Properties of Gases Notes
Properties of Gases: The Air We Breathe
Intended Learning Outcomes (ILOs)
- Apply the Ideal Gas Law to determine properties of gases.
Properties of Gases
- How are Gases unique?
- A gas expands spontaneously to fill its container.
- Gases are highly compressible.
- Gases form homogenous mixtures with other gases, regardless of identity or composition of the gas.
- At the same temperature and pressure:
- Equal volumes = equal number of molecules gas.
- These properties are unique to gases and do not apply to liquids or solids.
- Properties of gases:
- Volume is inversely proportional to pressure.
- Pressure is directly proportional to temperature.
- Pressure is directly proportional to quantity of gas.
- Gases are miscible.
- Densities are directly proportional to molar mass.
- Gases expand to occupy entire volume of their container.
- Gases effuse at rates inversely proportional to molar mass.
Pressure of Gases
- Pressure of a gas is the collective force particles exert on the walls of the container.
- More collisions cause more pressure.
- More particles cause more pressure.
- Smaller volume causes more pressure.
- Higher temperature causes more pressure.
Kinetic Molecular Theory (KMT)
- The behavior of gases can be described using kinetic molecular theory (KMT), which is based upon four assumptions:
- Gas molecules have tiny volumes compared with volume the gas occupies.
- Gases are mostly empty space
- Gas particles do not interact with each other.
- We assume there are NO interactions between gas particles, no IMFs
- Gas molecule collisions are elastic.
- Gas particles collide with each other constantly, but the collisions are elastic, meaning no loss of energy
- The average kinetic energy of the molecules in a gas are proportional to the absolute temperature.
- No matter the gas, if the temperature is the same, the average kinetic energy is the same.
Particle Speed
- u_m – The most likely speed for a gas particle. The speed the most particles are moving at.
- u{avg} – The average speed of all the particles. Slightly higher than the um.
- u_{rms} – Root mean square speed. Square root of the average of the squared speeds of all particles.
KE and u_{rms}
- KE: Kinetic Energy
- m: mass
- u_{rms}: root-mean-square speed
- At the same temperature, all gas particles, regardless of element, have the same KE_{avg}, but that does NOT mean, the same speed.
- Heavier molecules have slower average speeds.
- KE{avg} = \frac{1}{2} m u{rms}^2
Comparing Particle Speeds
- Let’s compare He and N2. According to KMT, all gases at the same temp have the same KE, but not the same speed
- KE{avg} = \frac{1}{2} m u{rms}^2
- KE{He} = KE{N_2}
- \frac{1}{2} m{He} u{rms,He}^2 = \frac{1}{2} m{N2} u{rms,N2}^2
- \frac{u{rms,He}}{u{rms,N2}} = \sqrt{\frac{m{N2}}{m{He}}}
- \frac{u{rms,He}}{u{rms,N2}} = \sqrt{\frac{\mathcal{M}{N2}}{\mathcal{M}{He}}}
- \mathcal{M} = molar \, mass
- He travels 2.646 times faster than N2.
- \frac{u{rms,He}}{u{rms,N2}} = \sqrt{\frac{\mathcal{M}{N2}}{\mathcal{M}{He}}} = \sqrt{\frac{28.02 \, g/mol}{4.003 \, g/mol}} = 2.646
Effusion
- Process by which gas molecules escape from a container through small holes to a region of lower pressure.
- Graham’s Law of Effusion
- \frac{effusion \, rateA}{effusion \, rateB} = \sqrt{\frac{\mathcal{M}B}{\mathcal{M}A}}
- \frac{u{rms,A}}{u{rms,B}} = \sqrt{\frac{\mathcal{M}B}{\mathcal{M}A}}
Determining Particle Speeds
- KE{avg} = \frac{1}{2} m u{rms}^2
- \frac{1}{2} m u{rms}^2 = \frac{3}{2} kB T
- KE{avg} = \frac{3}{2} kB T
- u{rms} = \sqrt{\frac{3kBT}{m}}
- u_{rms} = \sqrt{\frac{3RT}{\mathcal{M}}}
- R = 8.314 \frac{kg \cdot m^2}{s^2 \cdot mol \cdot K}
Practice
- Calculate the root-mean-square speed of Nitrogen gas molecules at 25℃.
- u_{rms} = \sqrt{\frac{3RT}{\mathcal{M}}}
- R = 8.314 \frac{kg \cdot m^2}{s^2 \cdot mol \cdot K}
Diffusion
- Spread of one substance (usually gas) through another substance.
- Mean Free Path: average distance a particle can travel through a gas before colliding with another particle.
- At 1 atm is about 6.8 x10^{-8} m (about 10^{10} collisions per second)
- Gases will spontaneously fill a space through diffusion and effusion, moving from high concentration to low concentration.
Atmospheric Pressure
- Produced by Earth’s gravity pulling with force on the air molecules.
- At higher altitudes, there are less gas molecules, and therefore lower pressure.
Atmospheric Pressure Effects
- Differences in pressure produces wind. Remember that gas will spontaneously move from high concentrations to lower concentrations.
- High pressure: usually associated with clear weather.
- Low pressure: usually associated with unstable weather.
Measuring Pressure
- Barometer
- Pressure of the atmosphere on the mercury pool pushes the mercury up the tube. Measuring that push up the tube determines the atmospheric pressure
- Manometer
- When the valve is closed, Hg levels are equal.
- Open valve and pressure will push against the atm pressure.
- Measure the difference between the two to determine the pressure of a sample
Measuring Pressure
- Standard pressure is the pressure at sea level:
Partial Pressures
- Gases create homogeneous mixtures
- So, when a gas exerts a pressure, all the component gases are actually individually exerting their own pressures.
- Dalton’s Law of Partial Pressures
- P{Total} = P1 + P2 + P3 …
- Where P1, P2, P_3, etc. are the partial pressures of the individual gases.
Partial Pressures
- Can also calculate the partial pressure for an individual gas if you know how many moles of it are present in the total mixture (mole fraction).
- P1 = \chi1 \times P_{Total}
- \chi_1 = \frac{moles \, of \, 1}{Total \, moles}
Partial Pressures
- One way to measure the partial pressure of a gas is over water.
- P{total} = P{dry \, gas} + P{H2O}
Simple Gas Laws
- Boyle’s Law – Pressure and Volume
- Charles’s Law – Volume and Temperature
- Amonton’s Law – Pressure and Temperature
- Avogadro’s Law – Volume and Moles
- When comparing two CHANGING properties, the others stay CONSTANT
Boyle’s Law
- The volume of a gas is inversely proportional to the pressure.
- As volume goes up, pressure goes down.
- P1V1 = P2V2
Charles’s Law
- The volume of a gas is directly proportional to temperature.
- As temperature goes up, volume goes up.
- \frac{V1}{T1} = \frac{V2}{T2}
Amonton’s Law
- The pressure of a gas is directly proportional to temperature.
- As temperature goes up, pressure goes up.
- \frac{P1}{T1} = \frac{P2}{T2}
Combined Gas Law
- Boyle’s, Charles’s, and Amonton’s Laws can be combined to produce the Combined Gas Law
- This is useful if P, V, and T are all changing. In the individual laws, 2 things can change while one has to be held constant.
- If you know the Combined Gas Law, you can derive Boyle’s, Charles’s, or Amonton’s
- \frac{P1V1}{T1} = \frac{P2V2}{T2}
Avogadro’s Law
- The volume of a gas is directly proportional to the number of moles of the gas.
- As the moles of a gas go up, the volume goes up.
- At STP: 22.4 L = 1 mole of gas
- Applies to ALL gases
- \frac{V1}{n1} = \frac{V2}{n2}
Ideal Gas Law
- Boyle’s, Charles’s, Amonton’s, and Avogadro’s Laws can be combined along with a constant to produce the Ideal Gas Law
- For this law, if you know 3 variables, you can determine the fourth.
- If you know the Idea Gas Law, you can derive Boyle’s, Charles’s, Amonton’s, or Avogadro’s.
- PV = nRT
What is an Ideal Gas
- 2 Main Assumptions:
- The volume of individual gas particles are insignificant.
- Gas particles DO NOT interact with each other.
- PV = nRT
Ideal Gas Constant
- R is known as the gas constant.
- Can have many combinations of units
- The units in the constant you use MUST match your units in your measurements/variables
- You can either choose your constant, or assure that your measurements are using the same units as your constant
- PV = nRT
Volume at STP
- 1 mole of ANY gas at STP is 22.4L
- PV = nRT
- \frac{V}{n} = \frac{RT}{P} = \frac{(0.0821 \frac{L \cdot atm}{mol \cdot K})(273 \, K)}{1 \, atm} = \frac{22.4 \, L}{1 \, mol}
Density of an Ideal Gas
- Density is mass/volume
- For an Ideal Gas, volume is the same from gas to gas, but mass is different, so therefore, Density is as well.
- Can use molar mass/molar volume instead of just mass/volume
- Density = \frac{molar \, mass}{molar \, volume} = \frac{mass/mol}{volume/mol}
Density of an Ideal Gas
- In the Ideal Gas Equation, we know that n (moles) is the same as grams/molar mass, so we can substitute it into the Ideal Gas equation to solve for density.
- Can also be rearranged to solve for molar mass:
- \frac{m}{V} = \frac{P (\mathcal{M})}{RT}
- \mathcal{M} = \frac{d (RT)}{P}
Stoichiometry Involving Gases
- You now have a new conversion factor you can use in stoichiometry calculations.
- H2 + O2 \rightarrow H_2O