## Isochoric Process – Ideal Gas Equation

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**.

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

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