Ideal Gas Law Notes

Ch. 14: Ideal Gas Law

Dr. Laura Saldarriaga, Chem 4

The Volume Amount

Equal volumes of gases at the same temperature (T) and pressure (P) contain the same number of molecules.

Volume (V) is proportional to the amount (n) for gases at the same T and P.

The Ideal Gas Law

The Ideal Gas Law is expressed as: $PV = nRT$

Where:

• $P$ is the pressure.

• $V$ is the volume.

• $n$ is the number of moles.

• $R$ is the Ideal Gas constant.

• $T$ is the temperature.

The value of $R$ depends on the units of $P$:

• $R = 0.08206 \frac{atm \cdot L}{K \cdot mole}$

• $R = 62.4 \frac{torr \cdot L}{K \cdot mole}$

• $R = 0.083145 \frac{bar \cdot L}{K \cdot mole}$

STP Conditions

STP stands for Standard Temperature and Pressure.

• Standard Temperature is $273.15 K$ ($0$ °C).

• Standard Pressure is $0.987 atm$ (1 bar) - IUPAC standard.

Note: Standard Pressure used to be 1 atm but the IUPAC standard pressure of 1 bar is now generally preferred.. We will use the IUPAC standard pressure of 1 bar for STP conditions.

Combined Gas Laws

The combined gas laws can be derived from the ideal gas law.

Given: $PV = nRT$

For two different states 1 and 2:

$P<em>1V</em>1 = n<em>1RT</em>1$

$P<em>2V</em>2 = n<em>2RT</em>2$

Depending on the variables held constant, different gas laws can be derived:

$\frac{P<em>1}{P</em>2} = \frac{V<em>2}{V</em>1}$

$\frac{V<em>1}{V</em>2} = \frac{T<em>1}{T</em>2}$

$\frac{P<em>1}{P</em>2} = \frac{T<em>1}{T</em>2}$

$\frac{P<em>1V</em>1}{T<em>1} = \frac{P</em>2V<em>2}{T</em>2}$

$\frac{P<em>1V</em>1}{n<em>1RT</em>1} = 1$

$\frac{P<em>2V</em>2}{n<em>2RT</em>2} = 1$

$\frac{P<em>1V</em>1}{n<em>1T</em>1} = \frac{P<em>2V</em>2}{n<em>2T</em>2}$

Ideal Gas Law Problems

To solve Ideal Gas Law problems, isolate the variable needed on the left side of the equation and plug in the known values.

Example 1: Calculate the pressure (in bar) of 3.5 moles of He which occupies 20.00 L at 100.0 °C.

Example 2: How many moles of fluorine occupy 9.8 L at 95.33 °C and 800.0 torr?

Molar Mass (MM)

When Molar Mass (MM) or grams appear in a problem, use a different form of the Ideal Gas Equation.

$MM = \frac{m}{n}$

Replacing $n$ in the Ideal Gas Equation:

$PV = \frac{m}{MM}RT$

$MM = \frac{mRT}{PV}$

Calculating Molar Mass of a Gas

Example 1: What is the molar mass of an unknown gas if 5.25 g occupies 8.25 L at 1.50 bar and 600.00 K?

Example 2: If 5.00 g of helium take up 3.50 L at 1.33 atm, find the temperature of the gas in °C.

Density of a Gas

Density is defined as mass/volume or $m/V$.

What is the density of krypton at 50.00 °C and 1500.0 torr?

Starting with:

$MM = \frac{mRT}{PV}$

$MM = \frac{DRT}{P}$

Where $D$ is the density. Solving for $D$:

$D = \frac{MM \cdot P}{RT}$

Calculating Density of a Gas

Example: The density of an unknown gas is 1.66 g/L at 293.5 K and 2.00 bar. What noble gas is it?

Molar Volume

Molar volume is the volume occupied by one mole of a gas.

Find the molar volume of a gas at STP:

$\frac{V}{n} = \frac{RT}{P}$

Stoichiometry: Ideal Gas Equation Method

Ammonia gas is formed from its elements:

a) Write a balanced equation.

b) How many liters of ammonia form from 3.00 g nitrogen at STP?

c) How many liters of ammonia form from 5.50 g hydrogen at 2.50 atm and 450.0 K?

Stoichiometry: Ideal Gas Equation Method (cont.)

d) How many grams of hydrogen are used when 6.80 L nitrogen are used at 650.0 torr and 50.00 °C?

$3H<em>2(g) + N</em>2(g) \rightarrow 2NH_3(g)$

Equations to Keep Handy

• When two sets of conditions are mentioned: $\frac{P<em>1V</em>1}{T<em>1} = \frac{P</em>2V<em>2}{T</em>2}$

• The Ideal Gas Equation: $PV = nRT$

• When molar mass or grams are mentioned: $MM = \frac{mRT}{PV}$

• When density is mentioned: $MM = \frac{DRT}{P}$

• For molar volume: $\frac{V}{n} = \frac{RT}{P}$

• For stoichiometry problems, if V, P, and T are given for a gas, calculate moles using the ideal gas equation first.

• If V is wanted in the problem, first do stoichiometry, then use the ideal gas equation to calculate V for the desired gas.

Stoichiometry Pathways

g A --> moles A --> moles B --> g B

(V, M)A --> moles A --> moles B --> (V, M)B

(P,V,T)A --> moles A --> moles B --> (P,V,T)B

units A --> moles A --> moles B --> units B

Examples

Given: $SO<em>2(g) + CaCO</em>3(s) + O<em>2(g) \rightarrow CaSO</em>4(s) + CO_2(g)$

a. 42.7 L of sulfur dioxide at 543 torr and 22.0 °C react with 33.8 L of oxygen at 287 K and 1.44 atm. How many dg of calcium sulfate can form?

b. How many grams of sulfur dioxide and oxygen are left at the end of the reaction?