Читать книгу Geochemistry - William M. White - Страница 82

1 For a pure olivine mantle, calculate the adiabatic temperature gradient (∂T/∂P)s at 0.1 MPa (1 atm) and 1000°C. Use the thermodynamic data in Table 2.2 for forsterite (Mg-olivine, Mg2SiO4), and , and .Note that: 1 cc/mol = 1 J/MPa/mol.

2 Complete the proof that V is a state variable by showing that for an ideal gas:

3 A quartz crystal has a volume of 7.5 ml at 298 K and 0.1 MPa. What is the volume of the crystal at 840K and 12.3 MPa ifα = 1.4654 × 10−5 K−1 and β = 2.276 × 10−11 Pa−1 and α and β are independent of T and P.α = 1.4310 × 10−5 K−1 + 1.1587 × 10−9 K−2T β = 1.8553 × 10−11Pa−1 + 7.9453 × 10−8 Pa−1

4 One mole of an ideal gas is allowed to expand against a piston at constant temperature of 0°C. The initial pressure is 1 MPa and the final pressure is 0.04 MPa. Assuming the reaction is reversible,What is the work done by the gas during the expansion?What is the change in the internal energy and enthalpy of the gas?How much heat is gained/lost during the expansion?

5 A typical eruption temperature of basaltic lava is about 1200°C. Assuming that basaltic magma travels from its source region in the mantle quickly enough so that negligible heat is lost to wall rocks, calculate the temperature of the magma at a depth of 40 km. The density of basaltic magma at 1200°C is 2610 kg/m3; the coefficient of thermal expansion is about 1 × 10−4/K. Assume a heat capacity of 850 J/kg-K and that pressure is related to depth as 1 km = 33 MPa (surface pressure is 0.1 MPa.).(HINT: “Negligible heat loss” means the system may be treated as adiabatic.)

6 Show that the Cp of an ideal monatomic gas is 5/2 R.

7 Show that:

8 Show that for a reversible process:(2.73) (Hint: Begin with the statement of the first law (eqn. 2.58), make use of the Maxwell relations, and your proof in problem 7.)

9 Imagine that there are 30 units of energy to distribute among three copper blocks.If the energy is distributed completely randomly, what is the probability of the first block having all the energy?If n1 is the number of units of energy of the first block, construct a graph (a histogram) showing the probability of a given value of n1 occurring as a function of n1.(HINT: Use eqn. 2.37, but modify it for the case where there are three blocks.)

10 Consider a box partitioned into equal volumes, with the left half containing 1 mole of Ne and the right half containing 1 mole of He. When the partition is removed, the gases mix. Show, using a classical thermodynamic approach (i.e., macroscopic), that the entropy change of this process is . Assume that He and Ne are ideal gases and that temperature is constant.

11 Find expressions for Cp and Cv for a van der Waals gas.

12 Show that β (the compressibility, defined in eqn. 2.12) of an ideal gas is equal to 1/P.

13 Show that Hint: Start with equations 2.47 and 2.36a using the approximation that .

14 Show that Hint: Begin with equation 2.63 and express dU as a function of temperature and volume change.

15 Helium at 298K and 1 atm has . Assume He is an ideal gas.Calculate V, H, G, α, β, Cp, Cv, for He at 298K and 1 atm.What are the values for these functions at 600K and 100 atm?What is the entropy at 600 K and 100 atm?

16 Using the enthalpies of formation given in Table 2.02, find ΔH in Joules for the reaction:

17 Using the data in Table 2.2, calculate the enthalpy and entropy change of diopside as it is heated at constant pressure from 600 K to 1000 K.

18 Calculate the total enthalpy upon heating of 100g of quartz from 25° C to 900° C. Quartz undergoes a phase transition from α-quartz to β-quartz at 575° C. The enthalpy of this phase transition is . Use the Maier−Kelly heat capacity data in Table 2.2.

19 Calcite and aragonite are two forms of CaCO3 that differ only their crystal lattice structure. The reaction between them is thus simply:Using the data in Table 2.2,Determine which of these forms is stable at the surface of the earth (25° C and 0.1 MPa).Which form is favored by increasing temperature?Which form is favored by increasing pressure?

20 Use the data in Table 2.2 to determine the pressure at which calcite and aragonite are in equilibrium at 300°C.

21 Suppose you found kyanite and andalusite coexisting in the same rock, that you had reason to believe this was an equilibrium assemblage, and that you could independently determine the temperature of equilibrium to be 400°C. Use the data in Table 2.2 to determine the pressure at which this rock equilibrated.