JEE Advanced Mix Test 49

Solution:

Given the diameter measurements of a spherical object using a micrometer screw gauge with a least count of 0.01 mm, the recorded measurements are:

• 24.12 mm
• 24.18 mm
• 24.14 mm
• 24.17 mm
• 24.16 mm

We need to find the diameter of the object with its absolute uncertainty.

Solution:

Given:

• Three charges are located on the circumference of a circle with radius R.
• Two charges, each with magnitude Q, are placed symmetrically at 90^∘ apart from each other on the circle.
• A third charge q is placed at a point symmetrically opposite to the resultant of the two charges Q, such that the electric field at the center of the circle is zero.

Positioning the Charges:

Let's place the charges on the circumference of a circle with radius R and determine their positions:

• Let the charge Q be at (R, 0).
• Let the other charge Q be at (0, R).
• Let the charge q be at (-R sin θ, -R cos θ) to cancel the resultant electric field at the center.

Net Electric Field at Center:

The electric field due to the third charge q placed at (−R sinθ, −R cosθ ) must cancel out the resultant electric field. Since θ = 45^∘, q placed symmetrically opposite to the resultant:

Solution:

Given:

We need to determine which of the following statements about ΔQ (the heat transferred in a quasistatic reversible thermodynamic process) is correct.

Concept of Perfect Differentials:

A perfect differential (or exact differential) in thermodynamics is a differential that is the exact derivative of a state function. State functions are properties that depend only on the current state of the system, not on the path taken to reach that state. Examples of state functions include internal energy U, enthalpy H, entropy S, and temperature T.

For a function f(x, y, z, . . . ), if df is an exact differential, then there exists a state function f such that df is the exact derivative of f.

Analyzing ΔQ:

According to the first law of thermodynamics:

dU = δ Q - δ W

where dU is the change in internal energy, δQ is the infinitesimal heat added to the system, and δW is the infinitesimal work done by the system.

Here, dU is an exact differential because internal energy U is a state function. However, δQ and δW are not exact differentials because they depend on the path taken by the process, not just on the initial and final states.

Instead:

dU = δQ - δW

This implies that δQ and δW are path functions, not state functions. Therefore, δQ (or ΔQ) is not an exact differential. Since it is a path function, it cannot be expressed in terms of partial derivatives of a single potential function like dU can.

Evaluating the Provided Statements:

ΔQ is a perfect differential if the process is isochoric (constant volume).

Since heat transferred in an isochoric process is related to changes in internal energy alone δ Q = dU and not δW = 0, but δQ is not generally a perfect differential in all conditions. Hence, incorrect.

ΔQ is a perfect differential if the process is isobaric (constant pressure).

While heat transfer with pressure put in perspective, not specifically sufficient correcting general exactivity differential context, hence incorrect.

ΔQ is always a perfect differential.

Incorrect because ΔQ being path-function, unbound process statally not performing perfectness

ΔQ cannot be a perfect differential.

Since δQ, δ W are path functions unbound exact differential function operational, inferring correctly they cannot be perfected.'

The correct statement about ΔQ is:

Option 4: Δ Q cannot be a perfect differential.

Concept:

An object's resistance to rotational motion about a particular axis is measured by its moment of inertia (I). It is mass in linear motion's rotational equivalent. The object's mass distribution with respect to the axis of rotation determines the moment of inertia.

The moment of inertia is defined as:

Concept:

Free Energy Change (ΔG) and Reduction Potentials

ΔG = −nFE^o

• Free energy change (ΔG) for a redox reaction is related to the standard cell potential (E^o) by the equation:
• Where ΔG is the free energy change, n is the number of electrons transferred, F is the Faraday constant (96,485 C/mol), and E^o is the standard reduction potential of the reaction.
• The overall cell potential is calculated by subtracting the anode potential from the cathode potential.

Concept:

Chromium Compounds and Reactions

• Chromite ore (FeCr₂O₄) is the primary source of chromium compounds.
• When chromite ore is fused with sodium carbonate (Na₂CO₃) in the presence of air, sodium chromate (Na₂CrO₄) is formed, which is a yellow compound (B).
• On acidification with sulfuric acid (H₂SO₄), sodium chromate (B) converts into sodium dichromate (Na₂Cr₂O₇) (compound C).
• When sodium dichromate is treated with potassium chloride (KCl), potassium dichromate (K₂Cr₂O₇) (compound D) crystallizes out as orange crystals.

Explanation:

• In the given process:
  • Compound (A) is chromite ore, FeCr₂O₄.
  • Compound (B) is sodium chromate, Na₂CrO₄, which is yellow in solution.
  • Compound (C) is sodium dichromate, Na₂Cr₂O₇, obtained by acidifying compound (B).
  • Compound (D) is potassium dichromate, K₂Cr₂O₇, obtained by treating (C) with KCl.
• Thus, the correct identification of compounds (A) and (D) is FeCr₂O₄ and K₂Cr₂O₇, respectively.

Therefore, the correct answer is: FeCr₂O₄, K₂Cr₂O₇.

Concept:

Factors Affecting Crystal Field Splitting (Δ)

• Oxidation state of metal ion: The higher the oxidation state, the stronger the attraction between the metal and the ligands, resulting in a larger Δ.
• Coordination number of metal ion: Higher coordination numbers usually result in larger crystal field splitting.
• Electronegativity (EN) of the donor atom of the ligand: A more electronegative donor atom leads to a stronger ligand field and greater splitting.
• Position of the metal ion in the d block: Transition metals in later periods experience greater splitting due to stronger interactions with ligands.

Explanation:

• All the factors (A), (B), (C), and (D) significantly influence the magnitude of crystal field splitting in metal complexes.
• The oxidation state of the metal, the coordination number, the electronegativity of the donor atom, and the position of the metal ion in the d-block all contribute to the strength and nature of the ligand field splitting.

Therefore, the correct answer is (4) (A), (B), (C), and (D).