# Gases - A Review

Skills Tested by This Quiz

- Identify the theory applicable to each problem, and calculate the desirable quantity from a given set of conditions.

## Properties of Gases

This quiz tests your comprehension and ability to apply the following topics to solve problems:

- Gases - the gaseous state of matter
- Gas laws - the ABCD gas laws
- Ideal gas law - a summary of gas laws
- Gas kinetics - motion of gas molecules
- Gas & stoichiometry - stoichiometry problems involving gases

A brief review is given again here, but it is your responsibility to **identify the theory applicable to each problem, and calculate the desirable quantity from a given set of conditions.** There is always more than one way to solve a problem, and a logical approach is a useful skill in itself. It is the process of problem solving, not the result, that is useful to you.

### Common Gas Properties

Gas is a state of matter. In this state, all substances behave alike. The properties of the gaseous state are:

- Amount of gas in moles,
*n*, number of molecules, mass, and volume. - The pressure,
*P*, exerted to the walls of the container by gas molecules or by the wall to contain the gas to a fixed volume*V*. - The temperature,
*T*, which is a measure of the kinetic energy of the molecules. - The volume,
*V*, occupied by a gas at temperature*T*, under the pressure*P*. - The average kinetic energy of a gas, and the root mean square speed of gas molecules.

These properties are often expressed in various units, and ability to convert units from one to another is always required.

### Gas Laws - Some Key Equations

You are expected to have mastered the ABCD laws of gases.

You should use the proper units for pressure, *P*, volume *V*, and temperature *T*. Letters *i* and *f* following these quantities refer to the **initial** and **final** states respectively. A summary of key equations is given below, but you should be able to derive one from each other, and understand the system from which these formulas apply.

#### Boyle's Law

\(P_i V_i = k\) (*k* is a constant)

\(P_i V_i = P_f V_f\)

#### Charles Law

\(T = 273.15 + t^{\textrm C}\)

\(\dfrac{V_i}{T_i}=\dfrac{V_f}{T_f}= k\)

\(\dfrac{P_i}{T_i}=\dfrac{P_f}{T_f}= k\)

#### The Ideal Gas Law

\(P V = n R T\)

where

\(\begin{align}

R &= \mathrm{\dfrac{1\: atm\: 22.4\: L}{1\: mol\: 273.15\: K}}\\

&= \mathrm{0.082058\: \dfrac{L\: atm}{mol\: K}}\\

&= \mathrm{8.3145\: \dfrac{J}{mol\: K}}

\end{align}\)

#### Avogadro's Law and Gas Density

\(n = \dfrac{P V}{R T}\)

\(P M =\dfrac{n M}{V}R T\) -- where *M* is the molecular weight of the gas

\(P V = n R T\) -- Ideal gas equation

\(P = \dfrac{d R T}{M}\) -- where *d* is the density of the gas

\(d = \dfrac{P M}{R T}\) -- density of gas is given by this equation

\(M = \dfrac{d R T}{P}\) -- hence, the Molecular Weight of a gas is given by this.

#### Dalton's Law of Partial Pressures

If *P _{total}* is the total pressure of a gas mixture, and

*P*, ... are partial pressures of gas A, B, C, ..., then

_{a}, P_{b}, P_{c}\(P_{total} = P_a + P_b + P_c + ...\)

If *n _{total}* is the total number of moles, and

*n*, ... are number of moles of gas A, B, C, ..., then

_{a}, n_{b}, n_{c}\(n_{total} = n_a + n_b + n_c ...\)

\(\begin{align}

n_t R T &= P V \\

&= (n_a + n_b + n_c + ...) R T\\

&= P_a + P_b + P_c + ...

\end{align}\)

#### Mole Fraction

The mole fraction of a gas A, *x _{a}* in a system containing

*n*mole of gas is

_{total}\(x_a = \dfrac{n_a}{n_{total}}\)

#### Non-Ideal Behavior of Gas

The ideal gas law has a limited precision for predicting the properties of gases. The imprecision is known as the non-ideal behavior of gas, and the **van der Waals equation**

\(\left(P + \dfrac{n^2a}{V^2}\right) (V - n b) = n R T\)

has been introduced to deal with non-ideal behavior of gases in Ideal gas law. For practical application in chemical manufacturing processes and in chemical reactions, the non-ideality has to be taken into account. For these applications, other methods of correction to the ideal gas law are also used.

## Practice Questions

**If 0.40 and 0.10 mol of**\(\ce{N2}\)**and**\(\ce{O2}\)**are enclosed in a 2-liter container at a temperature of 311 K, what is the pressure?**Hint: 6.4 atm

## Contributors

Chung (Peter) Chieh (Chemistry, University of Waterloo)