## 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?**

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22.4 L

**Step #2. What is standard pressure and temperature?**

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One atmosphere and 0 °C (273 K)

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

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6 x 10^{23} molecules

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

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22.4 L/ 6 x 10^{23} molecules = 4 x 10^{-23} L

**Step #5. Express this volume in cubic centimeters.**

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One liter is 1000 cm^{3}. Thus, the volume is:

4 x 10^{-23} L x 1000 cm^{3}/L = 4 x 10^{-20} cm^{3}

**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?**

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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)?**

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One mole of liquid methane has a volume of:

16 g / 0.8 g/mL = 20 mL = 20 cm^{3}. Thus:

20 cm^{3} / 6 x 10^{23} molecules = 3 x 10^{-23} cm^{3}/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?**

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volume, pressure, temperature, and number of moles

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

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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?**

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the volume decreases

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

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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?**

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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.**

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V = k/P

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

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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**

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

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

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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?**

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V = k/P. They are indirectly proportional.

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

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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?**

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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?**

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volume and temperature

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

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Volume is directly proportional to absolute temperature.

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

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50 K / 273 K = 0.18, or 18%

**Step #4. What percentage will the volume increase?**

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18%

**Step #5. What is the volume at 50 °C?**

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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?**

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temperature and pressure

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

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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?**

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760 torr / 500 torr = 1.52

**Step #4. Calculate the temperature required.**

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(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?**

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volume, temperature, and pressure

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

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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?**

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(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?**

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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?**

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n, the number of moles

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

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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.**

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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 (CH_{4})?

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

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1.6 g / 16 g/mole = 0.10 mole

**Step #2. Solve the ideal equation for P**

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PV = nRT, P = nRT/V

**Step #3. Determine the pressure**

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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**

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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.**

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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?**

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1.23 g

**Step #3. What is the molecular weight?**

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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.**

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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?**

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2.09 g

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

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2.09 g 0.040 moles = 52 g/mole

Problem Twelve

A 0.2 mole sample of N_{2} is mixed with 0.4 mole of O_{2} at a total pressure of 2 atm. What is the partial pressure of N_{2}?

**Step #1. Determine the mole fraction of N _{2}.**

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0.2 mole / 0.6 mole = 0.33

**Step #2. Determine the partial pressure of N _{2}.**

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P = 0.33 x 2 atm = 0.66 atm