Non-Ideal Gas Equation of State

• Lecture Date: Feb 28, 2023

Example Problem

• Considers a non-ideal gas equation of state: $p = \frac{NkT}{V} + B_2(T) (\frac{N}{V})^2$, where $B_2(T) < 0$ at low $T$ and $B_2(T) > 0$ at high $T$.

Questions

• What is the physical significance of $B_2(T)$?
• Plot isotherms of $p$ vs $V$ at different $T$.
• Find the critical Temperature ($T_c$), critical Pressure ($p_c$), and critical Volume ($V_c$).
• Evaluate the critical ratio $\frac{p_c V_c}{NkT_c}$.

Solutions

• $B_2(T)$ is the 2nd Virial coefficient, accounting for interactions between particle pairs.
  • $B_2(T) < 0$: Attraction dominates.
  • $B_2(T) > 0$: Repulsion dominates.
• Isotherms of $p$ vs $V$:
  • High $T$: $p$ decreases monotonically with $V$.
  • Low $T$: $p$ is non-monotonic, which is unphysical (system is unstable because $\frac{\partial p}{\partial V} > 0$).
• Solving for $T_c, p_c, V_c$: using $\frac{\partial p}{\partial V} = 0$ and $\frac{\partial^2 p}{\partial V^2} = 0$
  • First partial derivative: $\frac{\partial p}{\partial V} = -\frac{NkT}{V^2} + B_2(T) (-\frac{2N^2}{V^3}) = 0$, leading to $V = \frac{2NB_2(T)}{kT}$.
  • Second partial derivative: $\frac{\partial^2 p}{\partial V^2} = \frac{2NkT}{V^3} - B_2(T) (\frac{6N^2}{V^4}) = 0$, leading to $V = \frac{3NB_2(T)}{kT}$.
  • Setting the volume equations equal results in $B_2(T_c) = 0$.
• Estimating $T_c$ (using $B_2(T) = b(1 - e^{\epsilon/kT})$, with $b > 0, \epsilon > 0$):
  • $B_2(T_c) = 0$ indicates $T_c = \frac{\epsilon}{k}$.
  • At $T = T_c$, the gas behaves ideally: $p = \frac{NkT}{V}$. This model is not great.
• Model Considerations:
  • Testing the expression $B_2(T) = b - \frac{a}{kT}$
  • Deriving $\frac{\partial p}{\partial V} = -\frac{NkT}{V^2} + 2N^2 (b - \frac{a}{kT}) \frac{1}{V^3} = 0$ for $V = \frac{2N}{kT} (b - \frac{a}{kT})$
  • Deriving $\frac{\partial^2 p}{\partial V^2} = \frac{2NkT}{V^3} - 6N^2 (b - \frac{a}{kT}) \frac{1}{V^4} = 0$ for $V = \frac{3N}{kT} (b - \frac{a}{kT})$
  • Concluding $T_c = \frac{a}{bk}$
  • Deriving $V_c = 3Nb$

Analyse I: General Course Plan

Chapter 1. Numerical Sequences

• General Information
• Limited, Monotone, Convergent and Divergent Sequences
• Operations on Limits
• Indeterminate Forms
• Framing Theorem
• Extracted Sequences
• Bolzano-Weierstrass Theorem
• Cauchy Sequences

Chapter 2. Functions of a Real Variable

• General Information
• Limits and Continuity
• Theorem of Intermediate Values
• Derivability
• Rolle's Theorem
• Theorem of Finite Increases
• Monotone Functions and Derivatives
• L'Hôpital's Rule
• Higher Order Derivatives and Concavity
• Taylor's Formula

Chapter 3. Integrals

• Riemann integral
• Properties of the Integral
• Fundamental Theorem of Integral Calculation
• Integration by Parts
• Change Variable
• Generalized Integrals

Linear Algebra

Chapter 4. Vector Spaces

• Definitions and Examples
• Vector Subspaces
• Linear Combinations, Linear Envelope
• Free Families, Generating Families, Bases
• Dimension of a Vector Space

Chapter 5. Linear Applications

• Definitions and Examples
• Core and Image
• Range Theorem
• Matrix Representation
• Change of Base

Chapter 6. Systems of Linear Equations

• General Information
• Gauss Method
• Rouché-Frank Theorem
• Cramer Systems

Chapter 7. Determinations

• Definition and Properties
• Calculation of Determinants
• Applications of Determinants

Chapter 8. Diagonalization

• Own Values and Eigenvectors
• Characteristic Polynomial
• Cayley-Hamilton Theorem
• Diagonalizability
• Applications of Diagonalization

Statistics

Binomial Law

• Bernoulli trial definition: An experiment with two outcomes: Success (S) with probability $p$, and Failure (E) with probability $1-p = q$. The random variable $X$ takes the value 1 for success and 0 for failure, following a Bernoulli distribution $B(p)$.
• Bernoulli scheme of parameters $n$ and $p$: Repetition of $n$ independent Bernoulli trials.
• Binomial distribution: Random variable $X$ counts the number of successes in $n$ independent Bernoulli trials with parameter $p$, following a binomial distribution $B(n;p)$.
  • For $0 \leq k \leq n$: $P(X=k) = \begin{pmatrix} n \ k \end{pmatrix} p^k (1-p)^{n-k}$, where $\begin{pmatrix} n \ k \end{pmatrix} = \frac{n!}{k!(n-k)!}$
• Expectation: $E(X) = np$
• Variance: $V(X) = np(1-p)$

Fluctuation Interval

• Asymptotic fluctuation interval: For observed frequency $f$ in a sample size $n$, if $n \geq 30$, $np \geq 5$, and $n(1-p) \geq 5$, the interval $\displaystyle I = \left[ p - 1,96 \frac{\sqrt{p(1-p)}}{\sqrt{n}}~;~p + 1,96 \frac{\sqrt{p(1-p)}}{\sqrt{n}} \right]$ is the asymptotic fluctuation interval at the 95% threshold.
• Decision-making: If observed frequency $f$ is within the interval $I$, the hypothesis that the character proportion in the population is $p$ is accepted; otherwise, it is rejected.

Confidence Interval

• Confidence interval definition: For observed frequency $f$ in a sample size $n$, the confidence interval at level $1 - \alpha$ is $\displaystyle I = \left[ f - z_{\alpha/2} \frac{\sqrt{f(1-f)}}{\sqrt{n}}~;~f + z_{\alpha/2} \frac{\sqrt{f(1-f)}}{\sqrt{n}} \right]$, where $z_{\alpha/2}$ is the $1 - \alpha/2$ quantile of the standard normal distribution.
• Asymptotic confidence interval: If $n \geq 30$, $nf \geq 5$, and $n(1-f) \geq 5$, then $\displaystyle I = \left[ f - 1,96 \frac{\sqrt{f(1-f)}}{\sqrt{n}}~;~f + 1,96 \frac{\sqrt{f(1-f)}}{\sqrt{n}} \right]$ is the asymptotic confidence interval at the 95% confidence level.

Inference Rules

• Modus Ponens (MP): $(P \rightarrow Q), P \vdash Q$ (If P then Q; P; Therefore, Q)
• Modus Tollens (MT): $(P \rightarrow Q), \neg Q \vdash \neg P$ (If P then Q; Not Q; Therefore, not P)
• Hypothetical Syllogism (SH): $(P \rightarrow Q), (Q \rightarrow R) \vdash (P \rightarrow R)$ (If P then Q; If Q then R; Therefore, if P then R)
• Disjunctive Syllogism (SD): $P \lor Q, \neg P \vdash Q$ (P or Q; Not P; Therefore, Q)
• Simplification (Simp): $P \land Q \vdash P$ (P and Q; Therefore, P)
• Adjunction (Adj): $P, Q \vdash P \land Q$ (P; Q; Therefore P and Q)
• Addition (Ad): $P \vdash P \lor Q$ (P; Therefore, P or Q)
• Constructive Dilemma (DC): $(P \lor Q), (P \rightarrow R), (Q \rightarrow S) \vdash (R \lor S)$ (P or Q; If P then R; If Q then S; Therefore, R or S)

Demonstration Example

• Premises: $P \rightarrow Q$, $Q \rightarrow R$, $P$
• Proof:
  • Step 4: $Q$ (MP, 1, 3)
  • Step 5: $R$ (MP, 2, 4)

Conclusion

• $R$ (Derived conclusion)

Chemical Kinetics

• Chemical kinetics (reaction kinetics) is the study of chemical reaction rates, factors affecting these rates, and reaction mechanisms.

Reaction Rate

• For the reaction $aA + bB \rightarrow cC + dD$, the rate is expressed as: $Rate = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = \frac{1}{c}\frac{d[C]}{dt} = \frac{1}{d}\frac{d[D]}{dt}$
• Factors Affecting Reaction Rate:
  • Concentration of Reactants
  • Temperature
  • Physical State of Reactants
  • Presence of a Catalyst

Rate Laws

• Rate laws link reaction rate with reactant concentrations or pressures.
  • Differential Rate Law: rate dependence on concentration
  • Integrated Rate Law: concentration dependence on time
• Ways to determine rate laws:
  • Method of Initial Rates
  • Graphical Methods

Reaction Order

• Reaction Order: Describes how reactant concentrations affect rate.
• Common Reaction Orders:
  • Zero Order: Rate independent of concentration.
  • First Order: Rate directly proportional to concentration.
  • Second Order: Rate proportional to the square of concentration or product of two concentrations.

Temperature and Reaction Rate

• Arrhenius Equation: $k = Ae^{-\frac{E_a}{RT}}$
  • $k$ is the rate constant
  • $A$ is the pre-exponential factor
  • $E_a$ is the activation energy
  • $R$ is the gas constant
  • $T$ is the absolute temperature
• Activation Energy ($E_a$): Minimum energy for a reaction to occur.

Reaction Mechanisms

• Reaction Mechanism: Step-by-step sequence of elementary reactions.
  • Elementary Steps: Single-step reactions
  • Rate-Determining Step: Slowest step, controlling overall reaction rate.

Catalysis

• Types of Catalysis:
  • Homogeneous Catalysis: Catalyst in the same phase as reactants.
  • Heterogeneous Catalysis: Catalyst in a different phase.
  • Enzyme Catalysis: Biological catalyst

Chemistry

Chemical Equilibrium

• Reversible reactions, represented as $aA + bB \rightleftharpoons cC + dD$, reach dynamic equilibrium when the forward ($v_1$) and reverse ($v_2$) reaction rates are equal ($v_1 = v_2$), resulting in constant concentrations of reactants and products.
• The equilibrium constant, $K_c$, expresses the ratio of products to reactants at equilibrium:
  • $K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b}$
  • $K_c$ is constant at a given temperature and indicates the extent of a reaction.
    • When $K_c >> 1$, equilibrium favors products.
    • When $K_c << 1$, equilibrium favors reactants.
    • When $Q_c < K_c$, the reaction will shift towards the products.
    • $Q_c$ and $K_c$ can be compared to see if a process is in equilibrium
• In heterogeneous equilibria, solids and pure liquids are excluded from the $K_c$ expression due to their constant activity. Example: $CaCO_3(s) \rightleftharpoons CaO(s) + CO_2(g)$, thus $K_c = [CO_2]$.
• Le Chatelier's Principle states that a system at equilibrium under stress will shift to relieve that stress.
  • Increasing a reactant's concentration shifts the equilibrium to products.
  • Decreasing a product's concentration shifts the equilibrium to products.
• Pressure changes (for gases): Increased pressure shifts equilibrium to the side with fewer gas moles; decreased pressure shifts to the side with more gas moles.
• Temperature changes: In endothermic reactions ($\Delta H > 0$), higher temperatures favor products; in exothermic reactions ($\Delta H < 0$), higher temperatures favor reactants.
• Catalysts accelerate both forward and reverse reactions, reaching equilibrium faster but not altering the equilibrium position.
• Chemical equilibrium is crucial in industrial and biological processes, optimizing conditions for desired substance production.

Chapter 3. Vector functions

3.1 Parametric Curves

Definition 3.1.1

• A real variable vector function is an application $\overrightarrow{f} : I \subset \mathbb{R} \longrightarrow \mathbb{R}^n$ where $I$ is an interval of $\mathbb{R}$.
  • $\overrightarrow{f}(t) = (f_1(t), f_2(t),..., f_n(t))$ where the functions $f_i : I \longrightarrow \mathbb{R}$

Definition 3.1.2

• The set of points of $\mathbb{R}^n$ described by $\overrightarrow{f}(t)$ when $t$ varies in $I$ is called the parametric curve of parametric representation $\overrightarrow{f}(t)$, $t \in I$.
  • In $\mathbb{R}^2$, $\overrightarrow{f}(t) = x(t)\overrightarrow{i} + y(t)\overrightarrow{j} = (x(t), y(t))$.
  • In $\mathbb{R}^3$, $\overrightarrow{f}(t) = x(t)\overrightarrow{i} + y(t)\overrightarrow{j} + z(t)\overrightarrow{k} = (x(t), y(t), z(t))$.

Example 3.1.1

• $\overrightarrow{f}(t) = (t, t^2)$, $t \in \mathbb{R}$: A parametric representation of the parabola with equation $y = x^2$.

Example 3.1.2

• $\overrightarrow{f}(t) = (a\cos t, a\sin t)$, $t \in [0, 2\pi]$: A parametric representation of the circle with center $(0, 0)$ and radius $a > 0$.

Example 3.1.3

• $\overrightarrow{f}(t) = (a\cos t, b\sin t)$, $t \in [0, 2\pi]$: A parametric representation of the ellipse with equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $a, b > 0$.

Example 3.1.4

• $\overrightarrow{f}(t) = (t, t^2, t^3)$, $t \in \mathbb{R}$: A parametric representation of a curve in space $\mathbb{R}^3$.

Definition 3.1.3

• Let $\overrightarrow{f} : I \subset \mathbb{R} \longrightarrow \mathbb{R}^n$ be a vector function and let $t_0 \in I$. It is said that $\overrightarrow{f}$ admits a limit $\overrightarrow{L} \in \mathbb{R}^n$ in $t_0$ if $\forall \epsilon > 0, \exists \delta > 0 \text{ tel que } |t - t_0| < \delta \Longrightarrow ||\overrightarrow{f}(t) - \overrightarrow{L}|| < \epsilon$.
  • We write $\lim_{t \to t_0} \overrightarrow{f}(t) = \overrightarrow{L}$.

Theorem 3.1.1

• $\lim_{t \to t_0} \overrightarrow{f}(t) = \overrightarrow{L} = (L_1, L_2,..., L_n)$ if and only if $\lim_{t \to t_0} f_i(t) = L_i$ for all $i = 1,..., n$.

Definition 3.1.4

• Let $\overrightarrow{f} : I \subset \mathbb{R} \longrightarrow \mathbb{R}^n$ be a vector function and let $t_0 \in I$. It is said that $\overrightarrow{f}$ is continuous in $t_0$ if $\lim_{t \to t_0} \overrightarrow{f}(t) = \overrightarrow{f}(t_0)$.

Theorem 3.1.2

• $\overrightarrow{f}$ is continuous in $t_0$ if and only if $f_i$ is continuous in $t_0$ for all $i = 1,..., n$.

Definition 3.1.5

• Let $\overrightarrow{f} : I \subset \mathbb{R} \longrightarrow \mathbb{R}^n$ be a vector function and let $t_0 \in I$. It is said that $\overrightarrow{f}$ is differentiable in $t_0$ if the limit $\lim_{h \to 0} \frac{\overrightarrow{f}(t_0 + h) - \overrightarrow{f}(t_0)}{h}$ exists.
  • We then denote this limit $\overrightarrow{f}'(t_0)$ and we call it the derivative of $\overrightarrow{f}$ in $t_0$.
  • $\overrightarrow{f}'(t_0) = (f_1'(t_0), f_2'(t_0),..., f_n'(t_0))$.

Definition 3.1.6

• If $\overrightarrow{f} : I \subset \mathbb{R} \longrightarrow \mathbb{R}^n$ is differentiable at every point of $I$, we say that $\overrightarrow{f}$ is differentiable on $I$. The function $\overrightarrow{f}' : I \longrightarrow \mathbb{R}^n$ is called the derivative of $\overrightarrow{f}$.

Definition 3.1.7

• Let $\overrightarrow{f} : I \subset \mathbb{R} \longrightarrow \mathbb{R}^n$ be a vector function differentiable on $I$. If $\overrightarrow{f}'$ is continuous on $I$, we say that $\overrightarrow{f}$ is of class $C^1$ on $I$. If $\overrightarrow{f}'$ is differentiable on $I$, we say that $\overrightarrow{f}$ is twice differentiable on $I$ and we denote $\overrightarrow{f}''(t) = (\overrightarrow{f}'(t))'$ the second derivative of $\overrightarrow{f}$ in $t$.
  • $\overrightarrow{f}''(t) = (f_1''(t), f_2''(t),..., f_n''(t))$.

Example 3.1.5

• Let $\overrightarrow{f}(t) = (a\cos t, a\sin t)$, $t \in [0, 2\pi]$. Then $\overrightarrow{f}'(t) = (-a\sin t, a\cos t)$ and $\overrightarrow{f}''(t) = (-a\cos t, -a\sin t)$.

Theorem 3.1.3

• Let $\overrightarrow{f}, \overrightarrow{g} : I \subset \mathbb{R} \longrightarrow \mathbb{R}^n$ be two vector functions differentiable on $I$ and let $h : I \subset \mathbb{R} \longrightarrow \mathbb{R}$ be a real function differentiable on $I$. Then: $(\overrightarrow{f} + \overrightarrow{g})' = \overrightarrow{f}' + \overrightarrow{g}'$ $(h\overrightarrow{f})' = h'\overrightarrow{f} + h\overrightarrow{f}'$ $(\overrightarrow{f} \cdot \overrightarrow{g})' = \overrightarrow{f}' \cdot \overrightarrow{g} + \overrightarrow{f} \cdot \overrightarrow{g}'$ (if $n = 3$) $(\overrightarrow{f} \times \overrightarrow{g})' = \overrightarrow{f}' \times \overrightarrow{g} + \overrightarrow{f} \times \overrightarrow{g}'$ (if $n = 3$)