# 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 Ptotal is the total pressure of a gas mixture, and Pa , Pb , Pc , ... are partial pressures of gas A, B, C, ..., then

$$P_{total} = P_a + P_b + P_c + ...$$

If ntotal is the total number of moles, and na , nb , nc , ... are number of moles of gas A, B, C, ..., then

$$n_{total} = n_a + n_b + n_c ...$$

\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, xa in a system containing ntotal mole of gas is

$$x_a = \dfrac{n_a}{n_{total}}$$

#### 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

1. What does the variable P stand for in the ideal gas law,

$$P V = n R T$$?

Hint: pressure

Skill:
Describe the ideal gas law.

2. 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

Skill:
Apply the ideal gas law to solve a problem.

3. 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 partial pressure of $$\ce{N2}$$?

Hint: 5.1 atm

Skill:
Apply Dalton's partial pressure law to solve a problem.

4. A gas collected over water contains water vapor. At 23 °C, the vapor pressure of water is 2.8 kPa. Calculate the amount of water (molar mass 18) in mg contained in 3.0 L of a saturated air sample.

Hint: 61 mg or 0.061 g

Discussion
0.061 g of water = 37000000000000000000000 molecules.

5. In the van der Waals equation,

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

what unit should n b have if units for V is L?

Hint: L

Discussion:
Addition and subtraction can be carried out only on quantities having the same units.

6. At room temperature, $$\ce{CO2}$$ deviates from ideal gas behavior due to its high molecular mass and high boiling point. At 300 K, choose the gas that behaves most like an ideal gas from: $$\ce{CO}$$, $$\ce{CO2}$$, $$\ce{HF}$$, $$\ce{NH3}$$, $$\ce{H2S}$$, $$\ce{SO3}$$, $$\ce{NO2}$$, $$\ce{CH4}$$, $$\ce{H2}$$.

Hint: Hydrogen gas

Discussion:
Molecules of $$\ce{NH3}$$, $$\ce{H2S}$$ and $$\ce{HF}$$ are polar, and their molecules interact strongly. These gases are less ideal at room temperature compared to $$\ce{H2}$$, $$\ce{N2}$$ and $$\ce{O2}$$.

7. A weather balloon filled with $$\ce{He}$$ gas at 300.0 K has a volume of 2.0 m3 at ground level where the pressure is 1.0 atm. After the balloon rises to a height where the atmospheric pressure is 0.40 atm, its volume increases to 4.0 m3. Calculate the temperature in K of the $$\ce{He}$$ gas. R = 0.08205 L atm /(mol K)

Hint: 240 K

Skill:
Easily solved by the application of the formula
$$\dfrac{P_i V_i}{T_i} = \dfrac{P_f V_f}{T_f}$$

8. The van der Waals constants for $$\ce{CO2}$$ are: a = 3.592 L2 atm mol-2, b = 0.04267 L/mol. Use the van der Waals equation to estimate the pressure exerted by 0.500 mol of $$\ce{CO2}$$ in a 1.00 L tank at 298 K.

Hint: 11.6 atm

Skill:
Rearrange the van der Waals formula as

$$P =\dfrac{n R T}{V - n b}-\dfrac{n^2 a}{V^2}$$

9. A 1.0-L sample vapor of an unknown substance has a mass of 2.548 g at 101.3 kPa and 100 °C. Estimate the molar mass.

Hint: 78

Skill:
Apply the formula: $$M = \dfrac{d R T}{P}$$ to solve the problem.