Exercise 28

The Ideal Gas Law, $PV = RT$, relates the pressure ($P$, in pascals), temperature ($T$, in Kelvin), and volume ($V$, in cubic meters) of 1 mole of a gas ($R = 8.314 \ \frac{\text{J} }{\text{ mol } \ \text{K} }$ is the universal gas constant), and describes the behavior of gases that do not liquefy easily, such as oxygen and hydrogen. We can solve the ideal gas law for the volume and hence treat the volume as a function of the pressure and temperature: $$\begin{gathered} V(P,T) = \frac{8.314T}{P}. \end{gathered}$$

1. Explain in detail what the trace of $V$ with $P=1000$ tells us about a key relationship between two quantities.

2. Explain in detail what the trace of $V$ with $T=5$ tells us.

3. Explain in detail what the level curve $V = 0.5$ tells us.

4. Use 2 or three additional traces in each direction to make a rough sketch of the surface over the domain of $V$ where $P$ and $T$ are each nonnegative. Write at least one sentence that describes the way the surface looks.

5. Based on all your work above, write a couple of sentences that describe the effects that temperature and pressure have on volume.