tailieunhanh - Introduction to Elasticity Part 2

Tham khảo tài liệu 'introduction to elasticity part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | U n 1 ACe2 nr0 where A is the Madelung constant C is the appropriate units conversion factor and e is the ionic charge. 3. Measurements of bulk compressibility are valuable for probing the bond energy function because unlike simple tension hydrostatic pressure causes the interionic distance to decrease uniformly. The modulus of compressibility K of a solid is the ratio of the pressure p needed to induce a relative change in volume dV V K - d dV V The minus sign is needed because positive pressures induce reduced volumes volume change negative . a Use the relation dU pdV for the energy associated with pressure acting through a small volume change to show K _ d2U 1 V V0 where V0 is the crystal volume at the equilibrium interionic spacing r a0. b The volume of an ionic crystal containing N negative and N positive ions can be written as V cNr3 where c is a constant dependent on the type of lattice 2 for NaCl . Use this to obtain the relation K_ d2U 1 d _1dU V0 dV2 9c2N2r2 dr r2 dr V 0 c Carry out the indicated differentiation of the expression for binding energy to obtain the expression K _ K N 4ACe n n 3 B V0 cNr3 9c2 Nr0 .r0 r0n i Then using the expression B ACe2r0f 1 n obtain the formula for n in terms of compressibility n 1 9crịK ACe2 4. Complete the spreadsheet below filling in the values for repulsion exponent n and lattice energy U. 16 type ro pm K GPa A n U kJ mol Uexpt LiF 01 -1014 NaCl 01 -764 KBr 01 -663 The column labeled Uexpt lists experimentally obtained values of the lattice energy. 5. Given the definition of Helmholtz free energy A U - TS along with the first and second laws of thermodynamics dU dQ dW dQ TdS where U is the internal energy T is the temperature S is the entropy Q is the heat and W is the mechanical work show that the force F required to hold the ends of a tensile specimen a length L apart is related to the Helmholtz energy as F i dLJ T V 6. Show that the temperature dependence of .