## Constant Volume Heating

**Introduction:**

There is no work in constant volume heating.

**Description:**

- Measure δQ and dT, and define

Since there is no work, dU = δQ. The definition of entropy gives **δQ _{rev}=T dS** , and we have

**Special cases:**

Often, C_{V}= C_{V}(T) (e.g. ideal gas, incompressible substance), or C_{V}=constant. (perfect gas)

**EXAMPLE: C _{V }for an Ideal Gas**

• In general, for any pure substance, we can write u as a function of two independent properties, such as u = u(T,v) and it follows that

Introducing our definition gives

In general, we must realize that . However, in the case of an ideal gas, we know from Joule’s free expansion experiment that

, so that

**du = C _{v}dT**

Since u is a function only of temperature for an ideal gas, it follows that CV also must be a function of temperature for an ideal gas:

for an ideal gas.