Ideal Gas Law

The equality for the four variables involved in Boyle’s Law, Charles’ Law, Gay-Lussac’s Law and Avogadro’s law can be written

PV = nRT

R = ideal gas constant

Ideal Gases

Behave as described by the ideal gas equation; no real gas is actually ideal

Within a few %, ideal gas equation describes most real gases at room temperature and pressures of 1 atm or less

In real gases, particles attract each other reducing the pressure

Real gases behave more like ideal gases as pressure approaches zero.

PV = nRT

R is known as the universal gas constant

Using STP conditions P V

R = PV = (1.00 atm)(22.4 L) nT (1mol) (273K)

n T

= 0.0821 L-atm

mol-K

Learning Check G15

What is the value of R when the STP value for P is 760 mmHg of 1 mole of gas?

Solution G15

What is the value of R when the STP value for P is 760 mmHg?

R = PV = (760 mm Hg) (22.4 L)

nT (1mol) (273K)

= 62.4 L-mm Hg mol-K

Learning Check G16

Dinitrogen monoxide (N2O), laughing gas, is used by dentists as an anesthetic. If 2.86 mol of gas occupies a 20.0 L tank at 23°C, what is the pressure (mmHg) in the tank in the dentist office?

Solution G16

Set up data for 3 of the 4 gas variables

Adjust to match the units of R

V = 20.0 L 20.0 L

T = 23°C + 273 296 K

n = 2.86 mol 2.86 mol

P = ? ?

Rearrange ideal gas law for unknown P

P = nRT

V

Substitute values of n, R, T and V and solve for P

P = (2.86 mol)(62.4L-mmHg)(296 K)

(20.0 L) (K-mol)

= 2.64 x 103 mm Hg

Learning Check G17

A 5.0 L cylinder contains oxygen gas at 20.0°C and 735 mm Hg. How many grams of oxygen are in the cylinder?

Solution G17

Solve ideal gas equation for n (moles)n = PV

RT

= (735 mmHg)(5.0 L)(mol K) (62.4 mmHg L)(293 K)

= 0. 20 mol O2 x 32.0 g O2 = 6.4 g O2

1 mol O2

Molar Mass of a gas

What is the molar mass of a gas if 0.250 g of the gas occupy 215 mL at 0.813 atm and 30.0°C?

n = PV = (0.813 atm) (0.215 L) = 0.00703 mol

RT (0.0821 L-atm/molK) (303K)

Molar mass = g = 0.250 g = 35.6 g/mol

mol 0.00703 mol

Density of a Gas

Calculate the density in g/L of O2 gas at STP. From STP, we know the P and T.

P = 1.00 atm T = 273 K

Rearrange the ideal gas equation for moles/L

PV = nRT PV = nRT P = n

RTV RTV RT V

Substitute

(1.00 atm ) mol-K = 0.0446 mol O2/L

(0.0821 L-atm) (273 K)

Change moles/L to g/L

0.0446 mol O2 x 32.0 g O2 = 1.43 g/L

1 L 1 mol O2

Therefore the density of O2 gas at STP is

1.43 grams per liter

Formulas of Gases

A gas has a % composition by mass of 85.7% carbon and 14.3% hydrogen. At STP the density of the gas is 2.50 g/L. What is the molecular formula of the gas?

Formulas of Gases

Calculate Empirical formula

85.7 g C x 1 mol C = 7.14 mol C/7.14 = 1 C 12.0 g C

14.3 g H x 1 mol H = 14.3 mol H/ 7.14 = 2 H 1.0 g H

Empirical formula = CH2

EF mass = 12.0 + 2(1.0) = 14.0 g/EF

Using STP and density ( 1 L = 2.50 g)

2.50 g x 22.4 L = 56.0 g/mol

1 L 1 mol

n = EF/ mol = 56.0 g/mol = 4

14.0 g/EF

molecular formula

CH2 x 4 = C4H8

Gases in Chemical Equations

On December 1, 1783, Charles used 1.00 x 103 lb of iron filings to make the first ascent in a balloon filled with hydrogen

Fe(s) + H2SO4(aq) FeSO4(aq) + H2(g)

At STP, how many liters of hydrogen

gas were generated?

Solution

lb Fe g Fe mol Fe mol H2 L H2

1.00 x 103 lb x 453.6 g x 1 mol Fe x 1 mol H2

1 lb 55.9 g 1 mol Fe

x 22.4 L H2 = 1.82 x 105 L H2

1 mol H2

Charles generated 182,000 L of hydrogen to fill his air balloon.

Learning Check G18

How many L of O2 are need to react 28.0 g NH3 at 24°C and 0.950 atm?

4 NH3(g) + 5 O2(g) 4 NO(g) + 6 H2O(g)

Solution G18

Find mole of O2

28.0 g NH3 x 1 mol NH3 x 5 mol O2

17.0 g NH3 4 mol NH3

= 2.06 mol O2

V = nRT = (2.06 mol)(0.0821)(297K) = 52.9 L

P 0.950 atm

Summary of Conversions with Gases

Volume A Volume B

Grams A Moles A Moles B Grams B

Atoms or Atoms or

molecules A molecules B

Daltons’ Law of Partial Pressures

The % of gases in air Partial pressure (STP)

78.08% N2 593.4 mmHg

20.95% O2 159.2 mmHg

0.94% Ar 7.1 mmHg

0.03% CO2 0.2 mmHg

PAIR = PN + PO + PAr + PCO = 760 mmHg 2 2 2

Total Pressure 760 mm Hg

Learning Check G19

A.If the atmospheric pressure today is 745 mm Hg, what is the partial pressure (mm Hg) of O2 in the air?

1) 35.6 2) 156 3) 760

B. At an atmospheric pressure of 714, what is the partial pressure (mm Hg) N2 in the air?

1) 557 2) 9.14 3) 0.109

Solution G19

A.If the atmospheric pressure today is 745 mm Hg, what is the partial pressure (mm Hg) of O2 in the air?

2) 156

B. At an atmospheric pressure of 714, what is the partial pressure (mm Hg) N2 in the air?

1) 557

Partial Pressure

Partial Pressure

Pressure each gas in a mixture would exert if it were the only gas in the container

Dalton's Law of Partial PressuresThe total pressure exerted by a gas mixture is the sum of the partial pressures of the gases in that mixture.

PT = P1 + P2 + P3 + .....

Partial Pressures

The total pressure of a gas mixture depends

on the total number of gas particles, not on

the types of particles.

STP

P = 1.00 atm P = 1.00 atm

1.0 mol He0.50 mol O2

+ 0.20 mol He+ 0.30 mol N2

Health Note

When a scuba diver is several hundred feet

under water, the high pressures cause N2 from

the tank air to dissolve in the blood. If the diver rises too fast, the dissolved N2 will form bubbles

in the blood, a dangerous and painful condition called "the bends". Helium, which is inert, less dense, and does not dissolve in the blood, is mixed with O2 in scuba tanks used for deep

descents.

Learning Check G20

A 5.00 L scuba tank contains 1.05 mole of O2 and 0.418 mole He at 25°C. What is the partial pressure of each gas, and what is the total pressure in the tank?

Solution G20

P = nRT PT = PO + PHe

V 2

PT = 1.47 mol x 0.0821 L-atm x 298 K

5.00 L (K mol)

= 7.19 atm