Observe the following reactions (unbalanced):
\( P_2O_3 + H_2O \rightarrow X \)
\( P_4O_{10} + H_2O \rightarrow Y \)
The number of \( P=O \) bonds present in \( X, Y \) are respectively:
Step 1: Understanding the Hydrolysis of Phosphorus Oxides
- \( P_2O_3 \) (Phosphorus(III) oxide) reacts with water to form phosphorous acid (\( H_3PO_3 \)): \[ P_2O_3 + 3H_2O \rightarrow 2H_3PO_3 \] - In \( H_3PO_3 \), there is one \( P=O \) bond. - \( P_4O_{10} \) (Phosphorus(V) oxide) reacts with water to form phosphoric acid (\( H_3PO_4 \)): \[ P_4O_{10} + 6H_2O \rightarrow 4H_3PO_4 \] - In \( H_3PO_4 \), there is also one \( P=O \) bond per molecule.
Step 2: Evaluating the Given Options
- Option (1): Incorrect, as \( X = H_3PO_3 \) has 1 \( P=O \) bond, and \( Y = H_3PO_4 \) also has 1 \( P=O \) bond.
- Option (2): Incorrect, as \( Y \) does not have 2 \( P=O \) bonds.
- Option (3): Incorrect, as \( X \) does not have 2 \( P=O \) bonds.
- Option (4): Correct, as both \( X \) and \( Y \) have 1 \( P=O \) bond each.
Thus, the correct answer is
Option (4).
The products formed in the following reaction, A and B, are:
In Bohr model of hydrogen atom, if the difference between the radii of \( n^{th} \) and\( (n+1)^{th} \)orbits is equal to the radius of the \( (n-1)^{th} \) orbit, then the value of \( n \) is:
Given the function:
\[ f(x) = \frac{2x - 3}{3x - 2} \]
and if \( f_n(x) = (f \circ f \circ \ldots \circ f)(x) \) is applied \( n \) times, find \( f_{32}(x) \).
For \( n \in \mathbb{N} \), the largest positive integer that divides \( 81^n + 20n - 1 \) is \( k \). If \( S \) is the sum of all positive divisors of \( k \), then find \( S - k \).
If the real-valued function
\[ f(x) = \sin^{-1}(x^2 - 1) - 3\log_3(3^x - 2) \]is not defined for all \( x \in (-\infty, a] \cup (b, \infty) \), then what is \( 3^a + b^2 \)?