Gas Laws

Combined Gas Law Problems - Solutions

 

 

Boyle's Law 

1)         If I initially have a gas at a pressure of 12 atm, a volume of 23 liters, and a temperature of 200 K, and then I raise the pressure to 14 atm and increase the temperature to 300 K, what is the new volume of the gas? 29.6 L

 

2)         A gas takes up a volume of 17 liters, has a pressure of 2.3 atm, and a temperature of 299 K.  If I raise the temperature to 350 K and lower the pressure to 1.5 atm, what is the new volume of the gas?  30.5 L   

 

Combined Gas Laws

 

3)         1.00 L of a gas at standard temperature and pressure is compressed to 473 mL.  What is the new pressure of the gas?  2.11 atm

 

4)         In a thermonuclear device, the pressure of 0.050 liters of gas within the bomb casing reaches 4.0 x 106 atm.  When the bomb casing is destroyed by the explosion, the gas is released into the atmosphere where it reaches a pressure of 1­­­.00 atm.  What is the volume of the gas after the explosion?            2.0 x 10­5 L

 

Ideal Gas Law

 

5)         If I have 4 moles of a gas at a pressure of 5.6 atm and a volume of 12 liters, what is the temperature?  205 K

 

6)         If I have an unknown quantity of gas at a pressure of 1.2 atm, a volume of 31 liters, and a temperature of 87 0C, how many moles of gas do I have?            1.26 moles

 

Charles' Law

 

7)         The temperature inside my refrigerator is about 40 Celsius.  If I place a balloon in my fridge that initially has a temperature of 220 C and a volume of 0.5 liters, what will be the volume of the balloon when it is fully cooled by my refrigerator?  0.47 L

 

8)         A man heats a balloon in the oven.  If the balloon initially has a volume of 0.4 liters and a temperature of 20 0C, what will the volume of the balloon be after he heats it to a temperature of 250 0C?  0.71 L

 

 

 

 

THE IDEAL GAS

 

Lecture: The Ideal Gas (working with N,P,V, and T)

 

 

 

A gas is a substance that has neither a fixed volume nor a fixed shape. A gas is able to diffuse itself through any available space, even when another gas is present. This means that gas particles must be small and fast moving. (1 X 10-5 mm in diameter and move as fast as 1 mile per second).

 

The term gas in this course refers to any substance that is a gas at normal temperatures and normal pressures.

 

Measurable Properties of Gases (N,P,V,T)

 

1. Weight and number of molecules

 

a. We can measure the weight of a gas by weighing a container before and after removing the gas.

 

b. The number of molecules of the gas can be calculated by using the gram molecular weight value for the gas.

 

c. The relative number of molecules in samples of different gases can also be calculated.

 

1) The weights of two different gases containing the same number of molecules are directly proportional to their molecular weights. (Oxygen and hydrogen are a 16-1 ratio)

 

2) The numbers of molecules present in the same weights of two different gases are inversely proportional to the molecular weights of the gases.

 

2. Pressure exerted by a gas

 

a. Pressure is an expression of force per unit area and is a result of the molecules of the gas striking the sides of the container.

 

b. Gas pressures are commonly expressed in terms of atmospheres. 1 atmosphere (14.7 lb/in2) of pressure will support a column of mercury 76 cm high.

 

1) If the column of mercury is 1 cm2 in cross section, it will have a volume of 76 cm3.

 

2) Since the density of mercury is 13.6 g/cm3, the mass of the mercury is 1034 grams.

 

3) Since 1 gram of force = 980 dynes of force, the force of 1 atmosphere = 1034 g./cm.2 X 980 dynes/g. = 1.103 X 106 dynes/cm.2

 

  One dyne is the force that accelerates a mass of one gram at the rate of one centimeter per second per second. Expressed in SI units, the dyne equals 10-5newton.

 

  The pascal is the standard pressure unit in the MKS metric system, equal to one newton per square meter or one "kilogram per meter per second per second."

 

  Sounds impressive, but in traditional English terms a pascal is only 0.000 145 pounds per square inch (0.020 885 lbf/ft2 or 0.007 50 mmHg).

 

  Thus pressure is more commonly measured in kilopascals (kPa), with 1 kPa = 0.145 lbf/in2.

 

  Air pressure is also measured in hectopascals (hPa), with 1 hPa = 1 millibar.

 

  The unit is named for Blaise Pascal (1623-1662), French philosopher and mathematician, who was the first person to use a barometer to measure differences in altitude.

 

 

 

3. The volume of a gas

 

c. The volume occupied by a gas is determined by the size of the container and is measured in liters or cm.3.

 

4. The temperature of a gas

 

            a. We simply use a thermometer to measure the temperature of a gas.

 

b. The absolute scale of temperature is used when working with relationships among the N,P,V, and T of gases.

 

c. The absolute scale was determined experimentally when Charles determined that if the pressure and weight of a gas are kept constant and the temperature of the gas was increased from 0oC, the volume doubled after the temperature was increased by 273Co.

 

1) Lord Kelvin modified this scale into the present absolute or Kelvin temperature scale where absolute zero equals -273oK.

 

2) Charles’ Law states that ‘The volume of a gas is directly proportional to the absolute temperature when P and N are kept constant.’ Since there is no ideal gas, there is no way for a gas to reach absolute zero since they all liquefy before then.

 

 

 

The Kinetic Molecular Theory

 

1. Years of experimentation by many early scientists laid the groundwork for the development of a theory that would describe the behavior of matter in the gaseous state. This theory applies to what we call the ideal gas. Real gases deviate from this theory in varying degrees.

 

a. Gases are composed of extremely small molecules (billions in the tiniest possible volume that can be observed) separated by distances that are very large compared to their sizes. A volume of gas is mainly space.

 

b. Molecules of gas are in constant motion in straight lines except when they collide with each other or the walls of their container.

 

c. Collisions only change molecules directions and no energy is lost. If one molecule loses energy, it is taken on by the other molecule.

 

d. The average kinetic energy (K.E. = ½ mv2) of the molecules increases in direct proportion to the temperature of the gas on the absolute scale.

 

      e. Molecules of an ideal gas exert neither an attractive nor a repulsive force upon    each other.

 

2. Explaining the behavior of gases

 

a. Diffusion occurs because the molecules are moving fast in straight lines through lots of empty space.

 

b. Gas exerts pressure on a container because the molecules are constantly bombarding the surfaces inside the container and the pressure is the same for different masses of gases at the same temperature (more massive moves slower so hits with the same force).

 

1) Dalton’s Law of Partial Pressures says that- In any mixture of gases, the total pressure is equal to the sum of the partial pressures exerted by the individual gases. The partial pressure of any one gas is proportional to its percentage of molecules in the total mixture.

 

2) Boyle’s Law states that- The volume, V, of a fixed weight of gas, N, held at a constant temperature, T, varies inversely with the pressure. (Double the pressure, the volume is reduced by half so that the gas molecules can hit the surfaces twice as many times.)

 

3) Gay-Lussac’s Law states- If the volume remains constant, the pressure exerted by a fixed weight of gas is directly proportional to the temperature expressed on the absolute scale. (Doubling the kinetic energy of a gas molecule, doubles the v2 since the mass stays the same. So in a fixed volume, the pressure would be directly proportional to the absolute temperature.)

 

            a) P1/P2 = T1/T2

 

4) Charles’ Law kept the pressure constant so doubling the v2 meant the volume had to increase to make the molecules travel through more space before impacting a surface of the container.

 

            a) V1T2 =V2T1

 

5) Graham’s Law of Diffusion states- Under the same conditions of temperature and pressure, the rates of diffusion of gases are inversely proportional to the square roots of the molecular weights of the gases. (Comparing hydrogen and oxygen- H2 = 2 g. mol. wgt. while O2 = 32 g. mol. wgt. Their g. mol. wgt. differ by a factor of 16 so their rates of diffusion differ by a fact of 4. The lighter gas always diffuses fastest.

 

 

 

 

 

Non-Ideal Behavior

The Kinetic Theory makes several assumptions about an ideal gas. These cause problems because real gases are not ideal. The main causes of error are related to pressure and temperature.

Pressure
At high pressures, the behavior of real gases changes dramatically from that predicted by the Ideal Gas Law. Under 10 atmospheres of pressure or less, Ideal Gas Law predictions are very close to real amounts and do not generate serious error.

Temperature
When the temperature of a gas is close to its liquefaction point, the behavior is very different from Ideal Gas Law predictions. With increasing temperatures, the Ideal Gas Law predictions become close to real values.

Why?
The answer is simple: ideal gases have molecular volume and show no attraction between molecules at any distance; real gas molecules have volume and show attraction at short distances. Let us first consider what pressure does. Pressure at high degrees will bring the molecules very close together. This causes more collisions and also allows the weak attractive forces to come into play. With low temperatures, the molecules do not have enough energy to continue on their path to avoid that attraction.

 

 

 

 

 

 

Ideal Gas Law Problems

 

When calculating the effect of changes in N, P, V, and T, the effect of changing one factor on any other one factor can be calculated as long as the other two are kept unchanged.

 

http://jersey.uoregon.edu/vlab/Piston/

 

 

 

6) Universal Gas Law

 

            a) P1V1/T1 = P2V2/T2

 

                        When T1 = T2 (Boyle’s Law)                                    When P1 = P2 (Charle’s Law)                                    When V1 = V2 (Gay Lussac’s Law)

 

PV=nRT

 

 

First, we'll go over the parts of the equation, PV=nRT. P is pressure. Pressure can be in either atmospheres (atm) or kilopascals (kPa). V is volume in liters (L). n is the number of moles of the gas. Because moles of a substance are determined by mass divided by molecular mass, it can create an interesting variant we will discuss later. R is the Ideal Gas Constant. Depending on whether atmosphers or kilospascals were used, the value is either 0.0821 L-atm/mol-K or 8.31 L-kPa/mol-K, respectively. Temperature is in absolute degrees Kelvin.