of thermodynamics** dictates *∆U = ∆Q. *Therefore:

Q = *m c*_{v }*∆T*

See also: Specific Heat at Constant Volume and Constant Pressure

See also: Mayer’s formula

## Guy-Lussac’s Law

**Guy-Lussac’s Law** or the **Pressure Law, **one of the gas laws, states that:

*For a fixed mass of gas at constant volume, the pressure is directly proportional to the Kelvin temperature.*

That means that, for example, if you double the temperature, you will double the pressure. If you halve the temperature, you will halve the pressure.

You can express this mathematically as:

*p = constant . T*

Yes, it seems to be identical as** isochoric process** of ideal gas. These results are fully consistent with ideal gas law, which determinates, that the constant is equal to nR/V. If you rearrange the pV = nRT equation by dividing both sides by V, you will obtain:

*p = nR/V . T*

where nR/V is constant and:

*p*is the absolute pressure of the gas*n*is the amount of substance*T*is the absolute temperature*V*is the volume*R*is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant

## Example of Isochoric Process – Isochoric Heat Addition

Let assume the **Otto Cycle**, which is the one of most common **thermodynamic cycles** that can be found in **automobile engines**. This cycle assumes that the heat addition occurs instantaneously (**between 2 → 3**) while the piston is at top dead center. This process is considered to be isochoric.

Processes 2 → 3 and 4 → 1 are **isochoric processes**, in which the heat is transferred into the system between 2 → 3 and out of the system between 4 → 1. During these processes no work is done on the system or extracted from the system. The isochoric process 2 → 3 is intended to represent the ignition of the fuel–air mixture and the subsequent rapid burning.