Q: Material balances for a reacting system: For the reactor carrying out A + B → C (single reaction), how many independent material balance equations can be written around the reactor?

Introduction / Context: Writing and counting independent material balances is fundamental to setting up process calculations. For reacting systems, balances can be written on each molecular species and, optionally, on total mass. However, linear dependence must be considered to avoid overcounting equations. Given Data / Assumptions: Single reaction: A + B → C. Species present: A, B, C (three distinct components). Steady-state or unsteady does not change the count of independent component balances, only their forms. No side reactions and no additional inert species mentioned. Concept / Approach: Independent component balances can be written for each species: A, B, and C. An overall mass balance can also be written, but for single-reaction systems it is a linear combination of the component balances (hence dependent). The number of independent material balance equations equals the number of chemically independent species present (here, three). Step-by-Step Solution: List balances: species A, species B, species C.Recognize that an overall mass balance is not independent when component balances are already included (mass conservation is implied by summing components).Count of independent balances = 3. Verification / Alternative check: Rank analysis: For one reaction among three species, the stoichiometric matrix has rank 1. The number of independent species balances remains 3; including extent of reaction handles coupling, but independence among species balances persists. Why Other Options Are Wrong: 1 or 2: Underestimates; you can and should balance all three species separately. 4: Overestimates; the fourth equation (overall mass) is dependent on the three component balances. Common Pitfalls: Confusing “number of degrees of freedom” with “number of balance equations”; assuming overall mass balance adds independence in a single-reaction, single-phase system. Final Answer: 3

Q: Identify the incorrect temperature conversion relation: Choose the option that is wrong (others are correct reformulations).

Introduction / Context: Accurate temperature conversions are essential across thermodynamics, heat transfer, and instrumentation. Rømer, Rankine, Celsius, Fahrenheit, and Kelvin scales often appear in legacy data or vendor specifications; choosing the correct formula avoids significant calculation errors. Given Data / Assumptions: Kelvin–Celsius: K = °C + 273.15 (≈ 273 for quick estimates). Rankine–Fahrenheit: °R = °F + 459.67. Linear scale relations: °F = (9/5)°C + 32 and °C = (5/9)(°F − 32). Temperature differences: ΔK = Δ°C = (9/5)Δ°F. Concept / Approach: Check each formula against the standard relations. Remember that offsets differ between pairs: Rankine and Fahrenheit share degree size but different zero; Kelvin and Celsius share offset at zero. Differences (not absolute values) convert without offsets, but magnitudes scale by 9/5 between °F and °C/K. Step-by-Step Evaluation: (a) Claimed °R = °F + 273 is wrong; correct is °R = °F + 459.67.(b) Differences: 1 K = 1 °C = 1.8 °F → correct for temperature intervals.(c) °C = (°F − 32) × 5/9 ≈ (°F − 32) × 0.555 → correct.(d) °F = 1.8 × (°C + 17.778) is algebraically the same as °F = 1.8 °C + 32 → correct. Verification / Alternative check: Test with °C = 0: (d) gives °F = 1.8 × 17.778 ≈ 32, as expected. For (a), if °F = 0, formula would give °R = 273, but the actual Rankine at 0 °F is 459.67 °R. Why Other Options Are Wrong or Right: (a) Wrong due to incorrect zero offset. (b)–(d) are correct relations for differences or equivalent algebraic forms. Common Pitfalls: Confusing absolute temperatures with temperature differences; mixing the Kelvin and Rankine offsets; rounding offsets too aggressively. Final Answer: °R = °F + 273

Q: Baumé gravity scale (liquids lighter than water): Select the correct defining equation for °Bé when specific gravity G < 1.

Introduction / Context: The Baumé (°Bé) hydrometer scale predates API gravity and remains in use for some process liquids. Two distinct definitions exist: one for liquids lighter than water and another for heavier liquids. Correctly identifying the formula matters when converting between °Bé and specific gravity G for sizing, blending, or quality control. Given Data / Assumptions: G denotes specific gravity referenced to water at standard conditions. Different constants apply for lighter vs. heavier than water. We focus on the “lighter than water” case (G < 1). Concept / Approach: Historical definitions of the Baumé scale commonly used are: • For liquids lighter than water: °Bé = (140 / G) − 130. • For liquids heavier than water: °Bé = 145 − (145 / G). The constants reflect calibration of the original hydrometers and are empirical; modern practice often prefers API gravity for petroleum, but Baumé remains relevant in various industries. Step-by-Step Reasoning: Recognize the “lighter than water” formula by its 140 and 130 constants.Exclude expressions using 145 (these relate to the “heavier than water” definition).Exclude arbitrary-looking constants (e.g., 400) that do not correspond to standard Baumé definitions. Verification / Alternative check: If G = 1.0 (water), the lighter-than-water formula yields °Bé = 10, which aligns with traditional conventions (water near 10 °Bé on the “lighter” scale boundary definitions). Why Other Options Are Wrong: 200(G − 1): Not a standard Baumé relation. 145 − (145/G): This is the heavier-than-water relation, not applicable here. (400/G) − 400: Non-standard and dimensionally inconsistent with usual constants. Common Pitfalls: Mixing the lighter/heavier formulas; confusing Baumé with API gravity (API uses 141.5 and 131.5 constants and is defined only for petroleum liquids). Final Answer: °Bé = (140 / G) − 130

Q: Methods for estimating latent heat of vaporisation: Which of the following is NOT typically used to compute latent heat?

Introduction / Context: Latent heat of vaporisation is a key thermophysical property for distillation, evaporation, and energy balance calculations. Several correlations and thermodynamic relations allow its estimation when direct data is unavailable. Identifying which tools are appropriate speeds up preliminary design and troubleshooting. Given Data / Assumptions: Equilibrium thermodynamics applies; vapor and liquid phases are at saturation conditions during phase change. Engineering correlations are accepted approximations. Concept / Approach: Common approaches include: • Clausius–Clapeyron: Relates saturation pressure slope to latent heat; widely used for pure components at moderate ranges. • Dühring and Othmer plots: Reference-substance methods correlating boiling points/enthalpies; useful when a similar well-characterized fluid exists. • Kistyakowsky-type equations: Empirical correlations for enthalpy of vaporisation versus critical properties/temperature. In contrast, Hess’s law pertains to the additivity of reaction enthalpies in chemical reactions, not phase-change latent heats of pure substances. Step-by-Step Evaluation: Identify which options are phase-equilibrium-based or empirical latent-heat correlations (a, b, c).Recognize (d) as a reaction thermochemistry principle unrelated to vaporisation latent heat estimation. Verification / Alternative check: Applying Clausius–Clapeyron: λ ≈ R T^2 (d ln Psat/dT) for a pure fluid; this directly ties λ to the vapor-pressure curve slope—standard in thermodynamics texts. Why Other Options Are Wrong: (a), (b), (c) are established methods/correlations for latent heat; they are not incorrect choices. Common Pitfalls: Confusing chemical reaction enthalpy tools with phase-change properties; forgetting that latent heat refers to isothermal enthalpy change between phases at equilibrium. Final Answer: Hess’s law of constant heat summation

Question 1

Cooling an exactly eutectic solution: As a binary solution of eutectic composition is cooled, what happens before the eutectic temperature is reached?

Accepted Answer

Introduction / Context: Eutectic systems are crucial in alloy design, soldering, and crystallisation. A eutectic composition freezes at a single, sharp temperature where two solid phases form simultaneously from the liquid. Given Data / Assumptions: Binary system at exactly eutectic composition. Equilibrium cooling under negligible undercooling. No impurities shifting the phase diagram. Concept / Approach: At eutectic composition, the liquid remains entirely liquid until the eutectic temperature Te is reached. Then the eutectic reaction L → α + β occurs isothermally at Te, producing a characteristic fine microstructure. Step-by-Step Solution: Identify eutectic point on the phase diagram.Follow vertical line at eutectic composition during cooling.No intersection with liquidus lines before Te, so no solid appears earlier.At Te, liquid transforms to two solids simultaneously. Verification / Alternative check: Compare to hypoeutectic or hypereutectic compositions, where primary crystals appear before Te; that contrast confirms the unique “no change until Te” behavior at the eutectic composition. Why Other Options Are Wrong: Solvent freezing before Te: Occurs only off-eutectic (primary solid forms prior to Te). No solidification at or past Te: Contradicts the definition of the eutectic reaction. None: Not applicable since (b) is correct. Common Pitfalls: Confusing eutectic with azeotropic behavior or misreading phase diagrams; pay attention to composition versus temperature paths. Final Answer: It undergoes no phase change until the eutectic temperature is reached

Question 2

Fundamental units: Which of the following is not a unit of length in science and engineering usage?

Accepted Answer

Introduction / Context: Recognising physical units and their dimensions prevents serious calculation errors. While many specialised units exist for length at astronomical or subatomic scales, some similarly named units measure different physical quantities entirely. Given Data / Assumptions: We compare common physics units across astronomy and nuclear/atomic physics. We focus on dimensionality: length versus area. Concept / Approach: Parsec (pc), Angstrom (Å), and Fermi (fm) are all units of length. In contrast, a Barn (b) is a unit of area used for nuclear cross-sections, equal to 10^-28 m^2, not a length. Step-by-Step Solution: Parsec: Astronomical length unit ≈ 3.086 * 10^16 m.Angstrom: Atomic-scale length unit = 10^-10 m.Fermi: Nuclear-scale length unit = 10^-15 m.Barn: Cross-section area = 10^-28 m^2 (dimension of area, not length). Verification / Alternative check: Dimensional analysis confirms Barn has dimension L^2, unlike the others (dimension L). Why Other Options Are Wrong: Parsec, Angstrom, Fermi: All are indeed length units across different scales. Common Pitfalls: Assuming similarity by name (“Fermi” vs “Fermion”) or scale; always verify dimensionality to avoid mixing units such as area, volume, or energy with length. Final Answer: Barn

Question 3

Free expansion (Joule expansion) of an ideal gas: A rigid, perfectly insulated container of volume V is split into two equal halves. One side is vacuum; the other contains 1 mol ideal gas at 298 K. After the partition is broken, what is the final te…

Accepted Answer

Introduction / Context: Free expansion into vacuum (Joule expansion) is a classic thermodynamics scenario used to distinguish ideal from real gas behavior. It illustrates how internal energy depends on temperature for an ideal gas. Given Data / Assumptions: Rigid container (no boundary work). Perfectly insulated (adiabatic: Q = 0). Ideal gas with temperature-independent heat capacity. Initial T = 298 K; final volume doubles after partition removal. Concept / Approach: First law for a closed, adiabatic, rigid system: ΔU = Q − W = 0 − 0 = 0. For an ideal gas, U depends only on T. Therefore ΔT = 0 and the final temperature equals the initial temperature. Step-by-Step Solution: Apply ΔU = n * Cv * (T2 − T1) = 0.Since n and Cv are positive, T2 − T1 = 0.Hence T2 = 298 K. Verification / Alternative check: Real gases may show small Joule effects due to non-ideal interactions, but the ideal-gas model assumes zero; thus, temperature remains unchanged. Why Other Options Are Wrong: Greater/Less than 298 K: Would require net heat or work interactions, absent here, or non-ideal effects not assumed. Cannot be determined: It can, from the first law and ideal-gas property of U(T) only. Common Pitfalls: Confusing free expansion with throttling (Joule–Thomson) where enthalpy, not internal energy, is constant and real-gas effects matter. Final Answer: 298 K

Question 4

Crystallisation with hydrates: Cooling 100 kg of an aqueous Na2SO4 solution from 30% (w/w) to a mother-liquor concentration of 10% (w/w) precipitates Na2SO4·10H2O crystals. What is the mass of crystals formed?

Accepted Answer

Introduction / Context: Hydrate crystallisation problems are common in chemical process calculations. Correctly accounting for solute in the crystal lattice water is key to obtaining the right crystal mass and mother-liquor composition. Given Data / Assumptions: Feed: 100 kg solution at 30% w/w Na2SO4 (anhydrous basis). Final mother liquor: 10% w/w Na2SO4. Crystals: Na2SO4·10H2O (decahydrate). Molar masses (kg/kmol): Na2SO4 ≈ 142.04; 10H2O ≈ 180.16; crystal ≈ 322.20. Concept / Approach: Perform component balances on anhydrous Na2SO4. In the hydrate crystals, only a fraction of the mass is Na2SO4; the rest is water of crystallisation. Mass fraction of Na2SO4 in crystals = 142.04 / 322.20 ≈ 0.441. Use this to link crystal mass to solute removed from solution. Step-by-Step Solution: Initial Na2SO4 = 0.30 * 100 = 30 kg.Let crystal mass = Wc (kg). Na2SO4 in crystals = 0.441 * Wc.Mother liquor mass = 100 − Wc; Na2SO4 in liquor = 0.10 * (100 − Wc) = 10 − 0.10 Wc.Na2SO4 balance: 30 = (10 − 0.10 Wc) + (0.441 Wc).Solve: 30 − 10 = (0.441 − 0.10) Wc ⇒ 20 = 0.341 Wc ⇒ Wc ≈ 58.65 kg. Verification / Alternative check: Water balance (optional) would show the remaining water plus the 55.9% crystal water equals the original 70 kg less liquor water removal—consistent within rounding. Why Other Options Are Wrong: 20, 32.2, 45.35: Each fails the Na2SO4 balance when the hydrate fraction is applied. Common Pitfalls: Using 10% of the original solution instead of mother liquor mass, or forgetting that crystals contain substantial water; both lead to significant underestimation. Final Answer: 58.65

Question 5

Atomic counting in a sample of ammonia (NH3): Given 8.5 g of NH3, determine the total number of atoms present, expressed as X × 10^23. Choose the closest value for X.

Accepted Answer

Introduction / Context: Counting atoms in a given mass of a compound is a staple skill in chemical stoichiometry. This problem asks for the total number of atoms in a known mass of ammonia (NH3), reported in the convenient scientific form X × 10^23. Solving it reinforces conversions among grams, moles of molecules, and finally moles of atoms using Avogadro’s constant. Given Data / Assumptions: Molar mass of NH3 = 17 g/mol (14 for N and 1 each for 3 H). Sample mass m = 8.5 g NH3. Each NH3 molecule contains 4 atoms (1 N + 3 H). Avogadro’s constant NA ≈ 6.022 × 10^23 entities per mole. Concept / Approach: The workflow is: grams → moles of molecules → moles of atoms → number of atoms. First compute moles of NH3 from its molar mass. Multiply by the number of atoms per molecule to get moles of atoms. Finally, convert moles of atoms to number of atoms using NA. Report the result as X × 10^23 and pick X appropriately. Step-by-Step Solution: n(NH3) = m / M = 8.5 g / 17 g·mol^-1 = 0.5 mol NH3.Each molecule has 4 atoms ⇒ moles of atoms = 0.5 mol × 4 = 2.0 mol atoms.Number of atoms = 2.0 mol × 6.022 × 10^23 = 12.044 × 10^23 ≈ 12.04 × 10^23. Verification / Alternative check: If the mass were 17 g (one mole) of NH3, there would be 4 × 6.022 × 10^23 = 24.088 × 10^23 atoms. Halving the mass to 8.5 g halves the count to 12.044 × 10^23, matching the calculation. Why Other Options Are Wrong: 9.03, 6.02, 3.01, 1.204: These arise from common slip-ups such as forgetting the 4 atoms per molecule, using molecules instead of atoms, or mis-scaling by factors of 10. Common Pitfalls: Not multiplying by the number of atoms per molecule, rounding too early, or misreading the requested format (X × 10^23 rather than × 10^24). Work symbolically, then round only at the end. Final Answer: 12.04 × 10^23 atoms

Question 6

Atmospheric pressure as a hydrostatic head: Which of the following is the closest equivalent of 1 atm in terms of fluid column height?

Accepted Answer

Introduction / Context: Engineering calculations often require converting standard atmospheric pressure into an equivalent height of a liquid column. Knowing these equivalences helps when reading manometers, sizing barometric condensers, or interpreting pressure heads in piping systems. Given Data / Assumptions: 1 atm ≈ 101.325 kPa. Equivalent heads: about 10.33 m of water, 33.9–34.0 ft of water, and 760 mm (76 cm) of mercury at standard conditions. Densities and gravity at standard conditions are implied. Concept / Approach: The hydrostatic relation is ΔP = ρ g h. For a given pressure, required height h is inversely proportional to liquid density ρ. Thus, a denser fluid like mercury needs a much shorter column than water to balance atmospheric pressure. Recognized “rules of thumb” are typically used in quick selections. Step-by-Step Solution: For water: 1 atm ≈ 10.33 m H2O ≈ 33.9 ft H2O → rounded to 34 ft H2O.For mercury: 1 atm ≈ 760 mm Hg = 76 cm Hg, not 13.6 cm Hg (13.6 is mercury’s specific gravity).For 1 m H2O: 1 m is only about 0.097 atm, far less than 1 atm.For 13.6 m Hg: that would correspond to ~13.6 atm (clearly incorrect). Verification / Alternative check: Using ΔP = ρ g h with ρ_H2O ≈ 1000 kg/m^3 and g ≈ 9.81 m/s^2 yields h ≈ 10.33 m for 101.325 kPa. Converting 10.33 m to feet gives ~33.9 ft. Why Other Options Are Wrong: 13.6 cm Hg confuses mercury’s specific gravity with the required head. 1 m H2O is too small; it does not balance 1 atm. 13.6 m Hg is far too large. (Extra reference) 760 mm Hg is correct but was provided as an additional option; the best match among the listed head choices is 34 ft of H2O. Common Pitfalls: Mixing up 13.6 (SG of Hg) with 76 cm; ignoring unit conversions between metric and imperial; forgetting that heads scale inversely with density. Final Answer: 34 ft of H2O

Question 7

Trend of specific heat in aqueous solutions: As the solute concentration increases in a water-based solution, the specific heat capacity generally tends to…

Accepted Answer

Introduction / Context: Specific heat capacity (cp) is the heat required to raise the temperature of a unit mass by one degree. Water has a notably high specific heat, which is why aqueous solutions often exhibit lower cp values as solutes are added. Understanding this trend is important for heat exchanger design, reactor temperature control, and energy balances in process engineering. Given Data / Assumptions: Base solvent is liquid water, cp ≈ 4.18 kJ/kg·K near ambient conditions. Solutes typically have lower cp per unit mass than water (ionic and many molecular solutes). No unusual heat capacity anomalies (e.g., very low concentrations where measurement noise dominates). Concept / Approach: The mixture cp is a mass-fraction-weighted property, modified by interaction effects. When replacing part of the mass of water (high cp) with solute (often lower cp), the composite specific heat capacity usually decreases. This impacts steam demand, cooling water loads, and transient thermal response of equipment containing brines, syrups, or other concentrated solutions. Step-by-Step Discussion: Start with pure water at cp ≈ 4.18 kJ/kg·K (reference high value).Add solute with cp_s < cp_water; mixture cp ≈ w_watercp_water + w_solutecp_s (simplified).As w_solute increases, the lower cp contribution grows, typically decreasing the overall cp.Interaction terms and temperature dependence exist, but the first-order trend remains downward for most common aqueous solutions (salts, sugars, acids). Verification / Alternative check: Consult cp tables for sodium chloride solutions or sucrose syrups: increasing concentration correlates with lower cp compared to water at the same temperature. Why Other Options Are Wrong: Increase: Opposite to the typical behavior of replacing high-cp water with lower-cp solute. Remain unchanged / No predictable trend: Empirical data across many solutes shows a consistent decrease, aside from narrow exceptions. Common Pitfalls: Forgetting temperature dependence; assuming ideal linear mixing without acknowledging interaction effects (which usually strengthen the downward trend). Final Answer: Decrease

Question 8

Material balances for a reacting system: For the reactor carrying out A + B → C (single reaction), how many independent material balance equations can be written around the reactor?

Accepted Answer

Introduction / Context: Writing and counting independent material balances is fundamental to setting up process calculations. For reacting systems, balances can be written on each molecular species and, optionally, on total mass. However, linear dependence must be considered to avoid overcounting equations. Given Data / Assumptions: Single reaction: A + B → C. Species present: A, B, C (three distinct components). Steady-state or unsteady does not change the count of independent component balances, only their forms. No side reactions and no additional inert species mentioned. Concept / Approach: Independent component balances can be written for each species: A, B, and C. An overall mass balance can also be written, but for single-reaction systems it is a linear combination of the component balances (hence dependent). The number of independent material balance equations equals the number of chemically independent species present (here, three). Step-by-Step Solution: List balances: species A, species B, species C.Recognize that an overall mass balance is not independent when component balances are already included (mass conservation is implied by summing components).Count of independent balances = 3. Verification / Alternative check: Rank analysis: For one reaction among three species, the stoichiometric matrix has rank 1. The number of independent species balances remains 3; including extent of reaction handles coupling, but independence among species balances persists. Why Other Options Are Wrong: 1 or 2: Underestimates; you can and should balance all three species separately. 4: Overestimates; the fourth equation (overall mass) is dependent on the three component balances. Common Pitfalls: Confusing “number of degrees of freedom” with “number of balance equations”; assuming overall mass balance adds independence in a single-reaction, single-phase system. Final Answer: 3

Question 9

Identify the incorrect temperature conversion relation: Choose the option that is wrong (others are correct reformulations).

Accepted Answer

Introduction / Context: Accurate temperature conversions are essential across thermodynamics, heat transfer, and instrumentation. Rømer, Rankine, Celsius, Fahrenheit, and Kelvin scales often appear in legacy data or vendor specifications; choosing the correct formula avoids significant calculation errors. Given Data / Assumptions: Kelvin–Celsius: K = °C + 273.15 (≈ 273 for quick estimates). Rankine–Fahrenheit: °R = °F + 459.67. Linear scale relations: °F = (9/5)°C + 32 and °C = (5/9)(°F − 32). Temperature differences: ΔK = Δ°C = (9/5)Δ°F. Concept / Approach: Check each formula against the standard relations. Remember that offsets differ between pairs: Rankine and Fahrenheit share degree size but different zero; Kelvin and Celsius share offset at zero. Differences (not absolute values) convert without offsets, but magnitudes scale by 9/5 between °F and °C/K. Step-by-Step Evaluation: (a) Claimed °R = °F + 273 is wrong; correct is °R = °F + 459.67.(b) Differences: 1 K = 1 °C = 1.8 °F → correct for temperature intervals.(c) °C = (°F − 32) × 5/9 ≈ (°F − 32) × 0.555 → correct.(d) °F = 1.8 × (°C + 17.778) is algebraically the same as °F = 1.8 °C + 32 → correct. Verification / Alternative check: Test with °C = 0: (d) gives °F = 1.8 × 17.778 ≈ 32, as expected. For (a), if °F = 0, formula would give °R = 273, but the actual Rankine at 0 °F is 459.67 °R. Why Other Options Are Wrong or Right: (a) Wrong due to incorrect zero offset. (b)–(d) are correct relations for differences or equivalent algebraic forms. Common Pitfalls: Confusing absolute temperatures with temperature differences; mixing the Kelvin and Rankine offsets; rounding offsets too aggressively. Final Answer: °R = °F + 273

Question 10

Baumé gravity scale (liquids lighter than water): Select the correct defining equation for °Bé when specific gravity G < 1.

Accepted Answer

Introduction / Context: The Baumé (°Bé) hydrometer scale predates API gravity and remains in use for some process liquids. Two distinct definitions exist: one for liquids lighter than water and another for heavier liquids. Correctly identifying the formula matters when converting between °Bé and specific gravity G for sizing, blending, or quality control. Given Data / Assumptions: G denotes specific gravity referenced to water at standard conditions. Different constants apply for lighter vs. heavier than water. We focus on the “lighter than water” case (G < 1). Concept / Approach: Historical definitions of the Baumé scale commonly used are: • For liquids lighter than water: °Bé = (140 / G) − 130. • For liquids heavier than water: °Bé = 145 − (145 / G). The constants reflect calibration of the original hydrometers and are empirical; modern practice often prefers API gravity for petroleum, but Baumé remains relevant in various industries. Step-by-Step Reasoning: Recognize the “lighter than water” formula by its 140 and 130 constants.Exclude expressions using 145 (these relate to the “heavier than water” definition).Exclude arbitrary-looking constants (e.g., 400) that do not correspond to standard Baumé definitions. Verification / Alternative check: If G = 1.0 (water), the lighter-than-water formula yields °Bé = 10, which aligns with traditional conventions (water near 10 °Bé on the “lighter” scale boundary definitions). Why Other Options Are Wrong: 200(G − 1): Not a standard Baumé relation. 145 − (145/G): This is the heavier-than-water relation, not applicable here. (400/G) − 400: Non-standard and dimensionally inconsistent with usual constants. Common Pitfalls: Mixing the lighter/heavier formulas; confusing Baumé with API gravity (API uses 141.5 and 131.5 constants and is defined only for petroleum liquids). Final Answer: °Bé = (140 / G) − 130

Question 11

Methods for estimating latent heat of vaporisation: Which of the following is NOT typically used to compute latent heat?

Accepted Answer

Introduction / Context: Latent heat of vaporisation is a key thermophysical property for distillation, evaporation, and energy balance calculations. Several correlations and thermodynamic relations allow its estimation when direct data is unavailable. Identifying which tools are appropriate speeds up preliminary design and troubleshooting. Given Data / Assumptions: Equilibrium thermodynamics applies; vapor and liquid phases are at saturation conditions during phase change. Engineering correlations are accepted approximations. Concept / Approach: Common approaches include: • Clausius–Clapeyron: Relates saturation pressure slope to latent heat; widely used for pure components at moderate ranges. • Dühring and Othmer plots: Reference-substance methods correlating boiling points/enthalpies; useful when a similar well-characterized fluid exists. • Kistyakowsky-type equations: Empirical correlations for enthalpy of vaporisation versus critical properties/temperature. In contrast, Hess’s law pertains to the additivity of reaction enthalpies in chemical reactions, not phase-change latent heats of pure substances. Step-by-Step Evaluation: Identify which options are phase-equilibrium-based or empirical latent-heat correlations (a, b, c).Recognize (d) as a reaction thermochemistry principle unrelated to vaporisation latent heat estimation. Verification / Alternative check: Applying Clausius–Clapeyron: λ ≈ R T^2 (d ln Psat/dT) for a pure fluid; this directly ties λ to the vapor-pressure curve slope—standard in thermodynamics texts. Why Other Options Are Wrong: (a), (b), (c) are established methods/correlations for latent heat; they are not incorrect choices. Common Pitfalls: Confusing chemical reaction enthalpy tools with phase-change properties; forgetting that latent heat refers to isothermal enthalpy change between phases at equilibrium. Final Answer: Hess’s law of constant heat summation

Question 12

Find the incorrect statement (mix of blending, stoichiometry, vapor–liquid, and crystallisation): Select the option that is WRONG.

Accepted Answer

Introduction / Context: This composite question checks your grasp across several core topics: blending calculations, limiting reactant and percent excess, vapor–liquid equilibrium saturation constraints for gas bubbles, and crystallisation mass balances. Spotting the incorrect statement requires quick, reliable reasoning across these areas. Given Data / Assumptions: Linear mixing of salt solutions by mass. Reaction: C + O2 → CO2, with amounts given in gram-atoms (for elements) and gram-moles (for molecules). Gas bubble in contact with a nonvolatile, nonabsorbing carrier gas; solvent vapor pressure Ps at the contact temperature. Crystalliser fed with a supersaturated solution at solute concentration C (mass or molar units), with equilibrium solubility C1 at operating conditions. Concept / Approach: For blending, apply a one-component mass balance on solute. For limiting reactant, compare stoichiometric requirements. For the bubble, saturation implies the solvent partial pressure equals Ps; thus the gas-phase mole fraction equals Ps/Pg. For crystallisation, solids formation equals the excess solute over solubility—specifically (C − C1) multiplied by the feed volumetric rate V (with consistent units). Step-by-Step Checks: (a) Let x = kg of 25% solution; 0.25x + 0.50(100 − x) = 40 ⇒ x = 40 kg → statement is correct.(b) Stoichiometry requires 1 mol O2 per mol C. Available: 1.2 mol C and 1.5 mol O2 → carbon limiting, oxygen excess = (1.5 − 1.2)/1.2 × 100% = 25% → correct.(c) Long contact and insoluble carrier lead to y_solvent = Ps/Pg (saturation) → correct.(d) Correct expression is solids rate = (C − C1)·V, not (C + C1)·V → statement is wrong. Verification / Alternative check: Material balance over the crystalliser: solute in = C·V; solute out in mother liquor = C1·V; solids out = (C − C1)·V. Why Other Options Are Wrong (or Right): Only (d) is incorrect; (a)–(c) align with standard calculations. Common Pitfalls: Confusing gram-atoms with gram-moles; misapplying Raoult-type saturation to cases with soluble carrier gases; sign error in the crystallisation excess term. Final Answer: Option (d) is wrong; the correct solids rate is (C − C1)·V.

Question 13

API gravity of pure water: At standard conditions, the °API gravity value for water is approximately…

Accepted Answer

Introduction / Context: API gravity is a petroleum industry measure of how heavy or light a liquid is relative to water. It is defined to indicate that most crudes (lighter than water) have higher API numbers, while heavier liquids have lower values. Knowing the reference point for water helps interpret the scale correctly. Given Data / Assumptions: API gravity definition: °API = (141.5 / SG_60°F) − 131.5. Specific gravity of water at 60°F is approximately 1.000. Standard conditions (near NTP/60°F) are assumed for the definition. Concept / Approach: Substitute the specific gravity of water into the API formula. The calculation yields the benchmark value for water on the API scale, against which lighter or heavier liquids can be compared. Liquids with API > 10 float on water; those with API < 10 sink in water. Step-by-Step Solution: Use the definition: °API = (141.5 / SG) − 131.5.For water: SG = 1.000.°API(water) = (141.5 / 1.000) − 131.5 = 10.0. Verification / Alternative check: If SG slightly deviates from unity at different temperatures, API will shift modestly; however, by definition at 60°F, water maps to 10 °API exactly in this formula, anchoring the scale. Why Other Options Are Wrong: 0 or 1 would imply an SG far from 1 in the formula. 100 would correspond to an unrealistically low SG by the API relation. Common Pitfalls: Confusing API gravity with Baumé degrees or with mass-density units; forgetting the temperature reference (60°F) in the API definition. Final Answer: 10

Question 14

Process streams management: Rejecting a portion of the outlet stream to control impurity build-up in a recycle system is called…

Accepted Answer

Introduction / Context: Many continuous chemical processes include recycle loops to improve conversion or recover unreacted materials. However, inert or slowly reacting impurities can accumulate in the loop. A standard remedy is to intentionally remove a small slipstream to maintain impurity levels within specification. Knowing the correct terminology helps in design reviews and PFD/P&ID communication. Given Data / Assumptions: Process contains a recycle loop with a nonreactive impurity. Objective is to prevent impurity build-up to unacceptable levels. Unit operations and controls are otherwise steady. Concept / Approach: Definitions: • Purge: A deliberate withdrawal of a portion of the recycle or outlet stream to remove accumulated inerts or impurities. • Recycle: Returning part of an output stream to an upstream unit to improve utilization or stability. • Bypass: Diverting a portion of a feed around a unit without contacting it, usually for control (e.g., temperature or composition). • Recirculation: Often used synonymously with recycle in fluid systems, referring to repeated circulation without necessarily implying impurity control. Step-by-Step Reasoning: Identify the problem: impurity accumulation in a recycle loop.Apply the remedy: remove a fraction of the loop continuously or intermittently.The standard term for this practice is “purging.” Verification / Alternative check: Material balance on the impurity at steady state shows that the purge fraction sets the outlet impurity concentration by balancing impurity inflow against removal. Why Other Options Are Wrong: Recycling/Recirculation: These describe returning streams, not removing impurities. Bypassing: Skips a unit operation; it does not reduce impurity inventory in the loop. Common Pitfalls: Confusing “bypass” and “purge”; assuming that increasing recycle alone can control impurities (it cannot without a purge path). Final Answer: Purging

Question 15

Solutions with positive excess volume: For a solution (excluding solids dissolving in liquids) where the total volume after mixing is greater than the sum of the pure-component volumes, how does the solubility change with increasing pressure?

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Introduction / Context: Many liquid–liquid and gas–liquid solutions exhibit a non-ideal volume change on mixing. When the total volume of the solution exceeds the sum of the pure-component volumes, the mixture displays a positive excess molar volume. Le Chatelier’s principle and fundamental thermodynamics allow us to predict how pressure affects solubility under such conditions. Given Data / Assumptions: Solution shows positive excess volume (total volume after mixing > sum of component volumes). Temperature is fixed; only the effect of pressure on solubility is examined. System is not a solid–liquid dissolution case (excluded in the stem). Near-equilibrium, reversible considerations apply. Concept / Approach: For a process that increases volume (positive ΔV_mix > 0), an increase in pressure shifts equilibrium to reduce the total volume. According to dG = V dP − S dT and the pressure derivative of chemical potential, higher pressure penalizes states with larger partial molar volumes. Therefore, pressure opposes the extent of a mixing process that creates additional volume, decreasing the solubility of the solute in the solvent when ΔV_mix is positive. Step-by-Step Solution: Identify sign of volume change: ΔV_mix > 0 (expansive mixing).Apply Le Chatelier’s principle: increasing pressure favors the side with lower volume.Mixing that increases volume will be suppressed by higher pressure.Hence, solubility (extent of mixing) decreases as pressure rises. Verification / Alternative check: In partial molar terms, μ_i(P) increases with P roughly by ∫ V̄_i dP. A positive excess volume implies larger V̄_i upon mixing; increasing P raises μ_i of the mixed state more than that of unmixed components, reducing the driving force for dissolution and thus lowering solubility. Why Other Options Are Wrong: Independent of temperature: Irrelevant to pressure dependence; the stem fixes temperature. Increased with the increase in pressure: Opposite to the expected shift for ΔV_mix > 0. Unchanged by pressure: Contradicts thermodynamic response to volume changes. None of these: Not correct because a definite trend exists. Common Pitfalls: Confusing this case with gas solubility following Henry’s law (often increases with pressure). Here the key is the sign of excess volume of mixing for the specific solution in question. Final Answer: decreased with the increase in pressure

Question 16

At constant volume with a fixed number of moles of gas, why does the pressure increase when temperature rises?

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Introduction / Context: For a closed, rigid container (constant volume) holding a fixed amount of an ideal or near-ideal gas, the relation P ∝ T emerges from the microscopic kinetic theory of gases. Understanding the molecular basis of this macroscopic behavior is crucial in thermodynamics and transport phenomena. Given Data / Assumptions: Volume is constant; number of moles is fixed. Temperature increases by external heating. Gas behaves ideally or near-ideally over the range considered. Concept / Approach: Gas pressure arises from molecular impacts on container walls. The ideal-gas kinetic theory links temperature to the average translational kinetic energy. As temperature increases, molecules move faster on average, striking the walls more forcefully and somewhat more frequently, increasing momentum transfer per unit time and therefore the pressure at fixed volume and moles. Step-by-Step Solution: Average kinetic energy per molecule ∝ T.Higher T ⇒ higher molecular speeds.Momentum transfer per collision and collision frequency rise.At fixed V and n, this raises the macroscopic pressure. Verification / Alternative check: From the ideal gas law, PV = nRT. At constant V and n, P ∝ T. Kinetic theory gives P = (2/3)(N/V)⟨E_trans⟩; since ⟨E_trans⟩ ∝ T, the proportionality is confirmed. Why Other Options Are Wrong: Decrease in mean free path: Not the fundamental reason; mean free path often increases with temperature at fixed pressure; here pressure itself changes. Increased collision rate: A consequence of higher speeds, but the primary cause is increased average kinetic energy. Increase in molecular attraction: Intermolecular attractions do not cause the pressure rise; if anything, attractions reduce pressure from ideality. Generation of additional moles: Mole number is fixed by the stem. Common Pitfalls: Mixing cause and effect: collision rate rises because speed rises; speed rise is the temperature effect that fundamentally drives pressure upward. Final Answer: increase in average molecular speed

Question 17

Adiabatic humidification of unsaturated air at constant pressure: which property remains essentially constant while water evaporates into the air?

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Introduction / Context: Adiabatic humidification and cooling are central to air–water psychrometrics, evaporative coolers, and packed-tower humidifiers. When unsaturated air contacts liquid water in an adiabatic device at essentially constant pressure, the air warms or cools and its humidity changes in a coupled way along lines closely tied to the wet-bulb temperature. Given Data / Assumptions: Unsaturated air enters; water evaporates into the air. Process is adiabatic (no heat exchange with surroundings) and at nearly constant pressure. Negligible heat capacity of liquid compared with gas stream, or well-mixed water at constant temperature representative of the wet-bulb state. Concept / Approach: The wet-bulb temperature (T_wb) is the temperature a water-wetted thermometer reaches due to evaporative cooling under adiabatic conditions. In adiabatic saturation of air with water at constant pressure, the air state moves along a path of nearly constant T_wb: the reduction in sensible heat balances the gain in latent heat. The dry-bulb temperature (T_db) decreases as evaporation proceeds, while absolute humidity increases until an approach to the adiabatic saturation state is reached. Step-by-Step Solution: Start from an initial unsaturated state (T_db, humidity y).Water evaporates; latent heat is supplied by air’s sensible cooling.Along the adiabatic constant-pressure path, T_db drops while humidity rises.Wet-bulb temperature remains approximately constant and characterizes the path. Verification / Alternative check: On a standard psychrometric chart, adiabatic humidification trajectories are nearly lines of constant wet-bulb temperature, confirming the invariance of T_wb, while T_db shifts downward and humidity ratio increases toward saturation at the adiabatic saturation temperature. Why Other Options Are Wrong: Dry-bulb temperature: changes markedly (cools) during evaporation. Humidity and relative saturation: both increase, not constant. Dew-point temperature: increases as moisture is added. Common Pitfalls: Confusing isothermal humidification (contact with a very large water reservoir at fixed temperature) with adiabatic humidification; in the latter, the key invariant is the wet-bulb temperature, not dry-bulb or dew point. Final Answer: wet-bulb temperature

Question 18

Aqueous acidity–basicity: what is the hydroxide-ion concentration [OH−] in a solution with pH = 3 (at 25 °C)?

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Introduction / Context: In water at 25 °C, the autoionization equilibrium defines the ion product Kw = [H+][OH−] = 1.0 × 10^-14. Using pH and pOH relationships enables rapid interconversion of hydrogen- and hydroxide-ion concentrations for acid–base calculations in environmental, biochemical, and process applications. Given Data / Assumptions: Temperature is 25 °C (so pH + pOH = 14 holds). pH = 3 (strongly acidic solution). Activity coefficients taken as unity (dilute solution approximation). Concept / Approach: Use the relations pH = −log10[H+] and pOH = −log10[OH−], with pH + pOH = 14 at 25 °C. From pH we first get [H+], then compute pOH and [OH−]. Step-by-Step Solution: Given pH = 3 ⇒ [H+] = 10^-3 M.pOH = 14 − pH = 14 − 3 = 11.Therefore, [OH−] = 10^-pOH = 10^-11 M.Check: [H+][OH−] = 10^-3 × 10^-11 = 10^-14 = Kw at 25 °C. Verification / Alternative check: Directly from Kw: [OH−] = Kw/[H+] = 1.0 × 10^-14 / 10^-3 = 10^-11 M, identical to the pOH method. Why Other Options Are Wrong: 10^-3 M is [H+], not [OH−]. 10^-10 M and 10^-13 M do not satisfy the Kw product with pH = 3. 10^-7 M corresponds to neutrality at 25 °C, not an acidic solution with pH 3. Common Pitfalls: Forgetting the temperature dependence of Kw; at temperatures other than 25 °C, pH + pOH ≠ 14 exactly, though the qualitative approach remains valid. Final Answer: 10^-11 M

Question 19

Acid–base equivalent concept: For the reaction Na2CO3 + HCl → NaHCO3 + NaCl, what is the equivalent weight of Na2CO3 in this reaction?

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Introduction / Context: Sodium carbonate is a diprotic base (carbonate ion CO3^2− can accept two protons). Its equivalent weight depends on the specific reaction step: full neutralization to carbonic acid or partial neutralization to bicarbonate. Understanding n-factor for the actual stoichiometry is key in titration and industry-scale neutralization calculations. Given Data / Assumptions: Reaction: Na2CO3 + HCl → NaHCO3 + NaCl. Only one mole of HCl reacts per mole of Na2CO3 (partial neutralization). Molar mass of Na2CO3 = 106 g/mol. Concept / Approach: Equivalent weight = molar mass / n, where n is the number of equivalents reacted per mole in the given reaction. Here, carbonate is converted to bicarbonate, consuming one mole of H+ per mole of Na2CO3 (n = 1). If the reaction proceeded to Na2CO3 + 2 HCl → H2CO3 + 2 NaCl, then n would be 2, but that is not the reaction specified in the stem. Step-by-Step Solution: Determine n: 1 mol H+ reacted per mol Na2CO3 ⇒ n = 1.Molar mass M(Na2CO3) = 106 g/mol.Equivalent weight = M / n = 106 / 1 = 106 g per equivalent.Therefore, the correct answer is 106. Verification / Alternative check: Charge balance: CO3^2− + H+ → HCO3^− consumes one equivalent of acid per mole of carbonate, confirming n = 1 for this stepwise neutralization. Why Other Options Are Wrong: Values like 53 or 10.6 correspond to dividing by 2 or by 10, not applicable here. 5.3 and 26.5 are off by orders of magnitude. Common Pitfalls: Using the full-neutralization n = 2 when the reaction explicitly shows only one proton transfer; always match n to the actual reaction stoichiometry. Final Answer: 106

Question 20

Boiling of immiscible liquid mixtures: The boiling point of a heterogeneous mixture of immiscible liquids is __________ the boiling point of any single component.

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Introduction / Context: Steam distillation and co-distillation exploit the behavior of immiscible liquid mixtures, which boil when the sum of their vapor pressures equals the external pressure. This allows boiling at a temperature lower than the normal boiling point of either component, enabling gentle recovery of thermally sensitive compounds. Given Data / Assumptions: Two immiscible liquids in equilibrium with their vapors. External (usually atmospheric) pressure fixed. Ideal or near-ideal additivity of vapor pressures for immiscible systems. Concept / Approach: For immiscible liquids A and B, the boiling condition at temperature T is p_A^sat(T) + p_B^sat(T) = P_ext. Since each pure-component saturated vapor pressure is less than P_ext at T lower than their individual boiling points, their sum can reach P_ext at a temperature below either pure boiling point. Therefore, the mixture boils at a lower temperature than any component alone. Step-by-Step Solution: Write boiling criterion: p_A^sat(T) + p_B^sat(T) = P_ext.At T lower than each pure boiling point, both p_A^sat and p_B^sat are positive.Their sum can equal P_ext earlier than either single p_i^sat does.Hence, the mixture’s boiling point is lower than either pure component’s boiling point. Verification / Alternative check: Practical steam distillation of essential oils demonstrates boiling of oil–water mixtures well below the normal boiling point of the oil, confirming the rule. Why Other Options Are Wrong: Higher than or equal to: contradicts vapor-pressure additivity for immiscible pairs. Either (a) or (b) and undetermined: do not apply; the direction is definite for immiscible systems. Common Pitfalls: Confusing immiscible with miscible mixtures (Raoult’s law). For miscible systems, behavior depends on activity coefficients; for immiscible systems, the simple pressure-sum rule applies. Final Answer: lower than

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