Gases and Kinetic Molecular Theory

Gas Velocity (root mean square velocity)

Another useful quantity is known as the root-mean-square speed. This quantity is interesting because the definition is hidden in the name itself. The root-mean-square speed is the square root of the mean of the squares of the velocities. It might seem like this technique of finding an average value is unnecessarily complicated since we squared all the velocities, only to later take a square root. You might wonder, ""Why not just take an average of the velocities?" But remember that velocity is a vector and has a direction. The average gas molecule velocity is zero, since there are just as many gas molecules going right (+ velocity) as there are going left (- velocity). This is why we square the velocities first, making them all positive. This ensures that taking the mean (i.e. average value) will not give us zero. Physicists use this trick often to find average values over quantities that can take positive and negative values. This is the equation you need to use: v = (3RT/M)¹/² v = root mean square velocity in m/sec R = ideal gas constant = 8.3145 (kg·m²/sec²)/K·mol T = absolute temperature in Kelvin M = mass of a mole of the gas in kilograms. Really, the RMS calculation gives you root mean square speed, not velocity. This is because velocity is a vector quantity, which has magnitude and direction. The RMS calculation only gives the magnitude or speed.The temperature must be converted to Kelvin and the molar mass must be found in kg (for units to cancel) to complete this problem.

Avogadro's Law

Avogadro's law states that, "equal volumes of all gases, at the same temperature and pressure, have the same number of molecules." For a given mass of an ideal gas, the volume and amount (moles) of the gas are directly proportional if the temperature and pressure are constant. The equation is V=nk where k is constant. This can be rewritten as V/n=k which means that volume and the amount of gas present is directly proportional. It can also be written as V₁/n₁=V₂/n₂. So, at constant pressure and temperature, we would have to change the volume by the same factor if we also change the amount of particles, and vice versa. This is so that we can maintain the pressure (volume is inversely proportional to pressure) and the temperature (both are proportional to temperature).

Boyle's Law

Boyle's law states that at constant temperature for a fixed mass, the absolute pressure and the volume of a gas are inversely proportional. This means that pressure always increases when volume decreases and vice versa. For example, if we double the pressure, we have to divide the volume by 2 and vice versa. The equation is p=k/V where k is a proportionality constant. This can be arranged as pV=k. So, assuming that temperature is constant, pressure multiplied by volume would always be the same value. This can also be written as p₁V₁=p₂V₂, in which the former describes the scenario before the change while the latter describes the scenario after the change. For example, imagine a piston able to compress air inside of it. Assuming that temperature is the same, we can that the pressure is relatively low while the volume is relatively high before the change. After the change, we can say that the pressure would now be high and the volume is low. In both scenarios, the ratio is the same (k:k) since both pV equals k in both cases. Let's analyze whether the equation pV=k is consistent with the ideal gas equation, pV=nRT. The number of particles, n, is a fixed number. R is always constant. Boyle's law requires temperature, T, to be the same. That means that everything on the right-hand side of pV = nRT is constant, and so pV is constant - which is what we have just said is a result of Boyle's Law.

Dalton's Law

Dalton's law of partial pressures states that the total pressure of a mixture of non-reacting gases is equal to the sum of the partial pressures of the component gases: Pₜ=P₁+P₂+P₃... where each subscript refers to the pressure that each gas would exert if were alone in the gas. When we solve partial pressure problems, we disregard the mass of the element and we don't take that into account. This is because at the same temperature, the different elements still apply the same amount of pressure despite the difference in mass. Larger elements move slower as opposed to smaller elements, but the larger elements' larger size makes up for the low velocity. Likewise, smaller elements' faster speed makes up for its relatively small mass. Remember, KE and velocity are not the same thing. The equation that connects both of these is K=0.5mv^2. So, if K is constant (temperature is constant) and m (mass) becomes larger, then v (velocity) decreases. In other words, larger particles travel more slowly than smaller particles at the same amount of kinetic energy. This explains why all elements still apply the same amount of pressure despite the mass difference, given that the temperature is constant. In fact, pressure isn't affected by the identity of the gas particles. We can solve pressure problems by solving for P as if the gases are in the container alone. Then we add the pressures together to get the total pressure exerted. Another important thing to note is that a gas's mole/temperature ratio to the total number of moles/temperature in the container is directly related to that same gas's partial pressure ratio to the total pressure exerted, assuming that all the other conditions are the same. This is reasonable since we already established that moles and partial pressure are directly proportional.

Standard Temperature and Pressure (STP)

STP in chemistry is the abbreviation for Standard Temperature and Pressure. STP most commonly is used when performing calculations on gases, such as gas density. The standard temperature is 273 K (0° Celsius or 32° Fahrenheit) and the standard pressure is 1 atm pressure. In STP, 1 mole of any gas will take up 22.4 L of the volume of the container. We get this value when we plug in the conditions given (273 K and 1 atm) to the ideal gas equation and solve for volume. Remember that this value is an approximation since this equation assumes we are dealing with an ideal gas. However, it is very close to the volume we actually get from the real gases based on experimental data.

Graham's law of effusion

States that the ratio of the effusion rates of two gases is the square root of the inverse ratio of their molar masses. The relationship is based on the postulate that all gases at the same temperature have the same average kinetic energy (recall that a result of the Kinetic Theory of Gases is that the temperature, in degrees Kelvin, is directly proportional to the average kinetic energy of the molecules.). We can write the expression for the average kinetic energy of two gases with different molar masses: 1/2xm1xv1^2=1/2xm2xv2^2. They are equal since the postulate is that the two gases are at the same temperature. Next, we simply multiply two on both sides to eliminate the 1/2. Keep in mind that velocity, v, is essentially the rate of effusion of the gas. Next, we divide one v on both sides (v1^2 or v2^2) to get m1=m2x(v2^2/v1^2). Then we divide m2 on both sides to get m1/m2=v2^2/v1^2. Finally, we take the square root of both sides to get sqrtroot of m1/m2=v2/v1.

Van der Waals equation

The Ideal Gas Law is based on the assumptions that gases are composed of point masses that undergo perfectly elastic collisions. However, real gases deviate from those assumptions at low temperatures or high pressures. Imagine a container where the pressure is increased. As the pressure increases, the volume of the container decreases. The volume occupied by the gas particles is no longer negligible compared to the volume of the container and the volume of the gas particles needs to be taken into account. Imagine for the moment that the atoms or molecules in a gas were all clustered in one corner of a cylinder. At normal pressures, the volume occupied by these particles is a negligibly small fraction of the total volume of the gas. But at high pressures, this is no longer true. As a result, real gases are not as compressible at high pressures as an ideal gas. The volume of a real gas is therefore larger than expected from the ideal gas equation at high pressures since it doesn't consider this now-relatively large volume occupied by the gas molecules. At low temperatures, the gas particles have lower kinetic energy and do not move as fast. The gas particles are affected by the intermolecular forces acting on them, which leads to inelastic collisions between them. This leads to fewer collisions with the container and a lower pressure than what is expected from an ideal gas.

Ideal Gas Law

The Ideal Gas Law is simply the combination of all Simple Gas Laws (Boyle's Law, Charles' Law, and Avogadro's Law). The equation is pV=nRT where p=pressure, V=volume, n=amount of gas particles in moles, R=universal gas constant, and T=temperature. To understand how this equation was derived, we have to look at the three simple gas laws. According to Boyle's law, pressure is inversely proportional to volume when the temperature and amount of particles are constant, or p=k/V. According to the Charles' Law, at constant pressure, the volume of a fixed mass of a gas is directly proportional to its absolute temperature, or V=T. According to the Avogadro's Law, at same temperature and pressure, an equal volume of gases contain an equal number of molecules, or V=n. From these three equations, we can deduce that V=nT/P. We add the universal constant and the equations then becomes V=R(nT/P). This can be rearranged as pV=nRT, the standard arrangement of the equation. You can also manipulate this equation to include density and molar mass. This can be done by substituting n with m/M, where m=mass and M=molar mass, since dividing mass by MM is the same thing as moles. So, you get pV=(n/M)RT, which can be rearranged into PM=DRT.

Maxwell-Boltzmann distribution

The air molecules surrounding us are not all traveling at the same speed, even if the air is all at a single temperature. Some of the air molecules will be moving extremely fast, some will be moving with moderate speeds, and some of the air molecules will hardly be moving at all. Because of this, we can't ask questions like "What is the speed of an air molecule in a gas?" since a molecule in a gas could have any one of a huge number of possible speeds. So instead of asking about any one particular gas molecule, we ask questions like, "What is the distribution of speeds in a gas at a certain temperature?" In the mid to late 1800s, James Clerk Maxwell and Ludwig Boltzmann figured out the answer to this question. Their result is referred to as the Maxwell-Boltzmann distribution, because it shows how the speeds of molecules are distributed for an ideal gas. The Maxwell-Boltzmann distribution is often represented with a graph. The speed located directly under the peak is the most probable speed, since it is the speed that is most likely to be found for a molecule in a gas. The average speed of a molecule in the gas is actually located a bit to the right of the peak. The reason the average speed is located to the right of the peak is due to the longer "tail" on the right side of the Maxwell-Boltzmann distribution graph. This longer tail pulls the average speed slightly to the right of the peak of the graph. The y-axis of the Maxwell-Boltzmann distribution graph gives the number of molecules per unit speed. The total area under the entire curve is equal to the total number of molecules in the gas. If we heat the gas to a higher temperature, the peak of the graph will shift to the right (since the average molecular speed will increase). As the graph shifts to the right, the height of the graph has to decrease in order to maintain the same total area under the curve. Similarly, as a gas cools to a lower temperature, the peak of the graph shifts to the left. As the graph shifts to the left, the height of the graph has to increase in order to maintain the same area under the curve.

Barometer

The classic mercury barometer is designed as a glass tube about 3 feet high with one end open and the other end sealed. The tube is filled with mercury. This glass tube sits upside down in a container, called the reservoir, which also contains mercury. The mercury level in the glass tube falls, creating a vacuum at the top. (The first barometer of this type was devised by Italian physicist and mathematician Evangelista Torricelli in 1643.) The barometer works by balancing the weight of mercury in the glass tube against the atmospheric pressure. Atmospheric pressure is basically the weight of air in the atmosphere above the reservoir, so the level of mercury continues to change until the weight of mercury in the glass tube is exactly equal to the weight of air above the reservoir. Once the two have stopped moving and are balanced, the pressure is recorded by "reading" the value at the mercury's height in the vertical column. If the weight of mercury is less than the atmospheric pressure, the mercury level in the glass tube rises (high pressure). In areas of high pressure, air is sinking toward the surface of the earth more quickly than it can flow out to surrounding areas. Since the number of air molecules above the surface increases, there are more molecules to exert a force on that surface. With an increased weight of air above the reservoir, the mercury level rises to a higher level. If the weight of mercury is more than the atmospheric pressure, the mercury level falls (low pressure). In areas of low pressure, air is rising away from the surface of the earth more quickly than it can be replaced by air flowing in from surrounding areas. Since the number of air molecules above the area decreases, there are fewer molecules to exert a force on that surface. With a reduced weight of air above the reservoir, the mercury level drops to a lower level. the height that the mercury reaches can be determined using the formula, p=dhg where p=pressure exerted by mercury column (equals atm pressure), d=density of fluid (mercury), h=height of mercury, and g=acceleration of gravity (9.81) using this equation, we find out that pressure at sea level (~101,325 Pa) corresponds to the pressure exerted by a column of mercury that is about 760 mm high. This is why 760 mmHg=~101,325 Pa. The units also work out.

Collision Theory

The collision theory explains that gas-phase chemical reactions occur when molecules collide with sufficient kinetic energy. The collision theory is based on the kinetic theory of gases; therefore only dealing with gas-phase chemical reactions are dealt with. Ideal gas assumptions are applied. Furthermore, we also are assuming: -All molecules are traveling through space in a straight line. -All molecules are rigid spheres. -The reactions concerned are between only two molecules. -The molecules need to collide. -In order to effectively initiate a reaction, collisions must be sufficiently energetic (kinetic energy) to break chemical bonds; this energy is known as the activation energy. -As the temperature rises, molecules move faster and collide more vigorously, greatly increasing the likelihood of bond breakage upon collision. In summary: 1. Molecules must collide with sufficient energy, known as the activation energy, so that chemical bonds can break. 2. Molecules must collide with the proper orientation. 3. A collision that meets these two criteria, and that results in a chemical reaction, is known as a successful collision or an effective collision.

Universal Gas Constant

The constant 'R' is there in the equation to allow for a direct relationship among the other values, (P, V, n, T). It is there to account for all the different units of measure: atmosphere, liter, mole, and kelvin. Without a constant, one could only describe the proportionality among the values of pressure, volume, number of moles, and temperature. In other words, the universal gas constant is simply a combination of all the proportionality constants of the three laws into one. Example: As we increase the number of apple trees in the orchard, we find we get more apples. As the number of trees decrease, we get fewer apples. Only when we determine how many apples grow on a tree (the 'apple' constant), can we then calculate how many more apples we will get with a given increase/decrease in the number of trees. With PV=nRT, proportionality will indicate that when a value on the left of the equal sign increases, either some value on the right will also increase (except for 'R') or the other value on the left will decrease (inversely proportional). The experimentally derived constant R allows us to calculate the amount of the change. Value of R WILL change when dealing with different unit of pressure and volume (Temperature factor is overlooked because temperature will always be in Kelvin instead of Celsius when using the Ideal Gas equation). Only through appropriate value of R will you get the correct answer of the problem. It is simply a constant, and the different values of R correlates accordingly with the units given. When choosing a value of R, choose the one with the appropriate units of the given information (sometimes given units must be converted accordingly).

Manometer

The purpose of a manometer is to measure the pressure of a contained gas. It consists of a U-shaped glass tube with one end open to the atmosphere, mercury in the bottom of the U and the gas to be measured in the other side of the U-tube. The gas is added then the glass tube is sealed on the gas side so the gas is trapped on one side of the mercury. The liquid mercury is equalized in the two sides of the tube at atmospheric pressure. When the gas is added, it will exert pressure on the mercury on the other side. If the pressure of the gas is equal to atmospheric pressure the mercury will remain at the same levels on both sides. If the gas is exerting more pressure on the mercury than the atmosphere exerts, it will cause the mercury to be higher on the atmosphere side than on the sealed side. The difference between the two heights tells you how much higher the gas pressure is than atmospheric pressure (using p=dhg or just converting units to mmHg). Adding the difference in pressure to the atmospheric pressure (this is usually measured in mmHg as well) will give the gas pressure (this makes sense since atm pressure is lower, so adding the difference gives you the higher gas pressure). For example, if the difference in height is, say, 20 cmHg, then you can convert that to mmHg and add the result to the atm pressure (assuming its also in mmHg, which it usually is) to get the gas pressure. If the gas is at a pressure below that of the atmosphere then the mercury will be lower on the gas side and the difference in height is subtracted from atmospheric pressure to get the gas pressure. There is also a type of manometer that has gas sealed on one side of the U-tube, mercury in the middle of the U and the other side evacuated and sealed (closed-end). Since there is no air pressure in the evacuated side the pressure of the gas is given by the difference between the height of mercury on the two sides (in this case, it would be just like measuring a barometer since it also has a vacuum on the other side). For example, if the height difference is 20 cmHg, you can either use p=dhg or, even easier, convert that into mmHg and that'll be the pressure of the gas. https://opentextbc.ca/chemistry/chapter/9-1-gas-pressure/

Ideal Gas

There is no such thing as an ideal gas, of course, but many gases behave approximately as if they were ideal at ordinary working temperatures and pressures. The assumptions are: -All collisions, both between the molecules themselves, and between the molecules and the walls of the container, are perfectly elastic. (That means that there is no loss of kinetic energy during the collision.) -There are no (or entirely negligible) intermolecular forces between the gas molecules. -The volume occupied by the molecules themselves is entirely negligible relative to the volume of the container (don't take space).

Charles' Law

This law states that for a fixed mass of gas at constant pressure, the volume is directly proportional to the kelvin temperature. That means, for example, that if you double the kelvin temperature from, say to 300 K to 600 K, *at constant pressure*, the volume of a fixed mass of the gas will double as well. This can be written as V=kT where k is constant. This can be rewritten as V/T=k. This would mean that at constant pressure, V/T is always equal to the same value since they are directly proportional. So, V₁/T₁=V₂/T₂. If we change temperature, the volume must also change by the same factor. For example, multiplying temperature by 2 would also mean multiplying volume by 2 as well if pressure is constant. This makes sense because if we want pressure to be the same and we increase the volume, we must also increase the temperature (more collisions) to make up for the larger space.