Sample Solution

MCH-013 Solved Assignment 2024

GENERAL PHYSICAL CHEMISTRY

1. Answer any five of the following in brief.

2. (a) (i) An isothermal and isobaric process is accompanied by changes in enthalpy and entropy as $52\mathrm{kJ}{\mathrm{mol}}^{-1}$$52\mathrm{kJ}{\mathrm{mol}}^{-1}$52kJmol^(-1)52 \mathrm{~kJ} \mathrm{~mol}^{-1}$52\mathrm{kJ}{\mathrm{mol}}^{-1}$ and $165{\mathrm{JK}}^{-1}{\mathrm{mol}}^{-1}$$165{\mathrm{JK}}^{-1}{\mathrm{mol}}^{-1}$165JK^(-1)mol^(-1)165 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}$165{\mathrm{JK}}^{-1}{\mathrm{mol}}^{-1}$, respectively. Predict whether the process be spontaneous at 400 K .
   (ii) If the enthalpy and entropy changes are not affected by the change in temperature calculate the temperature at which the system will attain equilibrium

3. (a) Explain the difference between permutation and configuration. Calculate the number of permutations and configurations possible while selecting three days out of seven days in a week.

4. (a) The X-ray power diffraction angles from the molybdenum crystal are observed at ${20.26}^{\circ }$${20.26}^{\circ }$20.26^(@)20.26^{\circ}${20.26}^{\circ }$, ${29.30}^{\circ },{36.82}^{\circ },{43.82}^{\circ },{50.70}^{\circ },{58.80}^{\circ },{66.30}^{\circ }$${29.30}^{\circ },{36.82}^{\circ },{43.82}^{\circ },{50.70}^{\circ },{58.80}^{\circ },{66.30}^{\circ }$29.30^(@),36.82^(@),43.82^(@),50.70^(@),58.80^(@),66.30^(@)29.30^{\circ}, 36.82^{\circ}, 43.82^{\circ}, 50.70^{\circ}, 58.80^{\circ}, 66.30^{\circ}${29.30}^{\circ },{36.82}^{\circ },{43.82}^{\circ },{50.70}^{\circ },{58.80}^{\circ },{66.30}^{\circ }$. Determine the type of cubic crystal formed by molybdenum?

5. (a) Describe the basic premise of transition state theory of reaction rates and derive an expression by using this theory for the rate constant for the elementary reaction

6. Answer any five of the following in brief.
   $2\times 5$$2\times 5$2xx52 \times 5$2\times 5$
   (a) Discuss the role of solvent in reactions in solution phase.
   (b) Define fast reactions and give different strategies used for studying fast reactions.
   (c) Define enzyme inhibition and state it’s different types.
   (d) Define ionic strength and calculate the same for an aqueous solution of $0.1\mathrm{M}{\mathrm{MgCl}}_2$$0.1\mathrm{M}{\mathrm{MgCl}}_2$0.1MMgCl_(2)0.1 \mathrm{M} \mathrm{MgCl}_2$0.1\mathrm{M}{\mathrm{MgCl}}_2$.
   (e) Differentiate between true and potential electrolytes.
   (f) Differentiate between the phenomenon of Osmosis and Dialysis.
   (g) State Fick’s second law of diffusion and give its significance.
7. (a) Describe the encounter pair description of reaction in solution and derive the expression for the rate of a reaction in solution in terms of this description.

8. (a) Define homogeneous catalytic reactions and derive the expression for the initial reaction rate of a homogeneous catalytic reaction.

10. (a) Define coefficient of viscosity and derive the relationship between the coefficient of viscosity and mean free path.

Expert Answer:

MCH-013 Solved Assignment 2024

GENERAL PHYSICAL CHEMISTRY

The entropy change is not a good criterion for spontaneity of a thermodynamic process. Comment.

Answer:

• Entropy change of a system ($\mathrm{\Delta }{S}_{system}$$\mathrm{\Delta }{S}_{system}$DeltaS_(system)\Delta S_{system}$\mathrm{\Delta }{S}_{system}$) alone is not sufficient to determine spontaneity.
• Spontaneity requires a positive total entropy change of the universe ($\mathrm{\Delta }{S}_{universe}$$\mathrm{\Delta }{S}_{universe}$DeltaS_(universe)\Delta S_{universe}$\mathrm{\Delta }{S}_{universe}$).
• The Gibbs free energy change ($\mathrm{\Delta }G$$\mathrm{\Delta }G$Delta G\Delta G$\mathrm{\Delta }G$) is a better criterion for spontaneity, particularly at constant temperature and pressure.

Derive the relation between Gibbs energy and Helmholtz energy.

Answer:

Definitions

1. Helmholtz Free Energy (A):
   The Helmholtz free energy is defined as the energy available to do work at constant volume (V) and temperature (T). It is given by:
   $$A=U-TS$$$$A=U-TS$$A=U-TSA = U – TS$$A=U-TS$$
   where:
   • $U$$U$UU$U$ is the internal energy,
   • $T$$T$TT$T$ is the absolute temperature,
   • $S$$S$SS$S$ is the entropy.
2. Gibbs Free Energy (G):
   The Gibbs free energy is defined as the energy available to do work at constant pressure (P) and temperature (T). It is given by:
   $$G=H-TS$$$$G=H-TS$$G=H-TSG = H – TS$$G=H-TS$$
   where:
   • $H$$H$HH$H$ is the enthalpy,
   • $T$$T$TT$T$ is the absolute temperature,
   • $S$$S$SS$S$ is the entropy.

Relation Between Enthalpy (H) and Internal Energy (U)

Derivation of the Relation Between Gibbs Energy and Helmholtz Energy

Final Relation

Interpretation

• Helmholtz free energy ($A$$A$AA$A$) is used for processes at constant volume and temperature.
• Gibbs free energy ($G$$G$GG$G$) is used for processes at constant pressure and temperature.
• The relation $G=A+PV$$G=A+PV$G=A+PVG = A + PV$G=A+PV$ connects Gibbs and Helmholtz free energies, incorporating pressure and volume.

Differentiate between molar and partial molar properties.

Answer:

Molar Properties

1. Definition:
   Molar properties are extensive properties of a substance per mole. They describe the total property of a homogeneous substance when considered on a per-mole basis. Examples include molar volume ($V_m$$V_m$V_(m)V_m$V_m$), molar enthalpy ($H_m$$H_m$H_(m)H_m$H_m$), molar entropy ($S_m$$S_m$S_(m)S_m$S_m$), and molar Gibbs free energy ($G_m$$G_m$G_(m)G_m$G_m$).
2. Calculation:
   Molar properties are calculated by dividing the total property of the substance by the number of moles of the substance. For example, if $V$$V$VV$V$ is the volume of a substance and $n$$n$nn$n$ is the number of moles, then the molar volume $V_m$$V_m$V_(m)V_m$V_m$ is given by:
   $$V_m=\frac{V}{n}$$$$V_m=\frac{V}{n}$$V_(m)=(V)/(n)V_m = \frac{V}{n}$$V_m=\frac{V}{n}$$
3. Dependence on Composition:
   In a pure substance, molar properties are independent of the composition since there is only one component. For mixtures, molar properties are averaged based on the composition of the components, but they do not account for the interaction between different components.
4. Usage:
   Molar properties are typically used when dealing with pure substances or in scenarios where it’s practical to consider the properties on a per-mole basis, such as calculating the energy required for a chemical reaction involving known quantities of reactants.

Partial Molar Properties

1. Definition:
   Partial molar properties describe the change in an extensive property of a mixture when an infinitesimal amount of a component is added, keeping temperature, pressure, and the amounts of all other components constant. These properties indicate how the presence of each component affects the overall property of the mixture. Examples include partial molar volume (${\overline{V}}_i$${\overline{V}}_i$bar(V)_(i)\bar{V}_i${\overline{V}}_i$), partial molar enthalpy (${\overline{H}}_i$${\overline{H}}_i$bar(H)_(i)\bar{H}_i${\overline{H}}_i$), partial molar entropy (${\overline{S}}_i$${\overline{S}}_i$bar(S)_(i)\bar{S}_i${\overline{S}}_i$), and partial molar Gibbs free energy (${\overline{G}}_i$${\overline{G}}_i$bar(G)_(i)\bar{G}_i${\overline{G}}_i$).
2. Calculation:
   Partial molar properties are determined by differentiating the total property of the mixture with respect to the number of moles of a component, keeping other variables constant. For instance, the partial molar volume ${\overline{V}}_i$${\overline{V}}_i$bar(V)_(i)\bar{V}_i${\overline{V}}_i$ of a component $i$$i$ii$i$ in a mixture is given by:
   $${\overline{V}}_i={\left(\frac{\partial V}{\partial n_i}\right)}_{T,P,n_{j\ne i}}$$$${\overline{V}}_i={\left(\frac{\partial V}{\partial n_i}\right)}_{T,P,n_{j\ne i}}$$bar(V)_(i)=((del V)/(deln_(i)))_(T,P,n_(j!=i))\bar{V}_i = \left(\frac{\partial V}{\partial n_i}\right)_{T,P,n_{j \neq i}}$${\overline{V}}_i={\left(\frac{\partial V}{\partial n_i}\right)}_{T,P,n_{j\ne i}}$$
   where $V$$V$VV$V$ is the total volume of the mixture, $n_i$$n_i$n_(i)n_i$n_i$ is the number of moles of component $i$$i$ii$i$, and $n_{j\ne i}$$n_{j\ne i}$n_(j!=i)n_{j \neq i}$n_{j\ne i}$ represents the moles of all other components held constant.
3. Dependence on Composition:
   Partial molar properties depend on the composition of the mixture and how components interact with each other. They provide insight into how the properties of a mixture change with the addition or removal of a particular component.
4. Usage:
   Partial molar properties are crucial when dealing with mixtures and solutions because they account for interactions between different components. They are used to calculate the changes in thermodynamic properties during mixing, separation, or chemical reactions involving mixtures.

Key Differences

• Nature: Molar properties pertain to the per-mole basis of a pure substance or averaged property in a mixture, while partial molar properties represent the contribution of each component to a mixture’s total property.
• Dependence: Molar properties are straightforward for pure substances but averaged for mixtures, whereas partial molar properties vary with the composition and account for intermolecular interactions.
• Application: Molar properties are often used in calculations involving pure substances or stoichiometric mixtures, while partial molar properties are essential for analyzing solutions and mixtures, particularly when changes in composition occur.

• Molar properties represent total properties on a per-mole basis.
• Partial molar properties describe how a component’s addition affects the mixture’s properties.
• Partial molar properties are crucial for understanding mixture behavior and interactions.

Give Stirling’s approximation and outline its significance.

Answer:

Stirling’s Approximation

Significance of Stirling’s Approximation

1. Simplifying Calculations for Large $n$$n$nn$n$:
   Stirling’s approximation simplifies the computation of factorials for large numbers. Calculating the exact value of $n!$$n!$n!n!$n!$ for large $n$$n$nn$n$ can be computationally intensive and prone to errors due to overflow in digital computations. The approximation provides a quick and relatively accurate method to estimate the factorial without performing the full multiplication.
2. Applications in Statistical Mechanics:
   In statistical mechanics, Stirling’s approximation is frequently used in the derivation of the Boltzmann distribution, which describes the distribution of particles over various energy states in a system. The approximation simplifies the mathematics involved in finding the most probable distribution of particles.
3. Entropy and Thermodynamics:
   Stirling’s approximation is used in calculating entropy, particularly in deriving the entropy of an ideal gas. The entropy expression involves logarithms of factorials, which are simplified using Stirling’s approximation, making the calculations more manageable.
4. Probability and Combinatorics:
   In probability theory and combinatorics, Stirling’s approximation is often used to simplify expressions involving binomial coefficients, which include factorials. This is particularly useful in the analysis of large sample spaces, making it easier to derive probabilities and expected values.
5. Approximation in Asymptotic Analysis:
   In asymptotic analysis, where functions are approximated for large values of their variables, Stirling’s approximation helps to understand the behavior of factorial-related expressions as $n$$n$nn$n$ grows. It provides a way to approximate sums and products that include factorial terms, which is useful in various branches of mathematics.
6. Normal Approximation to the Poisson Distribution:
   Stirling’s approximation helps in deriving the normal approximation to the Poisson distribution, which is used when the mean of the Poisson distribution is large. This allows for a simpler and more tractable form of the probability distribution, useful in statistical modeling and inference.

Summary

• Stirling’s approximation is $n!\approx \sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n$$n!\approx \sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n$n!~~sqrt(2pi n)((n)/(e))^(n)n! \approx \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n$n!\approx \sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n$.
• It simplifies calculations involving large factorials.
• It is widely used in statistical mechanics, thermodynamics, probability, and combinatorics.

How can the diamagnetic and paramagnetic substances be distinguished using magnetic susceptibilities?

Answer:

Magnetic Susceptibility

Distinguishing Diamagnetic and Paramagnetic Substances

1. Diamagnetic Substances:
   • Definition: Diamagnetic substances are materials that create an induced magnetic field in a direction opposite to an applied magnetic field. This is due to the orbital motion of electrons generating small magnetic moments that oppose the external field.
   • Magnetic Susceptibility ($\chi$$\chi$chi\chi$\chi$): Diamagnetic materials have a negative magnetic susceptibility ($\chi <0$$\chi <0$chi < 0\chi < 0$\chi <0$). This means that when placed in a magnetic field, these materials exhibit a very weak and negative magnetization.
   • Behavior in Magnetic Fields: Diamagnetic materials are weakly repelled by magnetic fields. The negative susceptibility indicates that the induced magnetization opposes the applied magnetic field.
2. Paramagnetic Substances:
   • Definition: Paramagnetic substances are materials with unpaired electrons that have magnetic moments. These unpaired electrons cause the material to have a net magnetic moment, which aligns with an applied magnetic field.
   • Magnetic Susceptibility ($\chi$$\chi$chi\chi$\chi$): Paramagnetic materials have a positive magnetic susceptibility ($\chi >0$$\chi >0$chi > 0\chi > 0$\chi >0$). This indicates that when placed in a magnetic field, these materials become magnetized in the same direction as the applied field.
   • Behavior in Magnetic Fields: Paramagnetic materials are weakly attracted by magnetic fields. The positive susceptibility signifies that the induced magnetization enhances the external magnetic field.

Key Differences in Magnetic Susceptibility

• Sign of Susceptibility:
  • Diamagnetic Substances: $\chi <0$$\chi <0$chi < 0\chi < 0$\chi <0$ (Negative susceptibility)
  • Paramagnetic Substances: $\chi >0$$\chi >0$chi > 0\chi > 0$\chi >0$ (Positive susceptibility)
• Magnitude of Susceptibility:
  • Diamagnetic Substances: The magnitude of $\chi$$\chi$chi\chi$\chi$ is typically very small (close to zero but negative).
  • Paramagnetic Substances: The magnitude of $\chi$$\chi$chi\chi$\chi$ is also small but positive. It is usually much smaller than that of ferromagnetic materials but larger than diamagnetic susceptibility.

Practical Applications

• Magnetic Separation: In materials science and geology, magnetic susceptibility measurements can help differentiate minerals and materials based on their magnetic properties.
• Medical Diagnostics: In MRI (Magnetic Resonance Imaging), paramagnetic contrast agents are used because they enhance the magnetic signal due to their positive susceptibility, improving image contrast.

Summary

• Diamagnetic substances have a negative magnetic susceptibility ($\chi <0$$\chi <0$chi < 0\chi < 0$\chi <0$), indicating they are weakly repelled by magnetic fields.
• Paramagnetic substances have a positive magnetic susceptibility ($\chi >0$$\chi >0$chi > 0\chi > 0$\chi >0$), indicating they are weakly attracted to magnetic fields.

• Diamagnetic substances have negative magnetic susceptibility ($\chi <0$$\chi <0$chi < 0\chi < 0$\chi <0$).
• Paramagnetic substances have positive magnetic susceptibility ($\chi >0$$\chi >0$chi > 0\chi > 0$\chi >0$).
• Magnetic susceptibility indicates how a material responds to an external magnetic field.

Outline the limitations of collision theory.

Answer:

Limitations of Collision Theory

1. Assumes Ideal Behavior of Gases:
   • Collision theory is based on the kinetic theory of gases, which assumes that gases behave ideally. In real conditions, gases may deviate from ideal behavior due to intermolecular forces and finite molecular volumes, especially at high pressures and low temperatures. These deviations can affect the accuracy of predictions made by collision theory.
2. Overlooks Complex Molecular Interactions:
   • The theory assumes that only collisions with sufficient energy and proper orientation lead to a reaction. It does not account for complex interactions between molecules, such as quantum mechanical effects or the formation of intermediate complexes. These factors can significantly influence reaction rates and mechanisms but are not considered in collision theory.
3. Ignores Activation Energy Variability:
   • Collision theory presumes that there is a fixed activation energy ($E_a$$E_a$E_(a)E_a$E_a$) for a given reaction. However, in reality, the activation energy can vary depending on the distribution of energy among reactant molecules and other conditions such as pressure, concentration, and the presence of catalysts. The theory does not adequately explain how these factors influence the distribution of molecular energies and, consequently, the reaction rate.
4. Does Not Consider Reaction Dynamics:
   • The theory simplifies chemical reactions to be a result of simple elastic collisions. It does not account for the dynamic nature of molecular interactions, such as changes in energy distribution over time, vibration modes, and the potential energy surface of reacting molecules. These dynamic factors can significantly influence whether a collision leads to a successful reaction.
5. Limited Applicability to Complex Reactions:
   • Collision theory is most accurate for simple reactions involving small, gas-phase molecules. It is less effective for complex reactions involving larger molecules, condensed phases, or reactions in solution, where molecular interactions are more intricate and cannot be described by simple collision models.
6. Fails to Explain the Role of Catalysts:
   • While collision theory acknowledges that catalysts can lower the activation energy and increase reaction rates, it does not explain how catalysts achieve this at a molecular level. Catalysis often involves complex mechanisms that are beyond the scope of collision theory, including the formation of intermediate species and changes in the reaction pathway.
7. Neglects Temperature and Energy Distribution Impacts:
   • Although the theory accounts for temperature’s effect on reaction rates through the kinetic energy of molecules, it oversimplifies the impact of energy distribution. Real systems have a range of energy states, and not all collisions with sufficient energy result in a reaction due to improper orientation or other factors not considered by the theory.
8. Does Not Address Stereochemical Effects:
   • The theory does not consider the importance of molecular geometry and stereochemistry in determining reaction rates. The spatial arrangement of atoms within molecules can significantly affect the probability of a successful collision, which is not adequately addressed in the simple model of collision theory.