# General Chemistry/Gas Laws

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## Gas Laws

As the result of many different scientists and experiments, several gas laws have been discovered. These laws relate the various state variables of a gas.

State Variables of a Gas
• Pressure (P) in mmHg, atm, kPa or torr
• Volume (V) in L
• Temperature (T) in K
• Amount Of Substance (n)

These gas laws can be used to compare two different gases, or determine the properties of a gas after one of its state variables have changed.

 $\displaystyle \frac{V}{n} = constant$ $\displaystyle \frac{n_1}{V_1} = \frac{n_2}{V_2}$ Avogadro's Law states that equal volumes of all ideal gases (at the same temperature and pressure) contain the same number of molecules. $\displaystyle P \times V = constant$ $\displaystyle P_1 \times V_1 = P_2 \times V_2$ Boyle's Law states that equal pressure is inversely proportional to volume (when temperature is constant). $\displaystyle \frac{V}{T} = constant$ $\displaystyle \frac{V_1}{T_1} = \frac{V_2}{T_2}$ Charles's Law states that volume is proportional to temperature (when pressure is constant). Remember that temperature must be measured in Kelvin. $\displaystyle \frac{P}{T} = constant$ $\displaystyle \frac{P_1}{T_1} = \frac{P_2}{T_2}$ Gay-Lussac's Law states that pressure is proportional to temperature (when volume is constant).

### Combined Gas Law

Charles' Law, Boyle's Law, and Gay-Lussac's Law gives us the combined gas law.

 $\displaystyle \frac{P \times V}{T} = constant$ For a gas with constant molar mass, the three other state variables are interrelated. $\displaystyle \frac{P_1 \times V_1}{T_1} = \frac{P_2 \times V_2}{T_2}$ The Combined Gas Law can be used for comparisons between gases.

## Ideal Gas Law

When Avogadro's Law is considered, all four state variables can be combined into one equation. Furthermore, the "constant" that is used in the above gas laws becomes the Universal Gas Constant (R).

To better understand the Ideal Gas Law, you should first see how it is derived from the above gas laws.

 $\displaystyle V \propto n \,$ and $\displaystyle V \propto \frac{T}{P}$ This is simply a restatement of Avogadro's Law and the Combined Gas Law. $\displaystyle V \propto \frac{n \times T}{P}$ We can now combine the laws together. $\displaystyle \frac{V}{\frac{n \times T}{P}} = R$ Let R be a constant, and write the proportion in the form of an equation. $\displaystyle \frac{P \times V}{n \times T} = R$ Rearranging the fraction gives one form of the ideal gas law.

The ideal gas law is the most useful law, and it should be memorized. If you know the ideal gas law, you do not need to know any other gas laws, for it is a combination of all the other laws. If you know any three of the four state variables of a gas, the unknown can be found with this law. If you have two gases with different state variables, they can be compared.

There are three ways of writing the ideal gas law, but all of them are simply algebraic rearrangements of each other.

 $\displaystyle PV = nRT \,$ This is the most common form. $\displaystyle \frac{PV}{nT} = R$ This form is useful for predicting the effects of changing a state variable. To maintain a constant value of R, any change in the numerator must result in a proportional change in the denominator, and vice versa. If, for example, the pressure is decreased in a constant-volume container, you can use this form to easily predict that the temperature must decrease. $\displaystyle \frac{P_1V_1}{n_1T_1} = \frac{P_2V_2}{n_2T_2}$ Because R is the same constant for all gases, this equation can be used to relate two gases to each other.

Rules for Using the Ideal Gas Law

• Always convert the temperature to kelvins (K).
• Always convert mass to moles (mol).
• Always convert volume to liters (L).
• It is preferable to convert pressure to kilopascals (kPa). R, the Universal Gas Constant, would be 8.314 (L·kPa)/(mol·K).

## Kinetic Molecular Theory

The Kinetic Molecular Theory attempts to explain the gas laws. It describes the behavior of microscopic gas molecules to explain the macroscopic behavior of gases. According to this theory, an ideal gas is composed of continually moving molecules of negligible volume. The molecules move in straight lines unless they collide into each other or the walls of their container.

 $\displaystyle P = \frac{F}{A}$ The pressure of the gas on the container is explained as the force the molecules exert on the walls during a collision. Pressure is equal to the average force of collisions divided by the total surface area of the container. $\displaystyle T=\frac{2}{3k_B} K$ The temperature of the gas is proportional to the average kinetic energy of the molecules. $\displaystyle K$ denotes the average kinetic energy of the molecules, and $\displaystyle k_B$ is the Boltzmann constant (1.388 x 10-23).

The gas laws are now explained by the microscopic behavior of gas molecules:

• Boyle's Law: The pressure of a gas is inversely proportional to its volume. A container's volume and surface area are obviously proportional. Based on the pressure equation, an increase in volume (and thus surface area) will decrease pressure.
• Charles' Law: the volume of a gas is proportional to its temperature. As the volume (and surface area) increases, the pressure will decrease unless the force also increase. When pressure is constant, the volume and temperature must be proportional. The temperature equation above explains why: the energy of the molecules (and their collision force) is proportional to temperature.
• Gay-Lussac's Law: The temperature of a gas is directly proportional to its pressure. An increase in temperature will increase the kinetic energy of the molecules (shown by the temperature equation). Greater kinetic energy causes the molecules to move faster. Their collisions with the container will have more force, which increases pressure.
• Avogadro's Law: Equal volumes of all ideal gases (at the same temperature and pressure) contain the same number of molecules. According to the Kinetic Molecular Theory, the size of individual molecules is negligible compared to distances between molecules. Even though different gases have different sized molecules, the size difference is negligible, and the volumes are the same.

### Derivation of Ideal Gas Law

 Suppose there are $\displaystyle N$ molecules, each with mass $\displaystyle m$ , in a cubic container with side length $\displaystyle s$ . Even though the molecules are moving in all directions, we may assume, on average, that one third of the molecules are moving along the x-axis, one third along the y-axis, and one third along the z-axis. We may assume this because the motion of the molecules is random, so no direction is preferred. Suppose the average speed of the molecules is $\displaystyle u$ . Let a specific wall of the container be labeled A. Because the collisions in Kinetic Molecular Theory are perfectly elastic, the speed after a collision is $\displaystyle -u$ . Therefore, the average change in momentum (the product of mass and velocity) per collision is $\displaystyle 2mu$ . Each molecule, on average, travels a distance of $\displaystyle 2s$ between two consecutive collisions with wall A. Therefore, it will collide $\displaystyle u/2s$ times per second with wall A. $\displaystyle 2mu \times \frac{u}{2s}=\frac{mu^2}{s}$ The average change in momentum per molecule per second. $\displaystyle \frac{Nmu^2}{3s}$ Therefore, this is the total change in momentum per second for the $\displaystyle (1/3)N$ molecules that collide into wall A. This is the momentum per second that was exerted onto wall A. Because force equals the change in momentum over time, this value is the force exerted on wall A. $\displaystyle \frac{Nmu^2}{3s} \times \frac{1}{s^2}=\frac{Nmu^2}{3s^3}$ Pressure is defined as force per unit area, so this is the pressure $\displaystyle P$ of the gas. $\displaystyle PV=\frac{Nmu^2}{3}$ Because the volume of the container is $\displaystyle V=s^3$ , we can rearrange the equation. $\displaystyle K=\frac{1}{2}mu^2$ The kinetic energy of a single particle is given by this equation. $\displaystyle PV=\frac{2}{3}NK$ Substitute kinetic energy into the $\displaystyle PV$ equation. $\displaystyle PV=N k_B T$ Substitute the temperature equation (from the previous section). $\displaystyle PV = N_A n k_B T$ Avogadro's number $\displaystyle N_A$ is equal to the number of molecules per mole. $\displaystyle R = k_B N_A$ By definition, the ideal gas constant is equal to the Boltzmann constant times Avogadro's number. $\displaystyle PV=nRT$ The ideal gas law is derived from the Kinetic Molecular Theory.

## Deviations from the Ideal Gas Law

In an ideal gas, there are no intermolecular attractions, and the volume of the gas particles is negligible. However, there is no real gas that can perfectly fits this behavior, so the Ideal Gas Law only approximates the behavior of gases. This approximation is very good at high temperatures and low pressures.

At high temperature the molecules have high kinetic energy, so intermolecular attractions are minimized. At low pressure the gas occupies more volume, making the size of the individual molecules negligible. These two factors make the gas behave ideally.

At low temperature or high pressure, the size of the individual molecules and intermolecular attractions becomes significant, and the ideal gas approximation becomes inaccurate.

## Eudiometers and Water Vapor

In calculations for a gas above a liquid, the vapor pressure of the liquid must be considered.

A eudiometer is a device that measures the downward displacement of a gas. The apparatus for this procedure involves an inverted container or jar filled with water and submerged in a water basin. The lid of the jar has an opening for a tube through which the gas to be collected can pass. As the gas enters the inverted container, it forces water to leave the jar (displacing it downward). To fill the entire container with gas, there must enough gas pumped into the container to expel all of the water. Gas is created by burning a substance that releases methane. The eudiometer on the right was full of water before the gas was created. By measuring the change in volume, the amount of gas can be calculated.

As seen in this diagram, the downward displacement involves water. Therefore, in the container where the gas is collected, there is unwanted water vapor. To account for the water vapor, subtract the pressure of water vapor from the pressure of the gases in the container to find the pressure of the collected gas. This is simply a restatement of Dalton's Law of Partial Pressure:

$\displaystyle P_{total} = P_{water~vapor} + P_{gas}$

The pressure of water vapour can be found on this webpage.

## Gas Laws Practice Questions

1. Between the Combined Gas Law and the Ideal Gas Law, which one accounts for chemical change? Explain.
2. Calculate the density of hydrogen at a temperature of 298 K and pressure of 100.0 kPa.
3. What volume does 5.3 moles of oxygen take up at 313 K and 96.0 kPa?
4. Hydrogen and sulfur chemically combine to form the gas hydrogen sulfide, according to the reaction: H2 (g) + S(s) → H2S(g). How many liters of hydrogen are required to form 7.4 L of hydrogen sulfide (at STP: 273 K, 101.3 kPa)?