Class 12th

For the complex K? [Cr (oxalate)? ], the central metal ion is Cr³?

Electronic configuration of Cr (24) is [Ar] 4s¹3d?

Electronic configuration of Cr³? is [Ar] 4s?3d³.

The number of unpaired electrons in Cr³? is 3.

Given the function:
f(x) = { x(2 - sin(1/x)), if x ≠ 0
{ 0, if x = 0

For x < 0: f(x) = x(2 - sin(1/x))

For x > 0: f(x) = x(2 - sin(1/x))

The derivative f'(x) for x ≠ 0 is:
f'(x) = 1*(2 - sin(1/x)) + x*(-cos(1/x))*(-1/x²) = 2 - sin(1/x) + (1/x)cos(1/x)

The text calculates the derivative differently:
For x < 0: f'(x) = -2 + sin(1/x) - (1/x)cos(1/x)
For x > 0: f'(x) = 2 - sin(1/x) + (1/x)cos(1/x)

To check if f'(0) is defined, we would need to use the limit definition of the derivative at x=0. As x approaches 0, the term (1/x)cos(1/x) oscillates and does not approach a finite limit. Therefore, f'(0) is undefined.

Given K_f = 1.85 K kg mol? ¹ for a solution with molality of 2 m.

ΔT_f = I * K_f * m

3.885 = I * 1.85 * 2

The van't Hoff factor, I = 1.05.

i = 1 + (n-1)α. For an electrolyte dissociating into 2 ions, n=2.

1.05 = 1 + (2-1)α.

The degree of dissociation, α = 0.05 or 50 * 10? ³.

Please consider the following Image

Please consider the following Image

For an acidic buffer solution, pH = pKa + log ( [Base]/ [Acid]).

Given pH = 5.74 and pKa = 4.74.

5.74 = 4.74 + log ( [Base]/1).

1 = log ( [Base]).

[Base] = 10M.

For the reaction C? H? → C? H? + H? , calculate the enthalpy change (ΔH).

ΔH = [Bond energy (C-C) + 6 * Bond energy (C-H)] - [Bond energy (C=C) + 4 * Bond energy (C-H) + Bond energy (H-H)]

ΔH = 347 + 2 (414) - 611 - 436 = 128 kJ/mol.

When white light passes through a cool gas, atoms absorb specific wavelengths. This produces a continuous background with dark lines at those absorbed wavelengths. This is atomic absorption spectra. But when atoms are excited, they release photons at specific wavelengths, producing a dark background with bright lines. That is emission spectra. 

CH? (methane) is produced / generated from paddy fields. And methane leads to both global warming and photochemical smog.

CO? is used in photosynthesis, acid rain etc. but methane is not consumed.

So methane is a stronger global warming gas than CO?