Let G be an infinite group such that for any non-trivial subgroup H of G, |G : H | . Then prove that
i) H G
H = {e} or H is infinite;
ii) If g G, g
e, then o(g) is infinite.
Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that is not cyclic.
Which of the following statements are true? Give reasons for your answers.
i) If a group G is isomorphic to one of its proper subgroups, then G = .
ii) If x and y are elements of a non-abelian group (G, ) such that x
y = y
x, then x = e or y = e, where e is the identity of G with respect to
.
iii) There exists a unique non-abelian group of prime order.
iv) If (a, b) A
A, where A is a group, then o((a, b)) = o(a)o(b).
v) If H and K are normal subgroups of a group G, then hk = kh h
H, k
K.
Explain the packing efficiency
See Answer →What are Bravais lattices? Give the diagrams of any one type.
See Answer →Explain the seven primitive unit cells in crystals.
See Answer →Explain hydrogen bonding with suitable examples.
See Answer →The decomposition of acetaldehyde in gas-phase was studied at 791 K, using the initial concentrations of 7.77 10-3 M and 3.64
10-3 M. The half-life periods were determined to be 295 s and 515 s respectively. Determine the order of the reaction.
The decomposition of hydrogen iodide on gold was studied at 323 K with an initial concentration of HI to be 0.500 M. The reaction followed zero order kinetics with a rate constant of be 1.20 10-4 M s-1. Calculate the i) half-life of the reaction and ii) the time required for the concentration of HI to reduce to 0.10 M.
How is order of reaction determined by half-life method? Explain with the help of an example.
See Answer →Differentiate between the average and instantaneous rates of a reaction.
See Answer →Explain the pressure and volume correction terms to the ideal gas equation, and deduce van der Waals equation.
See Answer →) Define critical constants and give their expressions in terms of van der Waals constants.
See Answer →State the Dalton’s law of partial pressure and give its significance.
See Answer →Calculate the i) average speed ii) root mean square speed and iii) most probable speed of oxygen molecules at 515 K. (Given Mm(O2) = 0.016 kg mol-1)
See Answer →Give the pattern crystal field splitting energies for d0 to d10 high spin ions.
See Answer →Diagrammatically explain the crystal field splitting in octahedral complexes.
See Answer →Predict the geometry of the [Co(H2O)6] 2+ complex showing their hybridized orbitals.
See Answer →Explain the structure of [CoF6] 3- with the help of valence bond theory.
See Answer →Explain ligand isomerism with suitable example.
See Answer →