All gases have the following physical characteristics:
Elements that exist as gases at $25\,^\circ\text{C}$ and $1\,\text{atmosphere}$ are: noble gases ($\ce{He}$, $\ce{Ne}$, $\ce{Ar}$, $\ce{Kr}$, $\ce{Xe}$, $\ce{Rn}$), two halogens ($\ce{F2}$, $\ce{Cl2}$), oxygen ($\ce{O2}$), nitrogen ($\ce{N2}$) and hydrogen ($\ce{H2}$)
The unit for force is $\text{N}$ which is defined as $\text{kg}\cdot\text{m/s}^2$.
The unit for pressure is $\text{Pa}$ which is defined as $\text{N/m}^2$.
Other commonly used units for pressure are
Requirements¶
- Understand and remember the general properties of gases.
- Remember elements exist as gases at $25\,^\circ\text{C}$ and $1\,\text{atmosphere}$.
- Be familiar with the substances exist as gases at $25\,^\circ\text{C}$ and $1\,\text{atmosphere}$.
- Understand the defination of pressure and learn how to convert the units.
Boyle's Law: $P \propto \frac{1}{V}$ or $V \propto \frac{1}{P}$ (at constant $n$ and $T$)
Charles's Law: $V \propto T$ (at constant $n$ and $P$)
Avogadro's Law: $V \propto n$ (at constant $P$ and $T$)
Combine all three expressions
$$V \propto \frac{nT}{P}$$An ideal gas is a theoretical gas composed of many randomly moving point particles whose only interactions are perfectly elastic collisions. In another word, the molecules of an ideal gas do not attract or repel one another, and their volume is negligible.
$$V = R\frac{nT}{P}$$or
$$PV = nRT$$where $R$, the proportionality constant, is called the gas constant.
$$R = 8.314\,\text{J/K mol} = 8.314\,\text{N m/K mol} = 8.314\times10^{-3}\,\text{kJ/K mol} = 0.0821\,\text{L atm/K mol}$$Note
$$1\,\text{J} = 1\,\text{N m}$$$$1\,\text{Pa} = 1\,\text{N/m}^2$$For an ideal gas, at $0\,^\circ\text{C}$ ($273.15\,\text{K}$) and $1\,\text{atm}$, it occupies $22.14\,\text{L}$ volume. The conditions $0\,^\circ\text{C}$ and $1\,\text{atm}$ are called standard temperature and pressure,
where $m$ is the mass, $\mathcal{M}$ is the molar mass.
Thus,
$$PV = \frac{m}{\mathcal{M}}RT$$or
$$\frac{m}{\mathcal{M}V} = \frac{P}{RT}$$Since
$$\text{d} = \frac{m}{V}$$we get
$$\frac{d}{\mathcal{M}}= \frac{P}{RT}$$or
$$\mathcal{M}=\frac{dRT}{P}$$where $d$ is the density.
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# Watch the following video for a better understanding
# on gas laws.
Requirements¶
- Understand and remember the ideal gas equation. Konw the meaning of gas constant. Use ideal gas equation to do calculations.
- Learn how to calculate the molar mass or density of a gaseous substance.
In a mixture of several gaseous substances, the total moles of gasses is expressed as $n$, and the moles of each gas is expressed as $n_1$, $n_2$, etc. According to ideal gas equation
$$P = \frac{nRT}{V}$$where $P$ is the total pressure.
Define
$$P_1 = \frac{n_1RT}{V}$$$$P_2 = \frac{n_2RT}{V}$$and so on ...
The values of $P_1$, $P_2$, ... are regarded as the partial pressures of each gaseous substance.
The sum of all partial pressures must equal to the total pressure.
$$P = P_1 + P_2 + \dots$$Define
$$X_i = \frac{n_i}{n}$$as the molar fraction of one gas substance in the mixture. The sum of the fractions of every gaseous substances must equal to $1$.
We have
$$P_i = X_iP$$from IPython.display import YouTubeVideo
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# Watch the following video for a better understanding
# on partial pressures.
Requirements¶
- Understand what is partial pressure and learn how to do calculation with it.
Energy is defined as the capacity to do work or to produce change. In mechanics, work is defined as force times distance. Because energy can be measured as work, we can write
$$\text{energy}=\text{word done}=\text{force}\times\text{distance}$$The SI unit of energy is joule (J)
$$1\,\text{J} = 1\,\text{kg m}^2\text{/s}^2 = 1\,\text{N m}$$Kinetic energy (KE) is the type of energy expended by a moving object, or energy of motion.
The assumptions of the kinetic molecular theory of gases
An elastic collision is an encounter between two bodies in which the total kinetic energy of the two bodies after the encounter is equal to their total kinetic energy before the encounter.
Explanation of the gas laws
where $u_{rms}$is the root-mean-square speed which is an average molecular speed.
Gas diffusion is the gradual mixing of molecules of one gas with molecules of another by virtue of their kinetic properties.
is the process by which gas under pressure escapes from one compartment of a container to another by passing through a small opening.
In both diffusion and effusion the correlations among rate, time and molar mass are:
$$\frac{r_1}{r_2}=\frac{t_2}{t_1}=\sqrt{\frac{\mathcal{M}_2}{\mathcal{M}_1}}$$Requirements¶
- Understand the concepts and explanations of the kinetic molecular theory.
- Understand the square root rule. If the gas molecule is 100 times heavier, the average speed would reduce to 1/10.
Real gas molecules do have certain sizes and interactions! At higher pressure and/or lower temperature the deviations from ideal behavior become more significant.
One can use van der Waals equation to deal with real gases.
$$\left( P + \frac{an^2}{V^2}\right)\left(V-nb\right)=nRT$$The parameters $a$ and $b$ can be found in Table 5.3.
Requirements¶
- Understand why real gases do not behave exactly like ideal gases.
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