Gas Characteristics
Gases expand to fill their container, whereas solids and liquids do not. Additionally, gases are much more compressible than solids or liquids due to the larger interparticle space present in gases. In fact, the majority of a gas is empty space, reducing the impact of particles on one another.
The way that gases mix with each other also differs from that of solids and liquids. For example, two liquids may be immiscible, or not forming a homogenous mixture when added together, such as oil and water. Gases, on the other hand, form a homogenous mixture when combined, in which each gas can be treated independently. 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 gases that make up the mixture:
\[
P_{\text{total}} = P_1 + P_2 + P_3 + \ldots
\]
For example, a gas mixture contains O2 and N2 gases at a total pressure of 1.5 atm. The partial pressure of oxygen is 0.5 atm. What is the partial pressure of nitrogen in the mixture? Using Dalton's law of partial pressures we have:
\[
\text{P}_{\text{total}} = \text{P}_{\text{O}_2} + \text{P}_{\text{N}_2}
\]
\[
1.5 \, \text{atm} = 0.5 \, \text{atm} + \text{P}_{\text{N}_2}
\]
\[
\text{P}_{\text{N}_2} = 1.0 \, \text{atm}
\]
Pressure
Pressure is a force per unit area:
\(\ce{P = \frac{F}{A}}\)
The following conversion factors for units of pressure are useful:
1 atm = 760 mm Hg = 760 torr
Standard temperature and pressure, or STP, are a set of standard conditions for experiments, defined as a temperature of 0 C (273.15 K) and an absolute pressure of 1 atm.
Atmospheric pressure, also known as air pressure or barometic pressure, is the force exerted by the weight of the Earth's atmosphere. All objects with mass are influenced by Earth's gravity, g = 9.81 m/s2. Atmospheric pressure can be calculated by considering the force exercted by a column of air on an area of the Earth's surface. At sea level, atmospheric pressure is 1.013 x 105 Pascal or 1 atmosphere (atm).
The pascal (Pa), named after Blaise Pascal, is the SI unit of pressure, with units of N/m2, or base unit kg*m-1*s-2.
Gas Laws
Knowing temperature, pressure, volume, and the moles of gas can be used to describe its state. Scientists (Avogadro, Boyle, Charles, Gay-Lussac) investigated the effect of the variables on the state of the, establishing a series of gas laws.
Avogadro hypothesized that equal volumes of gas at the same temperatuer and pressure contain an equal number of particles. At conditions of STP (0 C and 1 atm pressure), 22.4 L of any gas contains 6 x 1023 molecules, or 1 mol. Following from this is Avogadro's law:
\[
V \propto n
\]
\[
\frac{V}{n} = k
\]
Where V is the volume of gas, n is the amount of gas (in moles), and k is a constant for a given temperature and pressure. Based on this realtionship, if temperature and pressure are held constant, what would be the effect of doubling the moles of gas on volume? The volume would double as well.
Jacques Charles investigated the relationship between volume and temperature. Consider the following graph demonstrating this relationship:
Mathematically, this is described as follows:
\( V \propto T \)
\( \ce{V = kT} \)
Where V is the volume of the gas, T is the temperature of the gas in kelvins, and k is a constant. Considering the same gas under two different conditions, the following relationship applies:
\[ \frac{{V_1}}{{T_1}} = \frac{{V_2}}{{T_2}} \]
Gay-Lussac investigated the relationship between temperature and pressure. At a constant volume, consider the following for a hypothetical gas:
Here, we see that pressure is directly proportional to temperature at a constant volume. Mathematically, this can be expressed as:
\(\ce{\frac{P}{T} = \text{constant}}\)
\(\ce{\frac{P_1}{T_1} = \frac{P_2}{T_2}}\)
Ideal Gas Law
Let's review the realtionship between pressure, temperature and volume that have been established by the Gas Laws thus far:
\(V \propto \frac{1}{P}\)
\(V \propto T\)
\(V \propto n\)
\(V \propto \frac{nT}{P}\)
The final statement, that volume is proportional to the product of nT over P, combines the relationships establisehd by the Gas Laws. A constant of propportionality, R, known as the molar gas constant or the ideal gas constant, is used. It is equal to:
\(\ce{R = N_Ak_B}\)
Where NA is the Avogadro constant and kB is the Boltzmann constant. Thus, the equation becomes:
\(V = R ( n \frac{T}{P} )\)
\(PV = nRT\)
This is known as the ideal gas law, a combination of the empircal laws described by Boyle, Charles, Avogadro, and Gay-Lussac. The ideal aspect of this law is very important recognize. An ideal gas is a theroetical gas in which:
- Gas molecules do not attract or repel one another, instead only interacting through elastic (no loss of KE in the system) collisions.
- The gas particles themselves are considered to have no volume.
While no ideal gas exists in reality, the error introduced by these assumptions is often small enough that they are acceptable. More accurate calculations require more advanced models to describe the true behavior of gases.
Applications
The ideal gas law can be used to solve a number of different problems. Consider the following question:
A gas sample occupies a volume of 2.5 L at a pressure of 3.00 atm and contains 0.500 moles of gas. What is the temperature of the gas in Celsius?