A mixture of PbS and PbO undergoes self-reduction to produce Pb and SO2.
In the extraction process of copper from copper pyrites, silica is added to produce copper silicate.
Partial oxidation of sulphide ore of copper by roasting, followed by self-reduction produces blister copper.
In the cyanide process, zinc powder is utilized to precipitate gold from Na[Au(CN)2].
PbS + 2PbO → 3Pb + SO2
The reaction involves self-reduction between PbS and PbO.
In the Bessemer converter, the raw material used is matte, consisting of Cu2S + a small amount of FeS. Initially, air blasting is conducted to facilitate slag formation, and SiO2 is introduced from an external source.
\(\text{FeS} + \frac{3}{2}\text{O}_2 \rightarrow \text{FeO} + \text{SO}_2 \uparrow\)
\(\text{SiO}_2 + \text{FeO} \rightarrow \text{FeSiO}_3\) (slag)
During slag formation, a characteristic green flame is observed at the mouth of the Bessemer converter, indicating the presence of iron in the form of FeO. The disappearance of this green flame signifies the completion of slag formation. Subsequently, air blasting is halted, and the slag is removed.
Air blasting is then resumed for partial roasting before self-reduction, continuing until approximately two-thirds of Cu2S is converted into Cu2O. Following this stage, only heating is maintained for the self-reduction process.
\(\text{Cu}_2\text{S} + \frac{3}{2}\text{O}_2 \rightarrow \text{Cu}_2\text{O} + \text{SO}_2 \uparrow\)
\(\text{Cu}_2\text{S} + 2\text{Cu}_2\text{O} \rightarrow 6\text{Cu(l)} + \text{SO}_2 \uparrow\)(self-reduction)
Additionally,
\(\text{Cu}_2\text{S} + 2\text{O}_2 \rightarrow \text{Cu}_2\text{SO}_4\)
\(\text{Cu}_2\text{S} + \text{Cu}_2\text{SO}_4 \rightarrow 4\text{Cu} + 2\text{SO}_2 \uparrow\) (self-reduction)
The resulting molten copper is poured into a large container and allowed to cool. During cooling, dissolved SO2 rises to the surface and forms blisters, resulting in the production of blister copper.
\(2\text{Na}[Au(CN)_2] + \text{Zn} \rightarrow \text{Na}_2[Zn(CN)_4] + 2\text{Au} \downarrow\)
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 $ \mathbb{R} $ denote the set of all real numbers. Then the area of the region $$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x},\ 5x - 4y - 1 > 0,\ 4x + 4y - 17 < 0 \right\} $$ is
The center of a disk of radius $ r $ and mass $ m $ is attached to a spring of spring constant $ k $, inside a ring of radius $ R>r $ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following Hooke’s law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as $ T = \frac{2\pi}{\omega} $. The correct expression for $ \omega $ is ( $ g $ is the acceleration due to gravity): 