Unit 10: Gases
Unit Outline I. Introduction II. Gas Pressure III. Gas Laws IV. Gas Law Problems V. Kinetic-Molecular Theory of Gases VI. Real Gases
I. Opening thoughts Have you ever: Seen a hot air balloon? Had a soda bottle spray all over you? Baked (or eaten) a nice, fluffy cake? These are all examples of gases at work!
4 Characteristics of Gases Gases are highly compressible and occupy the full volume of their containers. Gases exert pressure, P = F/A (force/area). Gases always form homogeneous mixtures with other gases.
Properties of Gases You can predict the behavior of gases based on the following properties: Pressure Volume Amount (moles) Temperature Lets review each of these briefly
Pressure Pressure is defined as the force the gas exerts on a given area of the container in which it is contained. The SI unit for pressure is the Pascal, Pa. If you ve ever inflated a tire, you ve probably made a pressure measurement in pounds (force) per square inch (area).
Volume Volume is the three-dimensional space inside the container holding the gas. The SI unit for volume is the cubic meter, m 3. A more common and convenient unit is the liter, L. Think of a 2-liter bottle of soda to get an idea of how big a liter is. (OK, how big two of them are )
Amount (moles) As we ve already learned, the SI unit for amount of substance is the mole, mol. Since we can t count molecules, we can convert measured mass to the number of moles, n, using the molecular or formula weight of the gas. By definition, one mole of a substance contains approximately 6.022 x 10 23 particles of the substance.
Temperature Temperature is the measurement of heat or how fast the particles are moving. Gases, at room temperature, have a lower boiling point than things that are liquid or solid at the same temperature. Remember: Not all substance freeze, melt or evaporate at the same temperature. Water will freeze at zero degrees Celsius. However Alcohol will not freeze at this temperature.
II. Pressure Pressure is simply a force exerted over a surface area.
Pressure 760 mm (at sea level) Hg If a tube is inserted into a container of mercury open to the atmosphere, the mercury will rise 760 mm up the tube (at sea level).
Atmospheric Pressure and the Barometer. Standard atmospheric pressure is the pressure required to support 760 mm of Hg in a column. Units: 1 atm = 760 mmhg = 760 torr = 1.01325 10 5 Pa = 101.325 kpa.
II. Atmospheric Pressure P atm is simply the weight of the earth s atmosphere pulled down by gravity. Barometers are used to monitor daily changes in P atm. Torricelli barometer was invented in 1643.
II. Units of Pressure The derived SI unit for pressure is N/m 2, known as the pascal (Pa). Standard conditions for gases (STP) occurs at 1 atm and 0 C. Under these conditions, 1 mole of gas occupies 22.4 L.
How do they all relate? Some relationships of gases may be easy to predict. Some are more subtle. Now that we understand the factors that affect the behavior of gases, we will study how those factors interact.
III. Gas Laws Robert Boyle 1627-1691. Boyle s Law. Jacques Charles 1746-1823. Charles Law. J. Charles 1783. First ascent in hydrogen balloon. 16
III. Gas Laws A sample of gas can be physically described by its pressure (P), temperature (T), volume (V), and amount of moles (n). If you know any 3 of these variables, you know the 4 th. We look at the history of how the ideal gas law was formulated.
Boyle s Law This law is named for Charles Boyle, who studied the relationship between pressure, p, and volume, V, in the mid-1600s. Boyle determined that for the same amount of a gas at constant temperature, results in an inverse relationship: when one goes up, the other comes down. pressure volume
The Pressures-Volume Relationship: Boyle s Law (P vs. V at constant T) 19
What does Boyle s Law mean? Suppose you have a cylinder with a piston in the top so you can change the volume. The cylinder has a gauge to measure pressure, is contained so the amount of gas is constant, and can be maintained at a constant temperature. A decrease in volume will result in increased pressure.
Boyle s Law at Work Doubling the pressure reduces the volume by half. Conversely, when the volume doubles, the pressure decreases by half. MAIN PREVIOUS NEXT
Breathe Deeply! It s Boyle s Law! When the diaphragm contracts, the volume of the thoracic cavity increases The lungs expand and pressure decreases. Since P air >P lungs, air enters. When the diaphragm relaxes, the volume of the thoracic cavity decreases. The lungs contract and the pressure in the lungs increases. P lungs >P air, so air is exhaled.
III. Volume and Temperature Charles s Law The volume of a gas is directly related to its temperature, i.e. if T is increased, V will increase.
Charles Law This law is named for Jacques Charles, who studied the relationship volume, V, and temperature, T, around the turn of the 19 th century. This defines a direct relationship: With the same amount of gas he found that as the volume increases the temperature also increases. If the temperature decreases than the volume also decreases. volume temperature
Charles s Law
The Temperature-Volume Relationship: Charles Law 26
The Absolute Temperature Scale Temperature measures average Kinetic Energy of particles: KE = 3 RT 2 When motion stops, particles have no kinetic energy. This means there must be an absolute zero temperature! The Kelvin temperature scale starts at Absolute Zero: K = C + 273.15
What does Charles Law mean? Suppose you have that same cylinder with a piston in the top allowing volume to change, and a heating/cooling element allowing for changing temperature. The force on the piston head is constant to maintain pressure, and the cylinder is contained so the amount of gas is constant. An increase in temperature results in increased volume.
Charles Law at Work As the temperature increases, the volume increases. Conversely, when the temperature decreases, volume decreases.
III. The Combined Gas Law Boyle s and Charles s Laws can be combined into a convenient form.
III. Volume and Moles Avogadro s Law The pressure of a gas is directly related to the number of moles of gas, i.e. if n increases, V will increase.
The Quantity-Volume Relationship: Avogadro s Law equal volumes of any gas at the same temperature and pressure will contain the same number of molecules. Same number of particles (same T and P) V = constant n at a constant P and T
33 V = constant n at a constant P and T 22.4 L = constant 1 mole at a 1 atm and 273 K
Avogadro s Law (n, V) Gay-Lussac s Law (T, P) Avogadro showed that the volume of a gas varies directly with the amount of gas (# of moles) Thus, a similar relationship exists as in Charles s Law: V V V = k or = n n n 1 2 1 2 Pressure & Temperature are held constant here! Gay-Lussac studied how temperature affects the pressure of a gas. He discovered a direct relationship! P P P = k or = T T T 1 2 1 2 Moles & Volume are held constant here!
Gas Laws Summary BOYLE CHARLES AVOGADRO GAY-LUSSAC
The Combined Gas Law The gas laws can be combined into one equation. Volume and pressure vary inversely, while volume varies directly with moles and temperature: PV 1 1 2 2 n T = P V n T 1 1 2 2 When variables are held constant, they can be deleted from the combined law this produces all four gas laws we studied earlier.
Ideal Gas Law, cont d We can rewrite the combined law in a form that is known as the Ideal Gas Law: PV = nrt The value of the Ideal Gas Law over the previous laws is that only ONE set of conditions is required if 3 of the variables are known, the 4 th can be calculated. Use substitution and some algebra to derive the related equations from the Ideal Gas Law: mrt M = and d = PV PM RT
Standard Molar Volume: 22.4 L @ STP
39 The Ideal Gas Equation Summarizing the Gas Laws Boyle: V 1/P (constant n, T) Charles: V T (constant n, P) Avogadro: V n (constant P, T). Combined: V nt P Ideal gas equation R = ideal gas constant V R nt P
III. The Ideal Gas Law The ideal gas law is a combination of the combined gas law and Avogadro s Law. R = 0.082058 L atm/k mole
41 The Ideal Gas Equation Ideal gas equation: PV = nrt R = gas constant = 0.08206 L atm/mol-k. We define STP (Standard Temperature and Pressure) = 0 C (273.15 K) = 1 atm. Volume of 1 mol of gas at STP is 22.4 L.
IV. Gas Law Problems There are many variations on gas law problems. A few things to keep in mind: 1) Temperature must be in Kelvin 2) If problem involves a set of initial and final conditions, use combined gas law. 3) If problem only gives information for one set of conditions, use ideal gas law.
IV. Sample Problem What s the final pressure of a sample of N 2 with a volume of 952 m 3 at 745 torr and 25 C if it s heated to 62 C with a final volume of 1150 m 3?
IV. Sample Problem What volume, in ml, does a 0.245 g sample of N 2 occupy at 21 C and 750 torr?
IV. Sample Problem A sample of N 2 has a volume of 880 ml and a pressure of 740 torr. What pressure will change the volume to 870 ml at the same temperature?
IV. Other Uses of Ideal Gas Law The ideal gas law can be used to find other physical values of a gas that are not as obvious. gas density, d = mass/volume gas molar mass, MW = mass/mole stoichiometry, via moles and a balanced equation
IV. Sample Problem Find the density of CO 2(g) at 0 C and 380 torr.
IV. Sample Problem How many ml of HCl (g) forms at STP when 0.117 kg of NaCl reacts with excess H 2 SO 4? H 2 SO 4(aq) + 2NaCl (s) Na 2 SO 4(aq) + 2HCl (g)
Dalton s Law The total pressure of a mixture of gases equals the sum of the partial pressures of the individual gases. P total = P 1 + P 2 +... When a H 2 gas is collected by water displacement, the gas in the collection bottle is actually a mixture of H 2 and water vapor.
Dalton s Law of Partial Pressures In a mixture of gases, the TOTAL pressure of gas is the sum of the pressures caused by each gas (the partial pressures): P Total = P 1 + P 2 + P 3 + The MOLE FRACTION ( )of a gas in a mixture can be calculated in two different ways, then: n P = = so we get: P = P A A A A A Total ntotal PTotal
Dalton s Law Illustrated
B. Dalton s Law Hydrogen gas is collected over water at 22.5 C. Find the pressure of the dry gas if the atmospheric pressure is 94.4 kpa. The total pressure in the collection bottle is equal to atmospheric pressure and is a mixture of H 2 and water vapor. GIVEN: P H2 =? P total = 94.4 kpa P H2O = 2.72 kpa Look up water-vapor pressure for 22.5 C. WORK: P total = P H2 + P H2O 94.4 kpa = P H2 + 2.72 kpa P H2 = 91.7 kpa Sig Figs: Round to least number of decimal places.
B. Dalton s Law A gas is collected over water at a temp of 35.0 C when the barometric pressure is 742.0 torr. What is the partial pressure of the dry gas? The total pressure in the collection bottle is equal to barometric pressure and is a mixture of the gas and water vapor. GIVEN: P gas =? P total = 742.0 torr P H2O = 42.2 torr Look up water-vapor pressure for 35.0 C. WORK: P total = P gas + P H2O 742.0 torr = P H2 + 42.2 torr P gas = 699.8 torr Sig Figs: Round to least number of decimal places.
V. Kinetic-Molecular Theory (KMT)
Kinetic Molecular Theory Particles in an ideal gas have no volume. have elastic collisions. are in constant, random, straight-line motion. don t attract or repel each other. have an avg. KE directly related to Kelvin temperature.
Imagining a Sample of Gas We imagine a sample of gas chaos, molecules bumping into each other constantly. After a collision, 2 molecules may stop completely until another collision makes them move again. Some molecules moving really fast, others really slow. But, there is an average speed.
57 Kinetic-Molecular Theory For gases, there is a range of velocities and energies at each temperature. N 2 0 o C 100 o C
Gas Molecular Speeds As temp increases, avg. speed increases. i.e. avg. KE is related to temp!! Any 2 gases at same temp will have same avg. KE!
Molecular Speeds The average kinetic energy per mole of gas can be calculated in two different ways: 1 3 2 2 2 KE = Mv = RT We can rearrange and solve for v, the velocity of a gas particle: In order to get a velocity in ms -1, we must use SI units for molar mass, kg mol -1. The gas constant (R) must be the SI value, 8.314. v = 3RT M Don t forget: Molar Mass in Kg for this equation!
Graham s Law Diffusion Spreading of gas molecules throughout a container until evenly distributed. Effusion Passing of gas molecules through a tiny opening in a container
Graham s Law Speed of diffusion/effusion Kinetic energy is determined by the temperature of the gas. At the same temp & KE, heavier molecules move more slowly. Larger m smaller v KE = ½mv 2
Graham s Law Graham s Law Rate of diffusion of a gas is inversely related to the square root of its molar mass. The equation shows the ratio of Gas A s speed to Gas B s speed. v v A B m m B A
Graham s Law Determine the relative rate of diffusion for krypton and bromine. The first gas is Gas A and the second gas is Gas B. Relative rate mean find the ratio v A /v B. v v A B m m B A vkr Br2 v Br 2 m m Kr 159.80 g/mol 1.381 83.80 g/mol Kr diffuses 1.381 times faster than Br 2.
Graham s Law A molecule of oxygen gas has an average speed of 12.3 m/s at a given temp and pressure. What is the average speed of hydrogen molecules at the same conditions? v v A B m m B A v H 2 12.3m/s 32.00 g/mol 2.02 g/mol v v H O 2 2 m m O H 2 2 Put the gas with the unknown speed as Gas A. v H 2 12.3m/s v H 49.0 2 3.980 m/s
v v A B An unknown gas diffuses 4.0 times faster than O 2. Find its molar mass. m m B A va O2 v O 2 m m A Graham s Law The first gas is Gas A and the second gas is Gas B. The ratio v A /v B is 4.0. Square both sides to get rid of the square root sign. m A 4.0 16 32.00 16 32.00 g/mol 32.00 g/mol m A m A g/mol 2.0 g/mol 2
Why is Diffusion so Slow?? If molecular speeds are so incredibly fast, why does a gas take so long to diffuse? The answer is in the completely random path a gas particle takes as it diffuses. The gas particle constantly changes direction when it collides with another particle This slows down its outward diffusion immensely! The MEAN FREE PATH is the distance a particle travels before colliding with another particle.
Effusion Gas moving through a pin-hole into a vacuum The rate of effusion: v = 3RT M Temp in Kelvin Molar mass in kg mol -1 Rate in ms -1
Graham s Law of Effusion Graham compared the rates of effusion for two gases at the same temperature. He derived the equation: v v = M M 1 2 2 1 Here, Molar Mass can be left in grams Can you explain why?? Graham s law is important because it can be used to determine the Molar Mass of an unknown gas if you compare its rate of effusion with the rate of a known gas under the same conditions!
Why Do Gas Laws Work So Well? Recall that the gas laws apply to any gas the chemical identity is not important. Gas particles only interact when they collide. Since this interaction is so short, chemical properties don t have time to take effect!!
VI. Deviations from PV=nRT Under extreme conditions (high P or low T), gases deviate from ideal gas law predictions. Why? What s so different about these conditions?
Real Gases Particles in a REAL gas have their own volume attract each other Gas behavior is most ideal at low pressures at high temperatures in nonpolar atoms/molecules
72 Real Gases: Deviations from Ideal Behavior (Temperature and Pressure Effects) As temperature increases, the gas molecules move faster and further apart. Also, higher temperatures mean more energy available to break intermolecular forces. Therefore, the higher the temperature, the more ideal the gas. As pressure increases, gas molecules are closer together making the gas less ideal. Therefore, the lower the pressure, the more ideal the gas.
Gas Particle Volume Gas molecules do take up space! When very close to one another, entire volume of container is not available for travel, so actual volume of gas is larger.
Intermolecular Forces Gas molecules interact if they are very close to one another
VI. van der Waals Equation Under extreme conditions, ideal gas law cannot be used. correction terms for P and V
76 Real Gases: Deviations from Ideal Behavior The van der Waals Equation We add two terms to the ideal gas equation, one to correct for volume of molecules, and the other to correct for intermolecular attractions The correction terms generate the van der Waals equation: nrt P nb V 2 V where a and b are empirical constants. n a 2