The molecular weight of \( C_8H_8O_2 \) is calculated as:
Molecular weight = \( 12 \times 8 + 1 \times 8 + 16 \times 2 = 96 + 8 + 32 = 136 \)
Next, we calculate the moles of product \( D \) formed:
Moles of product \( D \) = \( \frac{15}{3} \times 0.8 \times 0.5 = \frac{3}{3} = 1 \)
Therefore, the weight of product \( D \) is:
Weight of product \( D \) = \( 1 \times 136 = 136 \, \text{g} \)
Calculate the potential for half-cell containing 0.01 M K\(_2\)Cr\(_2\)O\(_7\)(aq), 0.01 M Cr\(^{3+}\)(aq), and 1.0 x 10\(^{-4}\) M H\(^+\)(aq).
Let $ S $ denote the locus of the point of intersection of the pair of lines $$ 4x - 3y = 12\alpha,\quad 4\alpha x + 3\alpha y = 12, $$ where $ \alpha $ varies over the set of non-zero real numbers. Let $ T $ be the tangent to $ S $ passing through the points $ (p, 0) $ and $ (0, q) $, $ q > 0 $, and parallel to the line $ 4x - \frac{3}{\sqrt{2}} y = 0 $.
Then the value of $ pq $ 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 ________.
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