(A) The extraction of aluminum is achieved through the Hall-Héroult process, where Al2O3 is obtained as a precipitate by bubbling CO2 through a solution of sodium aluminate, resulting in the reaction:
\(2\text{Na}[ \text{Al}(\text{OH})_4](\text{aq.}) + \text{CO}_2 \rightarrow \text{Na}_2\text{CO}_3 + \text{H}_2\text{O} + 2\text{Al}(\text{OH})_3(\downarrow) \text{ or } \text{Al}_2\text{O}_3 \cdot 2\text{H}_2\text{O} (\text{ppt})\)
(B) The electrolytic reduction of pure alumina occurs in a steel box lined with carbon (cathode), along with the presence of cryolite (Na3AlF6) and fluorspar (CaF2), which decrease the melting point and enhance the electrolyte's conductivity.
(C) In the electrolysis process of the Hall-Héroult method, graphite rods serve as anodes. At the cathode, aluminum ions are reduced:
\(\text{Al}^{3+} + 3e^- \rightarrow \text{Al}\)
At the anode, the liberated oxygen reacts with the carbon of the anode to form CO and CO2:
\(\text{C} + \text{O}_2^- \rightarrow \text{CO} + 2e^-\)
\(\text{C} + 2\text{O}_2^- \rightarrow \text{CO}_2 + 4e^-\)
(D) In this context, the cathode consists of a steel vessel with a carbon lining.
A block of mass 1 kg is pushed up a surface inclined to horizontal at an angle of \( 60^\circ \) by a force of 10 N parallel to the inclined surface. When the block is pushed up by 10 m along the inclined surface, the work done against frictional force is:
[Given: \( g = 10 \) m/s\( ^2 \), \( \mu_s = 0.1 \)]
A temperature difference can generate e.m.f. in some materials. Let $ S $ be the e.m.f. produced per unit temperature difference between the ends of a wire, $ \sigma $ the electrical conductivity and $ \kappa $ the thermal conductivity of the material of the wire. Taking $ M, L, T, I $ and $ K $ as dimensions of mass, length, time, current and temperature, respectively, the dimensional formula of the quantity $ Z = \frac{S^2 \sigma}{\kappa} $ is:
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.