Expansion, compression, and Isotherms

1. Expansion, compression, and Isotherms Pressure Volume Notice that the curves converge at large volume. A curve of constant temperature is called an isotherm. (“iso” is Greek for “same” while therm is for temperature).

2. Expansion, compression, and Isotherms Pressure Volume Suppose on the T2 isotherm we had two points at the same volume as point A and point B, how would that look? Let’s suppose we call these points on isotherm T2, C and D respectively. If we go between C and D, we are compressing the gas (the volume is being decreased).

3. Expansion, compression, and Isotherms Pressure Volume Suppose we go from point C to point D, what happens? Clearly the temperature changes since T2 is larger than T1. As the temperature increases and the volume is constant, the pressure must increase. The converse situation is true if we move from D to A. When we return the sample of gas to point A, is the gas the same as it was when we started?

4. weight piston gas What is work? When is the piston motionless?

5. weight Pressure piston Volume gas The piston is motionless when there is a BALANCE between the force down (due to the weight) and the force up (due too the pressure of the gas. This balance is the system in EQUILIBRIUM. Suppose I lean on the piston (keeping the temperature constant), then I compress the gas. Am I doing work on the system? Yes.

6. Suppose I now stop forcing the piston down. Does the system return to its original volume? (since P,T, and n are the same, it MUST). During this change, does the system do work on me? YES Chemists focus on what happens to the gas (the system). Physicists and engineers focus on what the gas can do to the world around it. The integral means that work is the area under the PV curve when a change is made.

7. On an isotherm. First expand from A to B, then compress. Pressure Pressure Volume Volume Expand P > 0 Vfinal > Vinitial W < 0 P > 0 Vinitial > Vfinal Compress W > 0 Work in both cases is the area under the curve. Total work = Wexpand + Wcompress = 0

8. Now suppose the expansion occurs at T1 and the compression at T2 > T1 Pressure Pressure Volume Volume Total work = Wexpand + Wcompress Wcompress > 0 ; Wexpand < 0 Wcompress > WexpandWtotal > 0

9. For those of you with a little more math background…

10. What we have just “invented” is called a PV cycle. The area enclosed in the PV loop is the work done in the cycle. The sign of W tells you whether work was done ON the system (gas) or BY the system (gas). Suppose V < 0 (this means Vfinal < Vinitial compression) W = -P V , Therefore W > 0. If we are compressing the gas (leaning on the piston) we are doing work ON the system. So in summary remember … W < 0  work is done BY the system. W > 0  work is done ON the system.

11. Gas Density and Molar Masses Where m is the mass of the gas and M is its molar mass. Rearranging, Since m/V = density we can write: Notice that the density is directly proportional to the molar mass!

12. Hot air ballooning and buoyancy Suppose we want to lift a young man and woman in a hot air balloon (for a romantic picnic). The couple plus the balloon basket weigh 400 lbs. Given that the volume of the balloon is 100 m3,suppose that the air in the balloon is heated to a temperature of 573 K, will the couple lift off? (Given that at 21C and 1 atm. air has a density of 1.2 g/L). 400 lbs x 454 g/lb = 1.816 x 105 g Ave. molar mass of air = 29 g/mol d=0.617 g/L Difference in density is 1.2 g/L - 0.617 g/L = 0.583 g/L The balloon at this temperature can lift: Calculate the temperature necessary to lift off!!!

13. Sect. 10.8 : Gas mixtures and Partial Pressures Partial pressure is the pressure of one gas in a mixture of gases. Total pressure is simply the sum of the partial pressures. (John Dalton) Make sure you can solve Problem Solving practice 10.12! Real Gases • How do "real" gases differ from "ideal" ones? • They do not obey PV=nRT • V is nonzero when T=0 • They condense to form liquids • They exhibit "critical behavior"

14. Critical Behavior in Gases Real gases liquefy. As P is increased at constant T, at some point liquid will form. The liquefaction occurs at constant P (horizontal line on the P-V plot.)

15. Ideal gas: Real gas: Z does not have to be (and is usually not) 1 Z is called the compression factor Z > 1  repulsion Z < 1  attraction

16. Consider the reason real gases deviate from ideality. • Molecules of real gases occupy volume (they are not point-like) • Molecules of real gases exert intermolecular interactions on each other. Van der Waals equation: The constants “a” and “b” are called the van der Waals constants for a gas. • Notice that the measured volume is decreased by the amount “nb”. Therefore “b” is the volume one mole of the gas actually occupies. • Observe that the measured pressure is increased by (n2a/V2) For an ideal gas a=0 and b=0 and the van der Waals equation is the same as the ideal gas law.

17. In “b” compare: • Ne and Ar • H2, N2, O2, and Cl2 • CH4, NH3, and H2O • In “a” compare: • Ne and Ar • H2, N2, O2, and Cl2 • CH4, NH3, and H2O