### "Heat Capacity"

Consider two closed systems to which you add heat ...

$\Delta u + \Delta e_{k} + \Delta e_{p} = w + q $

- For the constant volume process:

$\Delta e_{k} = \Delta e_{p} = w = 0 $

$\Delta u = q $

- For the constant pressure process:

$\Delta e_{k} = \Delta e_{p} = 0 $

$\Delta u = q + w = q - P\Delta v$

$\Delta h = \Delta u + P\Delta v = q $

For convenience the "other" intensive property is chosen to be
temperature ... why?

##### Definition:

The heat capacity at constant volume is defined as:

$c_v \equiv \left ( \frac{\partial u}{\partial T} \right
)_v$

##### Definition:

The heat capacity at constant pressure is defined as:

$c_P \equiv \left ( \frac{\partial h}{\partial T} \right
)_P$

##### Important:

For solids and liquids: $c_v \approx c_P$ why?

For ideal gases: $c_P = c_v + R$ why?

For real gases: $c_P > c_v$

##### Note:

Heat capacities are state functions (depend on T!).

##### Test Yourself:

Verify if the "cold water diet" would work. Some joker proposed
that drinking 0.5kg of ice-cold water at each meal would be an
effective diet method. Determine the internal energy change this
would cause. What if you used ice cubes instead of water?

##### Definition:

The **latent heat** is the specific enthalpy change
associated with a change in state of aggregation (phase).

##### Note:

The latent heat is often known only at a single temperature, yet
this state function is, of course, dependent on temperature. How do
we handle that?.