## Methodology

Problem One
What is the effective volume of a gaseous methane molecule at STP?

Step #1. What is the volume of one mole of a gas at STP?

22.4 L

Step #2. What is standard pressure and temperature?

One atmosphere and 0 °C (273 K)

Step #3. How many molecules of methane are there in 22.4 L at STP?

6 x 1023 molecules

Step #4. What is the effective volume of one molecule?

22.4 L/ 6 x 1023 molecules = 4 x 10-23 L

Step #5. Express this volume in cubic centimeters.

One liter is 1000 cm3. Thus, the volume is:
4 x 10-23 L x 1000 cm3/L = 4 x 10-20 cm3

Step #6. Our term "effective" volume is useful because it gives some idea of how much space can be a assigned to a molecule in the gaseous state. Is a molecule confined to this space?

No, each and every molecule can move throughout the entire container. The effective volume is the space each molecule would occupy if it did not move about through the container.

Step #7. What is the effective volume of methane in the liquid state (assume a density of 0.8 g/mL)?

One mole of liquid methane has a volume of:
16 g / 0.8 g/mL = 20 mL = 20 cm3. Thus:
20 cm3 / 6 x 1023 molecules = 3 x 10-23 cm3/molecule

Problem Two
At a certain temperature, 1.0 L of a gas has a pressure of 760 torr. What will the volume be if the pressure changes to 380 torr?

Step #1. What are the four variables that we use to measure a sample of gas?

volume, pressure, temperature, and number of moles

Step #2. What is the relationship between volume and pressure?

If the temperature and number of moles remain constant, volume is inversely proportional to pressure.

Step #3. Qualitatively, if the pressure of a gas increases (at constant temperature and number of moles), what happens to the volume?

the volume decreases

Step #4. If the pressure doubles, how much will the volume decrease?

If the pressure goes up by a factor of 2, the volume will decrease by a factor of 2.

Step #5. For our 1.0 L sample of gas, what will the new volume be?

760 torr / 380 torr x 1.0 L = 2.0 L

Problem Three
The indirect proportionality between volume and temperature can be expressed as:
V µ 1/P

Step #1. Write this as an equation.

V = k/P

Step #2. For the gas in question 2, what is the value of k?

Using the fact that 1.0 L is the volume at 760 torr,
V = k/P = 1.0 L = k/760 torr, or,
k = 760 L-torr.

Step #3. Use this value of k to calculate the volume at 380 torr

V = 760 L-torr/P = 760 L-torr/380 torr = 2.0 L

Step #4. What factors will affect the value of k?

the number of moles of gas and its temperature.

Problem Four
What pressure is required to change the volume of a gas at constant temperature from 2.0 L at 1.0 atm to 0.50 L?

Step #1. What is the relationships between volume and pressure?

V = k/P. They are indirectly proportional.

Step #2. If the volume decreases by a factor of 4, what will happen to the pressure?

It will increase by a factor of 4.

Step #3. What pressure is required to change from a volume of 2.0 L at 1.0 atm to 0.50 L?

2.0 L / 0.50 L x 1.0 atm = 4.0 atm

Problem Five
A 1.0 g sample of gas occupies a volume of 1.0 L at a temperature of 0 °C. What volume will it occupy at a temperature of 50 °C if it is maintained at a constant pressure?

Step #1. What variables are changing in this problem?

volume and temperature

Step #2. What is the relationship between volume and temperature?

Volume is directly proportional to absolute temperature.

Step #3. What percentage does the absolute temperature increase in this problem?

50 K / 273 K = 0.18, or 18%

Step #4. What percentage will the volume increase?

18%

Step #5. What is the volume at 50 °C?

323 K / 273 K x 1.0 L = 1.18 L
This is an 18% increase in volume.

Problem Six
A 10 L sample of a gas is at a pressure of 500 torr and 100 °C. If the volume is kept constant at 10 L, what temperature will produce a pressure of 760 torr?

Step #1. What variables are changing in this problem?

temperature and pressure

Step #2. What is the relationship between temperature and pressure at constant volume?

The pressure is directly proportional to the absolute temperature.

Step #3. By what factor must the absolute temperature increase in order to produce a pressure change from 500 torr to 760 torr?

760 torr / 500 torr = 1.52

Step #4. Calculate the temperature required.

(760 torr / 500 torr) x 373 K = 567 K

Problem Seven
A 10 L sample of a gas is at a pressure of 500 torr and 100 °C. What will the volume be at a pressure of 760 torr and 0.0 °C?

Step #1. What variables are changing in this problem?

volume, temperature, and pressure

Step #2. In previous problems, only two variables were changing. How do we deal with three?

Specifically, we will first change the temperature and look at the change in volume, and then apply the pressure change to the new volume.

Step #3. How will the volume of 10 L be affected by a change from 100 °C to 0 °C?

(10 L x 273 K) / 373 K = 7.3 L

Step #4. How will this volume of 7.3 L be affected by a change from 500 torr to 760 torr?

7.3 L x (500 torr / 760 torr) = 4.8 L

Problem Eight
A 10 L sample of a gas is at a pressure of 500 torr and 100 °C. How many moles of gas are present?

Step #1. What variable is unknown?

n, the number of moles

Step #2. When three of the four variables are known, how do we find the fourth?

by using the ideal gas equation, PV = nRT

Step #3. If R = 0.0821 L-atm/K-mole, then volume must be in units of liters, pressure in atmospheres, temperature in K. Solve for n.

n = PV/RT
= (500 torr/760 torr)(10 L) (0.0821 L-atm/K-mole)(373 K)
= 0.21 moles

Problem Nine
What pressure is required to produce a volume of 10 L at 100 °C for a sample of 1.6 g of methane (CH4)?

Step #1. How many moles of methane are we dealing with?

1.6 g / 16 g/mole = 0.10 mole

Step #2. Solve the ideal equation for P

PV = nRT, P = nRT/V

Step #3. Determine the pressure

P = nRTV = (0.10 mole)(0.0821 L-atm/K-mole)(373 L)10 L = 0.31 atm

Step #4. Express the pressure in torr

0.31 atm x 760 torr/atm = 233 torr

Problem Ten
What is the molecular weight of a 1.23 g sample of a gas that occupies a volume of 1.28 L at a temperature of 730 torr and 10.5 °C?

Step #1. Determine the number of moles of gas using the ideal gas equation.

PV = nRT
n = PVRT
= (730 torr/760 torr)(1.28 L)(0.0821 L-atm/K-mole)(283.5 K)
= 0.0528 moles

Step #2. How much does this 0.053 moles weigh?

1.23 g

Step #3. What is the molecular weight?

1.23 g / 0.053 moles = 23.3 g/mole

Problem Eleven
What is the molecular weight of a gas with a density of 2.09 g/L at 500 torr and 200 K?

Step #1. The density provides the mass of one liter. Calculate the number of moles of gas in one liter.

n = PVRT
= (500 torr,760 torr)(1.00 L)(0.0821 L-atm/K-mole)(200 K)
= 0.040 moles

Step #2. What is the mass of this number of moles?

2.09 g

Step #3. What is the molar mass (molecular weight)?

2.09 g 0.040 moles = 52 g/mole

Problem Twelve
A 0.2 mole sample of N2 is mixed with 0.4 mole of O2 at a total pressure of 2 atm. What is the partial pressure of N2?

Step #1. Determine the mole fraction of N2.