Properties of Gases

Chemical Principles: The Properties of Gases

• Gases are highly compressible.
• Gases occupy the full volume of their container.
• Gases mix completely with each other.
• Gases have much smaller densities than solids or liquids.
• Gas properties are largely independent of their chemical identity.

A Model for Gases

• Gases are composed of randomly moving particles (atoms or molecules).
• Particle size is negligible compared to the space between them.
• Particles do not interact with each other.

The Ideal Gas Law

Boyle's Law

• For a fixed amount of gas at constant temperature, pressure is inversely proportional to volume.
• P ∝ 1/V, where P × V = constant.

Charles's Law

• For a fixed amount of gas at constant pressure, volume is directly proportional to temperature.
• V ∝ T, where V/T = constant.

Avogadro's Hypothesis

• At a fixed temperature and pressure, the volume of a gas is directly proportional to the amount of gas.
• V ∝ n, where V/n = constant.

The Ideal Gas Law Formula

• PV = nRT
• P is pressure in pascals, V is volume in $m^3$, n is moles, R is the gas constant ($8.314 \mathrm{~J} \cdot \mathrm{mol}^{-1} \cdot \mathrm{K}^{-1}$), T is temperature in kelvin.

Using the Ideal Gas Law

• $\frac{P_{1} V_{1}}{n_{1} T_{1}}=\frac{P_{2} V_{2}}{n_{2} T_{2}}$
• $\frac{P_{1} V_{1}}{T_{1}}=\frac{P_{2} V_{2}}{T_{2}}$ for a fixed amount of gas.

Mixtures of Gases

• For a mixture of gases: $P_{\text {total }}=P_{1}+P_{2}+P_{3}+\ldots$
• $P_1$, $P_2$ etc., are the partial pressures of individual gases.
• Mole fraction: $X_{1}=\frac{n_{1}}{n_{\text {total }}}$
• Partial pressure: $P_{1}=X_{1} P_{\text {total }}$

The Kinetic Molecular Theory of Gases

• $P V=\frac{1}{3} N m \overline{u^{2}}$
• N is the number of molecules, m is the mass of each molecule, $\overline{u^{2}}$ is the mean square speed.

Average Translational Kinetic Energy

• $\overline{E_{k}}=\frac{3}{2} R T$
• $\overline{E_{k}}=\frac{1}{2} m \overline{u^{2}}$

Root Mean Square Speed

• $u_{r m s}=\sqrt{\overline{u^{2}}}$
• $u_{r m s}=\sqrt{\frac{3 R T}{M}}$
• M is the molar mass.

Diffusion and Effusion

Diffusion

• Gas molecules spread out in response to a concentration gradient.

Effusion

• Gas molecules escape through a tiny hole into a vacuum.

Graham's Law

• $\frac{\text { Rate of effusion of gas } A}{\text { Rate of effusion of gas } B}=\sqrt{\frac{M_{B}}{M_{A}}}$

Real Gases

Van der Waals Equation

• $(P+\frac{a n^{2}}{V^{2}})(V-n b)=n R T$
• $a$ and $b$ are empirical constants, dependent on the gas.

Radiative Heat Transfer: Radiation Properties

Emissivity (ϵ)

• Measure of how efficiently a surface emits radiation.
• Formula: ϵ = E/Eb
• Range: 0 ≤ ϵ ≤ 1

Absorptivity (α)

• Measure of how much incident radiation is absorbed by a surface.
• Formula: α = (Absorbed radiation) / (Incident radiation)
• Range: 0 ≤ α ≤ 1

Reflectivity (ρ)

• Measure of how much incident radiation is reflected by a surface.
• Formula: ρ = (Reflected radiation) / (Incident radiation)
• Range: 0 ≤ ρ ≤ 1

Transmissivity (τ)

• Measure of how much incident radiation is transmitted through a surface.
• Formula: τ = (Transmitted radiation) / (Incident radiation)
• Range: 0 ≤ τ ≤ 1

Key Relations

• α + ρ + τ = 1
• Opaque Surface: τ = 0, therefore α + ρ = 1
• Kirchhoff's Law: ϵ = α at thermal equilibrium

Blackbody Radiation

Planck's Law

• Formula: $E_b(\lambda, T) = \frac{C_1}{\lambda^5 (\exp(C_2/\lambda T) - 1)}$
  • $E_b(\lambda, T)$: Blackbody emissive power
  • λ: Wavelength
  • T: Absolute temperature
  • $C_1 = 2\pi hc_o^2 = 3.742 \times 10^8 \frac{W \cdot \mu m^4}{m^2}$
  • $C_2 = \frac{hc_o}{k} = 1.439 \times 10^4 \mu m \cdot K$
  • h: Planck's constant
  • $c_o$: Speed of light in a vacuum
  • k: Boltzmann's constant

Wien's Displacement Law

• $\lambda_{max} T = 2898 \mu m \cdot K$

Stefan-Boltzmann Law

• $E_b = \sigma T^4$
  • $E_b$: Total emissive power of a blackbody
  • σ: Stefan-Boltzmann constant, $\sigma = 5.67 \times 10^{-8} \frac{W}{m^2 \cdot K^4}$
  • T: Absolute temperature

Band Emission

• $E_{b, \lambda_1 - \lambda_2} = \int_{\lambda_1}^{\lambda_2} E_b(\lambda, T) d\lambda$
• $f_{\lambda_1 - \lambda_2} = \frac{E_{b, \lambda_1 - \lambda_2}}{E_b} = \frac{\int_{\lambda_1}^{\lambda_2} E_b(\lambda, T) d\lambda}{\sigma T^4} = f_{0-\lambda_2} - f_{0-\lambda_1}$
• $f_{0-\lambda} = \frac{\int_0^{\lambda} E_b(\lambda, T) d\lambda}{\sigma T^4}$ is obtained from the blackbody radiation function table.

Radiative Exchange

View Factor

• $F_{i \to j} = \frac{\text{Radiation leaving surface i that strikes surface j}}{\text{Total radiation leaving surface i}}$

Key Relations

• Summation Rule: $\sum_{j=1}^{N} F_{i \to j} = 1$
• Reciprocity Rule: $A_i F_{i \to j} = A_j F_{j \to i}$
• Superposition Rule: $F_{i \to (j+k)} = F_{i \to j} + F_{i \to k}$
• Symmetry Rule: If surfaces j and k are symmetric with respect to surface i, then $F_{i \to j} = F_{i \to k}$

Two-Surface Enclosure

• $q_{12} = \frac{\sigma (T_1^4 - T_2^4)}{\frac{1 - \epsilon_1}{A_1 \epsilon_1} + \frac{1}{A_1 F_{12}} + \frac{1 - \epsilon_2}{A_2 \epsilon_2}}$

Three-Surface Enclosure

• $q_{12} = \frac{\sigma (T_1^4 - T_2^4)}{\frac{1 - \epsilon_1}{A_1 \epsilon_1} + \frac{1}{A_1 F_{12}} + \frac{1 - \epsilon_2}{A_2 \epsilon_2}}$
• $q_{13} = \frac{\sigma (T_1^4 - T_3^4)}{\frac{1 - \epsilon_1}{A_1 \epsilon_1} + \frac{1}{A_1 F_{13}} + \frac{1 - \epsilon_3}{A_3 \epsilon_3}}$
• $q_{32} = -q_{23} = \frac{\sigma (T_3^4 - T_2^4)}{\frac{1 - \epsilon_3}{A_3 \epsilon_3} + \frac{1}{A_3 F_{32}} + \frac{1 - \epsilon_2}{A_2 \epsilon_2}}$
• $q_1 = q_{12} + q_{13}$
• $q_1 + q_2 + q_3 = 0$

Simplified Cases

Large Enclosure

• $q_{12} = A_1 \epsilon_1 \sigma (T_1^4 - T_2^4)$

Long Concentric Cylinders

• $\frac{q_1}{A_1} = \frac{\sigma (T_1^4 - T_2^4)}{\frac{1}{\epsilon_1} + \frac{1 - \epsilon_2}{\epsilon_2} (\frac{r_1}{r_2})}$

Concentric Spheres

• $\frac{q_1}{A_1} = \frac{\sigma (T_1^4 - T_2^4)}{\frac{1}{\epsilon_1} + \frac{1 - \epsilon_2}{\epsilon_2} (\frac{r_1}{r_2})^2}$

Radiation Shield

• $q_{12, \text{with shield}} = \frac{\sigma (T_1^4 - T_2^4)}{\frac{1}{\epsilon_1} + \frac{1}{\epsilon_2} + \frac{2}{\epsilon_s} - 2}$

Introduction to Differential Equations

Definition

• A differential equation is any equation containing derivatives of one or more dependent variables with respect to one or more independent variables.

Classifications of Differential Equations

Type

• Ordinary Differential Equation (ODE): Involves ordinary derivatives with respect to a single independent variable. Example: $\frac{dy}{dx} + 5y = e^x$
• Partial Differential Equation (PDE): Involves partial derivatives with respect to two or more independent variables. Example: $\frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} - 2\frac{\partial u}{\partial t}$

Order

• The order is determined by the highest derivative in the equation.
• Example: $\frac{d^2y}{dx^2} + 5(\frac{dy}{dx})^3 - 4y = e^x$ is a second-order ODE.

Linearity

• An n-th order ODE is linear if it can be written as: $a_n(x)\frac{d^ny}{dx^n} + a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}} + \cdots + a_1(x)\frac{dy}{dx} + a_0(x)y = g(x)$
• Non-linear differential equations are usually challenging to solve.
  • Example linear: $x^2y'' - xy' + y = \cos(x)$
  • Example nonlinear: $(1-y)y'' + xy' + 5y = \cos(x)$ (due to $(1-y)y''$)

Solution

• Explicit Solution: Dependent variable expressed solely in terms of the independent variable and constants. Example: $y = \phi(x)$

• Implicit Solution: A relation $G(x, y) = 0$ is an implicit solution if it defines at least one function $\phi$ that is an explicit solution. Example: $x^2 + y^2 = 25$ is an implicit solution to $\frac{dy}{dx} = -\frac{x}{y}$ on the interval $(-5, 5)$.

• Initial-Value Problem (IVP): Solving a differential equation subject to initial conditions, where the number of conditions aligns with the equation's order.

  • Solve: $\frac{dy}{dx} = g(x)$
  • Subject to: $y(x_0) = y_0$

Interval of Definition

• The interval $I$ on which the solution is defined.

Initial-Value Problems

Existence and Uniqueness

• For the first-order IVP: $\frac{dy}{dx} = f(x, y), \quad y(x_0) = y_0$
• Theorem: If $f(x, y)$ and $\frac{\partial f}{\partial y}$ are continuous on an open rectangle $R$ containing $(x_0, y_0)$, then there exists an interval $I$ centered at $x_0$ for which the IVP has a unique solution.
• Example:
  • $\frac{dy}{dx} = xy^{1/2}, \quad y(x_0) = y_0$
  • $f(x, y) = xy^{1/2}$
  • $\frac{\partial f}{\partial y} = \frac{x}{2y^{1/2}}$.
  • If $y_0 \neq 0$, a unique solution exists.
  • If $y_0 = 0$, multiple solutions may exist.
• General Notes:
  • You can't guarantee a solution to a nonlinear IVP, but you can try to find a region in which a unique solution might exist.
  • The IVP has to have a solution before you can reasonably consider whether it is unique.
  • The interval of existence does not have to be symmetric around $x_0$.
  • The hypotheses of the theorem are sufficient but not necessary.

Plan Estratégico de Tecnologías de la Información y Comunicación (PETIC)

What is PETIC?

• A plan that aligns ICT with the institutional strategy.

Why is PETIC Important?

• Strategic Alignment: Ensures that ICT drives institutional objectives.
• Resource Optimization: Enables efficient and focused investment in ICT.
• Improved Management: Facilitates decision-making and tracking of ICT initiatives.
• Risk Reduction: Identifies and mitigates risks associated with ICT.
• Transparency and Accountability: Promotes transparency in ICT management.

PETIC Development Process

1. Diagnosis:

   • Environmental analysis
   • SWOT analysis of ICT
   • Evaluation of the current situation of ICT

2. Strategy Definition:

   • Definition of the vision and mission of ICT
   • Definition of the strategic objectives of ICT
   • Definition of lines of action

3. Action Plan Development:

   • Identification of ICT initiatives
   • Definition of responsibilities and resources
   • Preparation of the schedule

4. Implementation and Monitoring:

   • Execution of ICT initiatives
   • Monitoring the progress of the plan
   • Evaluation of results

Key Elements of PETIC

• Vision: Desired future state for ICT in the institution.
• Mission: Fundamental purpose of ICT in the institution.
• Strategic Objectives: Results expected to be achieved with ICT.
• Lines of Action: Strategies to achieve objectives.
• ICT Initiatives: Specific projects and activities.
• Indicators: Metrics to measure progress and achievement of objectives.

Benefits of PETIC

• Improves the efficiency and effectiveness of institutional processes.
• Strengthens decision-making.
• Drives innovation and development.
• Increases user satisfaction.
• Contributes to the achievement of institutional objectives.
• PETIC is a fundamental tool for the strategic management of ICT.