### Given Data:
– Total examinees = 1350
– Passed in Accounts (\(A\)) = 700
– Passed in Economics (\(E\)) = 600
– Passed in Statistics (\(S\)) = 350
– Failed in all subjects = 50
– Passed in Economics and Statistics (\(E \cap S\)) = 200
– Passed in Accounts and Statistics (\(A \cap S\)) = 150
– Passed in Statistics only = 50

### Calculations:

1. **Total Passed in at least one subject**:
\[
\text{Total passed} = 1350 – 50 = 1300
\]

2. **Using Venn Diagram and Principle of Inclusion-Exclusion**:

\[
|A \cup E \cup S| = |A| + |E| + |S| – |A \cap E| – |E \cap S| – |A \cap S| + |A \cap E \cap S|
\]

Let’s denote \( x = |A \cap E \cap S| \).

3. **Substitute given values**:

\[
1300 = 700 + 600 + 350 – |A \cap E| – 200 – 150 + x
\]

Simplify the equation:

\[
1300 = 1650 – |A \cap E| – 350 + x
\]

\[
1300 = 1300 – |A \cap E| + x
\]

\[
0 = – |A \cap E| + x
\]

\[
x = |A \cap E|
\]

To find \( |A \cap E| \):

4. **Determine students passing only Economics**:

From the image:
\[
\text{Students passing Economics only} = 600 – 200 – x = 450
\]

5. **Determine students passing only Accounts**:

From the image:
\[
\text{Students passing Accounts only} = 700 – 150 – x = 550
\]

### Correct Results:

1. **Passed in at least one subject**:
\[
1300
\]

2. **Passed in all three subjects**:
\[
50
\]

3. **Passed in Economics only**:
\[
450
\]

4. **Failed in Accounts**:
\[
1350 – 700 = 650
\]

Analytical estimating is a method used in construction and project management to estimate the cost, time, and resources required for a project based on historical data, mathematical formulas, and statistical analysis. Here are key points about analytical estimating:

• Historical Data: Analytical estimating relies on historical data from previous projects to develop mathematical models and formulas for estimating. This data includes costs, durations, and resource usage for various project activities.

• Mathematical Formulas: Analytical estimating uses mathematical formulas to calculate estimates based on project parameters such as project size, complexity, and location. These formulas are derived from historical data and are often refined over time to improve accuracy.

• Statistical Analysis: Analytical estimating involves statistical analysis to identify trends, patterns, and uncertainties in project data. This helps in developing probabilistic estimates that account for risks and uncertainties in the project.

• Types of Estimates: Analytical estimating can be used to develop different types of estimates, including preliminary estimates, budget estimates, and detailed estimates. The level of detail and accuracy of the estimate depend on the stage of the project and the availability of data.

• Benefits: Analytical estimating offers several benefits, including increased accuracy compared to other estimation methods, the ability to quickly generate estimates for complex projects, and the ability to identify and mitigate risks early in the project lifecycle.

• Challenges: Despite its benefits, analytical estimating has some challenges, such as the need for accurate historical data, the complexity of developing and maintaining mathematical models, and the difficulty of accounting for all project variables and uncertainties.

Overall, analytical estimating is a valuable tool in project management, providing a systematic and data-driven approach to estimating project costs, time, and resources.

Definition of Molecular Partition Functions

Molecular partition function is a fundamental concept in statistical mechanics that quantifies the number of available quantum states for a molecule at a given temperature. It plays a crucial role in linking the microscopic quantum states of a system to its macroscopic thermodynamic properties. The partition function is a sum over all possible energy states of a system, weighted by the Boltzmann factor, \( e^{-\beta E} \), where \( \beta = \frac{1}{k_BT} \) (with \( k_B \) being the Boltzmann constant and \( T \) the temperature), and \( E \) represents the energy levels of the system.

Derivation of the Translational Partition Function for Motion Along the X-direction

1. Setting the Framework

The translational partition function quantifies the number of ways a particle can be distributed in space, considering its kinetic energy due to motion. For a single particle in a one-dimensional box (along the x-axis), the energy states can be described by the quantum mechanics of a particle in a box:

\[
E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}
\]

where:

• \( n \) is the quantum number,
• \( \hbar \) is the reduced Planck’s constant,
• \( m \) is the mass of the particle,
• \( L \) is the length of the box.

2. Expression for the Partition Function

The translational partition function for a particle confined to move along the x-axis in a box of length \( L \) is given by:

\[
q_x = \sum_{n=1}^{\infty} e^{-\beta E_n}
\]

Substituting the expression for \( E_n \):

\[
q_x = \sum_{n=1}^{\infty} e^{-\beta \frac{n^2 \pi^2 \hbar^2}{2mL^2}}
\]

3. Approximating the Summation

For high temperatures or large boxes, the energy levels are closely spaced, allowing the summation to be approximated by an integral:

\[
q_x \approx \int_{0}^{\infty} e^{-\beta \frac{\pi^2 \hbar^2 x^2}{2mL^2}} dx
\]

To solve the integral, we perform a change of variables \( u = \frac{\pi \hbar x}{\sqrt{2mL^2 \beta}} \), which simplifies the integral:

\[
q_x \approx \frac{\sqrt{2mL^2 \beta}}{\pi \hbar} \int_{0}^{\infty} e^{-u^2} du
\]

The integral of \( e^{-u^2} \) from 0 to \( \infty \) is \( \frac{\sqrt{\pi}}{2} \), thus:

\[
q_x \approx \frac{\sqrt{2mL^2 \beta}}{\pi \hbar} \cdot \frac{\sqrt{\pi}}{2} = \frac{\sqrt{2\pi mk_BT}}{h}L
\]

This result shows that the translational partition function for motion along the x-direction is proportional to the length of the box and depends on the mass of the particle, the temperature, and the Boltzmann constant.

Conclusion

The translational partition function for motion in one dimension provides insight into how quantum mechanical properties of particles contribute to macroscopic thermodynamic quantities. It illustrates the dependency of statistical properties on physical dimensions and conditions of the system, such as temperature and size. This concept is extensible to three dimensions and forms the foundation for understanding molecular behavior in gases and other phases.

Definition of Chemical Potential

Chemical potential is a thermodynamic quantity that represents the potential energy change of a system when an infinitesimal amount of substance is added to the system, keeping temperature and pressure constant. It can be thought of as the energy change per unit addition or removal of a particle from a system. Essentially, it indicates how the Gibbs free energy of the system changes as the amount of substance changes, making it a crucial factor in understanding chemical reactions and phase changes.

Effects of Temperature on Chemical Potential

1. Fundamental Relationship

The chemical potential (\(\mu\)) of a substance is not only a function of its concentration but also of temperature and pressure. The relationship between chemical potential and temperature at constant pressure can be derived from the fundamental thermodynamic equations:

\[
\left(\frac{\partial \mu}{\partial T}\right)_P = -S
\]

where \(S\) is the molar entropy of the substance. This equation implies that the chemical potential of a substance decreases with an increase in temperature at constant pressure, provided the entropy (\(S\)) is positive, which is generally the case.

2. Practical Implications

• Influence on Reaction Direction: As temperature increases, the chemical potentials of the reactants and products change, influencing the direction of the chemical reaction. According to Le Chatelier’s principle, if a reaction produces heat (exothermic), increasing the temperature will increase the chemical potentials of the products more than the reactants, potentially driving the reaction in the reverse direction.
• Impact on Phase Transitions: The chemical potentials of different phases (solid, liquid, gas) of a substance also depend on temperature. For example, the melting of ice into water as temperature increases is a result of the chemical potential of water becoming more favorable (lower) than that of ice at temperatures above 0°C.

3. Dependence on Entropy

The change in chemical potential with temperature is directly tied to the entropy of the substance. A higher entropy value indicates a greater decrease in chemical potential with temperature. This relationship underlines the intrinsic connection between the disorder within a system and its energetic favorability under varying thermal conditions.

Conclusion

Understanding the chemical potential and its temperature dependence is essential in the fields of chemical thermodynamics and reaction engineering. It provides insights into how substances behave under different thermal conditions, influencing everything from industrial synthesis to environmental processes. The decrease in chemical potential with temperature, due to positive entropy, plays a critical role in determining the direction of chemical reactions and the stability of various phases in a substance.

To determine if the process is spontaneous at a given temperature, we can use the Gibbs free energy change (\(\Delta G\)), which is related to the changes in enthalpy (\(\Delta H\)) and entropy (\(\Delta S\)) through the equation:

\[
\Delta G = \Delta H – T\Delta S
\]

Where:
– \(\Delta H\) is the change in enthalpy,
– \(T\) is the temperature in Kelvin,
– \(\Delta S\) is the change in entropy.

Given:
– \(\Delta H = 52 \text{ kJ mol}^{-1} = 52000 \text{ J mol}^{-1}\) (since 1 kJ = 1000 J)
– \(\Delta S = 165 \text{ JK}^{-1} \text{ mol}^{-1}\)
– \(T = 400 \text{ K}\)

Plugging these values into the equation for \(\Delta G\):

\[
\Delta G = 52000 \text{ J mol}^{-1} – 400 \text{ K} \times 165 \text{ JK}^{-1} \text{ mol}^{-1}
\]

We can calculate \(\Delta G\) as follows:

\[
\Delta G = 52000 \text{ J mol}^{-1} – 66000 \text{ J mol}^{-1}
\]

\[
\Delta G = -14000 \text{ J mol}^{-1}
\]

Since \(\Delta G\) is negative, the process is spontaneous at 400 K.