4.3 Gas

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4.3.1 Introduction

I There are three properties of the gaseous state that you should know:

1. the gaseous state much more compressible than solids or liquids

   1. there is more space between gas particles than solid or liquid particles.

2. gas molecules move very rapidly and therefore respond quickly to changes

3. the motion of gas molecules is chaotic and random

   1. the pressure of gases is homogenous (the force exerted by the gas on the walls is almost always constant)

4.3.2 Pressure

II pressure (P): defined as the force of the gas over the area the force is exerted on $P = \frac {F}{A}$ (Eq.31)

1. force can be simply thought of the “push” that gas particles exert on the surroundings

2. the atmospheric pressure (pressure exerted by air) was famously measured by Torricelli using a mercury barometer (Ref. Figure 46)

3. the atmospheric pressure can be determined using Eq. 32 for any barometer (d = density): $P = \textrm{d} \, \textrm{h} \, \textrm{g}$ (Eq. 32)

   1. using water (that is less dense than mercury) would result in a higher height compared to a mercury barometer at the same atmospheric pressure

III a manometer is used to compare the pressure of two different regions

1. An open manometer is used to compare the pressure of the sample with the atmosphere

2. closed manometer is used to compare the pressure of the sample with a vacuum

   1. think of a barometer that can measure pressures besides the atmospheric pressure

      

IV There are many different units of pressure:

1. pascal (Pa): The SI unit for pressure ($\textrm{kg} \, \textrm{m}^{-1} \, \textrm{s}^{-2}$)

2. standard pressure ($P^\varnothing$): 1 bar (= 105 Pa = 100 kPa)
3. atmosphere (atm): 1 atm = 1.013 25 $\times$ 105 Pa = 101.325 kPa
4. millimeters of mercury (mmHg): 760 mmHg = 1 atm
5. torr (torr): 760 torr = 760 mmHg = 1 atm

4.3.3 The Ideal Gas Law

V Boyle's Law: at constant temperature and amount of gas, $(\textrm{Volume}) \displaystyle \propto \frac{1}{(\textrm{Pressure})}$

$P \, V = k$ (Eq.33)

VI Charles' Law: at constant pressure and amount of gas, $(\textrm{Volume}) \displaystyle \propto (\textrm{Temperature})$

$\frac{V}{T} = b$ (Eq.34)

1. extrapolation is essential for graphs related to Charles' law; all gases liquify before absolute zero (0 K), so it cannot be reached by practice (Ref. Figure 48)

2. because volume cannot be negative, absolute zero is the lowest possible temperature

VII Gay-Lussac's Law: at constant volume and amount of gas, $(\textrm{Pressure}) \displaystyle \propto T$

$\frac{P}{T} = k$ (Eq.35)

VIII Avogadro's principle: at constant pressure and temperature, a given number of gas molecules will occupy the same volume regardless of its chemical properties

$V = n V_{m}$ (Eq.36)

1. the molar volume, or $V_{m}$, is the volume a mole of gas will occupy

2. real gases will not have the same molar volume due to intermolecular interactions

IX ideal gas: a hypothetical gas that obeys the ideal gas law under all conditions

1. an ideal gas does not have any intermolecular interactions
2. an ideal gas is moving point particles, and do not have any volume

X The combination of these empirical laws leads to the ideal gas law:

$P \, V \, = \, n \, R \, T$ (Eq.37)

1. the ideal gas law is a limiting law; the ideal gas law resembles a real gas as $P \rightarrow 0$.

2. at high pressures and lower temperatures, the ideal gas model fails as molecular size and intermolecular forces become important

3. for calculations, the gas law constant R is $\frac{0.08206 \, \textrm{L} \, \textrm{atm}}{\textrm{K} \, \textrm{mol}}$ (when pressure is in atm)

   1. do not use the 8.314 value! It is crucial that you always check your units.

4.3.4 Applications of the Ideal Gas Law

XI Standard Ambient Temperature and Pressure (SATP): 25 $^\circ C$ (298.15 K) and 1 bar pressure

1. the molar volume ($V_{m}$) of an ideal gas at SATP is $24.79 \, \textrm{L } \textrm{mol}^{-1}$.

XII Standard Temperature and Pressure (STP): 0 $^\circ C$ (273.15 K) and 1 atm pressure

1. the molar volume ($V_{m}$) of an ideal gas at STP is $22.41 \, \textrm{L} \, \textrm{mol}^{-1}$.

XIII The molar volume of an ideal gas can be calculated by:

1. $V_{m} = \frac{V} {n} = \frac{nRT / P} {n} = \frac{RT} {P}$ (Eq.38)

XIV The density of an ideal gas can be calculated by ($M_{r}$ is the molar mass of the gas)

1. $d = \frac{m}{V} = \frac{n M_{r}}{n V_{m}} = \frac{M_{r}}{V_{m}} = \frac{M_{r} P}{R T}$ (Eq.39)

XV The molar concentration of a gas can be calculated by:

1. $c = \frac{n}{V} = \frac{PV/RT}{V} = \frac{P}{RT} = \frac{1}{V_{m}}$ (Eq.40)

4.3.5 Mixture of Gases

XVI A mixture of gases that do not react with one another behaves like a single, pure gas

1. Dalton's Law of Partial pressures: the total pressure of a mixture of gases is the sum of the partial pressures of its components

   $P_{tot} = \sum_{i=1}^{n}{P_i}$ (Eq.41)

2. alternatively, this can be simplified by using this equation ($n_{tot}$ is the total moles of the mixture of the gas)

   $P_{tot} = \frac{n_{tot} RT}{V}$ (Eq.42)

XVII mole fraction (xi): the fraction a component contributes to the total amount of molecules in a mixture

$x_{A} = \frac{n_{A}}{n_{tot}}$ (Eq.43)

1. the sum of all mole fractions in a mixture must equal 1.

2. Dalton's law of partial pressures can be redefined using mole fractions; $P_ {A} + P_{B} = x_{A} P + x_{B} P = (x_{A} + x_{B}) P = P$

4.3.6 Diffusion and Effusion

XVIII Two qualitative observations were made that illustrated the relationship between the average speed of gas molecules, molar mass, and temperature

1. diffusion: the gradual dispersal of one substance through another substance

   1. a bottle of vinegar gradually dispersing its pungent smell when it is opened

2. effusion: the escape of a gas through a small hole into a vacuum or region of lower pressure

   1. air escaping from a bicycle tire valve

XIX Graham's Law of Effusion: At constant temperature,

$(\textrm{Rate \, of \, effusion}) = \displaystyle \propto = \sqrt{\frac{1}{M_{r}}} \frac{\textrm{Rate \, of \, effusion \, of \, A \, molecules}}{\textrm{Rate \, of \, effusion \, of \, B \, molecules}} = \sqrt{\frac{M_{B}}{M_{A}}} = \frac{t_B}{t_A}$ (Eq.44)

1. effusions at different temperatures reveal that $(\textrm{rate \, of \, effusion}) = \displaystyle \propto \sqrt{T}$

2. these two observations can be combined to reveal that $(\textrm{average \, speed}) = \displaystyle \propto \sqrt{\frac{T}{M_{r}}}$

4.3.7 The Kinetic Molecular Theory (KMT)

XX Kinetic Molecular Theory: a model that predicts behavior of gas molecules with consideration of the ideal gas model and effusion observations

1. The model is based on five rules:

   1. the volume of gas molecules are negligible

   2. A gas consists of a collection of molecules in ceaseless random motion

   3. Molecules move in straight lines (trajectories) until they collide

   4. the molecules do not influence one another except during collisions

   5. all collisions are elastic (momentum of gases is conserved)

2. the kinetic molecular theory can be used to derive the equation to calculate the root mean speed of gas molecules through temperature: $v-{rms} = \sqrt{v_{1}^{2} + v_{2}^{2} + v_{3}^{3} + ...}$

   $PV = \frac{1}{3} n M v_{rms}^{2}$ (Eq.45)

3. This equation can be rearranged in terms of temperature:

   $T= \frac{M_{r} v_{rms}^{2}}{3R}$ (Eq.46)

4. the equation can be substituted with the ideal gas law:

   $v_{rms} = \sqrt{\frac{3RT}{M_{r}}}$ (Eq.47)

5. rearrangement of Eq. 17 gives:

   $E_{k_avg} = \frac{3}{2} RT$ (Eq.48)

XXI The empirical gas laws can be explained conceptually using the kinetic molecular theorem:

1. Boyle's law: as volume $\downarrow$, # of molecules in a given region $\uparrow$, # of collisions $\uparrow$, pressure $\uparrow$

2. Charles' law: as temperature $\uparrow$, the average speed of a molecule $\uparrow$, pressure $\uparrow$, increase of pressure is counteracted by volume $\uparrow$

3. Avogadro's Principle: as # of molecules $\uparrow$, the volume increases in order for the pressure of the gas to remain constant

XXII like all models, the kinetic model theorem has fundamental flaws:

1. it does not properly account for the volume of the molecules

2. it does not account for the interaction between molecules (intermolecular forces)

3. relativistic and quantum-mechanical effects are ignored.

XXIII Despite this, the kinetic molecular theory works fairly well even for polar gas molecules as they rotate; their intermolecular forces are much weaker than solids or liquids (Ref. 4.2.3)

4.3.8 The Maxwell Distribution of Speeds

XXIV Maxwell-Boltzmann distribution: the distribution of the speeds of gas particles at a certain temperature

$f(v) = 4 \pi (\frac{M_{r}}{2 \pi RT})^{\frac{3}{2}} v^{2} e^{-\frac{M v^{2}}{2RT}}$ (Eq. 49)

1. derived from the kinetic molecular theory
2. lighter molecules have higher average speeds but also wider intervals

3. as temperature increases, the average molecular speed increases, and the interval becomes wider

4.3.9 Deviations from Ideality

XXV real gases: gases in the real world that deviate from the ideal gas model

1. the ideal gas law is a limiting law; it is only valid as $P \rightarrow 0.$

2. the effects of intermolecular forces on its deviations can be illustrated using the compression factor (z) where $z = \frac{V_{m}}{V_{m,ideal}}$ (Eq.50)

   1. the compression factor of an ideal gas is 1 as it has no intermolecular forces

   2. if z > 1, repulsive forces dominate and it tends to drive the molecules apart; $V_{m}$ increases

   3. if z < 1, attractive forces dominate and it tends to draw the molecules together; $V_{m}$ decreases

   

3. At high pressures, the separation between molecules are smaller and therefore repulsive forces dominate, leading to a higher compression factor. (Ref. Figure 50)

4. at low pressures, the separation between molecules are big and therefore attractive forces dominate. (Ref. Figure 50)

4.3.10 Real Gases and Liquefaction

XXVI virial equation:

$PV = nRT(1 + \frac{B}{V_{m}} + \frac{C}{V_{m}} + ...)$ (Eq.51)

1. the coefficients in the equation (B, C, ...) are known as virial coefficients

   1. all of these coefficients are determined experimentally and dependent on temperature

2. the virial equation predicts real gases very accurately, but it is extremely difficult to interpret the data without advanced analysis

XXVII van der Waals equation:

$(P + \frac{a n^{2}}{V^{2}})(V - nb) = nRT$ (Eq.52)

1. van der Waals parameters (a and b) are temperature independent and unique to each gas

   1. a coefficient represents attractions (considers intermolecular forces)

   2. b coefficient represents repulsions (considers size of molecules)

XXVIII molecules have low kinetic energy at low temperatures (Ref. Eq. 48); molecules stick together when they do not have enough kinetic energy to overcome IMFs.

1. Joule-Thomson effect: provided that attractive forces are dominant, a real gas cools as it expands.

   1. expansion forces gas molecules to overcome attractive forces, lowering their kinetic energy (and therefore temperature) and increasing their potential energy (collapse of intermolecular forces)

   2. hydrogen and helium have weak attractive interactions and get warmer as they expand.

2. the lowering of the average speed of molecules is equivalent to lowering its temperature (Ref. Eq. 48)