CHEM 1411 Concept Reviews: Gases

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Atmospheric Pressure:  The pressure due to the column of air weighing down on an object at any given time.  Atmospheric pressure changes with the weather (Temperature, air density, etc.)

Standard Atmospheric Pressure:  The typical atmospheric pressure at sea level, abbreviated as STP.  The values for standard atmospheric pressure in the different units of pressure are as follows:

 

 

The Gas Laws

(NOTE:  For all of the gas laws, the temperature used MUST be in Kelvin)

Boyle’s Law:  States that pressure (P) and volume (V) have an inverse relationship.  As one increases, the other decreases.  This relationship can be written as follows:

$V=constant\times\frac{1}{P}$        or        $P_1V_1=P_2V_2$

 

 

Charles’ Law:  States that volume (V) and temperature (T) have a direct relationship.  As one increases, the other increases.  The relationship can be written as follows:

$V=constant\times T$       or        $\frac{V_1}{T_1}=\frac{V_2}{T_2}$

 

 

Gay-Lussac’s Law:  States that pressure (P) and temperature (T) have a direct relationship.  As one increases, the other increases.  The relationship can be written as follows:

$P=constant\times T$         or         $\frac{P_1}{T_1}=\frac{P_2}{T_2}$

 

 

Avogadro’s Law:  States that volume (V) and number of moles (n) have a direct relationship.  As one increases, the other increases.  The relationship can be written as follows:

$V=constant\times n$         or        $\frac{V_1}{n_1}=\frac{V_2}{n_2}$

 

 

 

All of these gas laws can be integrated into two laws; one for comparing a sample of gas under two sets of conditions called The Combined Gas Law and one for determining the properties of a gas under one set of conditions called The Ideal Gas Law.

 

The Combined Gas Law shown below can be used any time two sets of conditions are listed.  If a variable is not listed or held constant, it can be removed from the equation to give the equation in the form you need.

   $\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$        (Combined Gas Law)

where the “1” represents the initial set of conditions and the “2” represents the final set of conditions.

 

 

 

The Ideal Gas Law shown below can be used anytime a single set of conditions for a gas are given:

$PV=nRT$          (Ideal Gas Law)

Where “R” is the ideal gas constant, the value of which depends on the units used for pressure and volume:

 $R=0.08206\:\frac{L\cdot atm}{K\cdot mol}=8.314\:\frac{J}{K\cdot mol}=8.314\:\frac{m^3\cdot Pa}{K\cdot mol}=8.314\:\frac{L\cdot kPa}{K\cdot mol}=62.36\:\frac{L\cdot torr}{K\cdot mol}$

 

The Ideal Gas Law can also be rearranged to solve for other properties of gases as well.  Other forms of the Ideal Gas Law are as follows:

PVM = mRT                                    PM = DRT

where “M” is the molar mass of the gas, “m” is the mass of the gas in the sample, and D is the density of the gas.

 

 

 

Dalton’s Law of Partial Pressures:  states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the individual gases.  This is expressed in the following formula:

$P_{total}=P_1+P_2+P_3+...$

Since pressure and the number of moles are directly related, the following relationship can be determined:

$\frac{P_1}{P_{total}}=\frac{n_1}{n_{total}}$    which can be rearranged to      $P_1=X_1P_{total}$

where P₁ is the partial pressure of gas “1”, and X₁ is the mole fraction of gas “1” $\left(mole\:fraction=\frac{n_1}{n_{total}}\right)$

 

 

 

The Kinetic Molecular Theory of Gases

This theory can be summarized in the following 5 statements:

1. Gases consist of large numbers of particles (molecules or atoms) that are in continuous, random motion.
   
   
2. The combined volume of all the particles of the gas is negligible relative to the total volume of the container.
   
   
3. Attractive and repulsive forces between gas particles are negligible.
   
   
4. Energy can be transferred between molecules during collisions but, as long as temperature remains constant, the average kinetic energy of the particles remains the same.
   
   
5. The average kinetic energy of the particles is proportional to the absolute temperature. At any given temperature, the particles of all gases have the same average kinetic energy.

 

 

 

Graham’s Law of Effusion:  The rate of effusion of a gas is inversely proportional to the square root of its molar mass.  When comparing two gases, this relationship can be written as follows:

$\frac{r_1}{r_2}=\sqrt{\frac{M_2}{M_1}}$     which is derived from the formula       $\mu_{rms}=\sqrt{\frac{3RT}{M}}$

where “r₁” and “r₂” are the rates of effusion of gases “1” and “2”, “M₁” and “M₂”are the molar masses of gases “1” and “2”, and μ_rms is root mean square speed of the gas molecules in a sample with molar mass “M” at temperature “T”.

 

 

 

Deviations from Ideal Gas Behavior

At very high pressures, gas particles occupy a significant amount of volume when compared to their container.  At very low temperatures, gas particles move slowly enough for their forces of attraction to be significant.  These deviations are accounted for (though imperfectly) using the van der Waals equation:

$\left(P+\frac{n^2a}{V^2}\right)\left(V-nb\right)=nRT$

with the term “$\frac{n^2a}{V^2}$” accounting for the attractive forces of gas particles and the term “nb” accounting for the volume occupied by the gas particles.  “a” and “b” are van der Waals constants that can be looked up.