The internal energy of an ideal gas is dependent on temperature only. However, for real gases, the internal energy also depends on the volume. Starting with the differential of the internal energy (U), we get:

$$dU=TdS-PdV$$

At constant temperature:

$${ \left(\frac { \partial U }{ \partial V } \right) }_{ T }=T{ \left(\frac { \partial S }{ \partial V } \right) }_{ T }-P$$

What is the molar change in internal energy for a ${ CO }_{ 2 }$ gas that obeys the van der Waals equation when it expands from 2.00 to 10.0 L at 298 K?

The van der Waals constants for ${ CO }_{ 2 }$ are:

- $a = 3.6551 \ { { \text{L} }^{ 2 }-\text{bar} }/{ { \text{mol} }^{ 2 } }$
- $b = 0.042816 \ { \text{L} }/{ \text{mol} }$

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