Problem Set #1
Ideal or Van der Waals Gases
1. A gas mixture containing 5 mol % butane and 95 mol % argon is prepared by allowing gaseous butane to fill an evacuated 40.0 L cylinder at 1.00 atm pressure. Calculate the mass of argon that gives the desired composition if the temperature is maintained at 25°C. Calculate the total pressure of the final mixture. The atomic mass of argon is 39.95 g/ mol. Assume ideal gas behavior.
2. Five moles of methane are confined to a container with volume 20 liters at 300 K. Estimate the pressure using the ideal gas and van der Waals equations of state given that the van der Waals constants a and b are 2.283 atm L2 mol-2 and 0.04278 L mol-1 respectively.
Isothermal Compressibility, Coefficient of Thermal Expansion
3. When a mercury-in-glass thermometer is overheated, the top will break off because of the expansion of mercury. A thick, well-tempered glass tube may withstand ~50 atm pressure without breaking. How far can a thermometer be heated past the temperature at which the capillary is filled before the pressure becomes this large? [kT = 3.9 x 10-6 atm-1, a = 1.8 x 10-4 K-1].
4. The specific volume of H2O (liquid) is given by the following empirical formula:
Derive from this a formula for the coefficient of thermal expansion (a).
5. Derive a formula for the coefficient of thermal expansion (a) for a gas with
6. Derive an expression for the isothermal compressibility of a van der Waals gas as a function of P, Vm, and the constants a and b.
7. Show that for an ideal gas and .
Van der Waals, Virial Equations of State, Compression Factor
8. A gas obeys the van der Waals equation with Pc = 3.040 x 106 Pa and Tc = 473 K. Calculate the value of the van der Waals constant b for this gas.
9. Show that Zreal = Zideal in the limit of low pressures.
10. a) Estimate the volume occupied by 2.76 kg of methane at 325 K, 10 atm, using the van der Waals equation. [a = 2.283 atm L2 mol-2, b = 0.04278 L mol-1].
b) Using the van der Waals equation, calculate the pressure exerted by 1 mole of CO2 at 0° C in a volume of 1.00 L. [a = 3.640 L2 bar mol-2; b = 0.04267 L mol-1].
11. Show that for a van der Waals gas the second B and third C virial coefficients are given by and .
12. In the volume virial equation, the second virial coefficient B of methyl isobutyl ketone is –1580 cm3 mol-1 at 120° C and 1 bar. Calculate its compressibility factor.
13. The equation of state of a certain gas is given by , where a and b are constants. Find .
14. The following equations of state are sometimes used for approximate calculations on gases: and . Assuming that there were gases that actually obeyed these equations of state, would it be possible to liquefy either gas A or B? Would they have a critical temperature? Explain.
15. The barometric formula relates the pressure of a gas of molar mass M at an altitude h to its pressure Po at sea level. Derive this relation by showing that the change in pressure dP for an infinitesimal change in altitude dh is . Remember that the density, r, depends on pressure. Evaluate the percent change in pressure between the top and bottom of the World Trade Center (412 m). Ignore temperature variations.
The Dieterici equation of state can be written as , where a and b are constants not necessarily equal to the van der Waals constants. Explain (in a few words) how we could derive the relationship of a and b to the critical volume and temperature using the Dieterici equation.