WBCHSE Class 11 Chemistry Notes On Kinetic Molecular Theory Of Gases
The gas laws that we have studied so far, such as Boyle’s law, Charles’s law, and Dalton’s law, describe the relationships between the macroscopic properties of gases, and their behavior as observed in experiments.
They tell us how gases behave, but not why they behave the way they do. Experiments are a vital aspect of science because they tell us how nature behaves.
- But science has another aspect. The scientist is curious to know why or what makes nature behave in a particular way. To answer the whys of science, scientists construct theories or models. Often, a theory is built around an inspired guess. Such theories make certain assumptions and attempt to explain the behavior of nature observed in experiments. If the predictions made by a theory fit experimental observations, the theory is accepted.
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“WBCHSE Class 11 Chemistry, kinetic molecular theory of gases, notes and key concepts”
- The kinetic molecular theory of gases was developed to explain the (observed) behavior of gases. J C Maxwell (British physicist), L E Boltzmann (Austrian physicist), R J E Clausius (German physicist), and James Joule (British physicist) were some of the scientists who contributed to building this theory. The model we will consider was an improvement of an earlier model of gases put forward by a Swiss mathematician called Bernoulli.
Assumptions of kinetic theory The kinetic theory is based on the microscopic model of a gas. It makes certain assumptions about the particles (atoms or molecules) that constitute gases.
1. The particles (molecules) that constitute a gas are so small and so far apart that the actual volume of these particles is negligible compared to the volume occupied by the gas. The fact that gases are highly compressible seems to support this assumption. When a gas is compressed, the spaces between its molecules (or atoms) become smaller.
2. The molecules of a gas are so far apart that the force of attraction between them is negligible and they are completely independent of each other. Gases are known to expand and occupy all the space available. This justifies the assumption that the attractive force between the molecules is negligible.
Read and Learn More WBCHSE For Class11 Basic Chemistry Notes
3. The molecules of a gas are in a state of constant motion. They move in straight lines but in any random direction. In doing so they collide with each other and against the walls of the container.
The pressure exerted by a gas is a result of these collisions. When a molecule collides with another molecule or with a wall of the container, its path changes. It still moves in a straight line but in another direction.
Molecules are too small to be observed even through sophisticated microscopes, so it is not possible to verify whether they are in a state of constant motion. However, it is possible to observe the movement of particles (much larger than molecules) caused by collisions with molecules.
You may have noticed that the dust particles caught in a beam of sunlight entering a darkened room seem to be moving in zig-zag lines. The dust particles cannot be moving on their own. The movement must be caused by collisions with randomly moving air molecules.
“Kinetic molecular theory of gases, WBCHSE Class 11, chemistry notes, and explanation”
4. The collisions between molecules are elastic, i.e., there is no net loss of kinetic energy during such a collision. There may be a transfer of energy from one molecule to another, however.
Therefore, the momenta of the molecules are conserved. Had the collisions between molecules been inelastic, there would be some loss of energy during each collision and the molecules would ultimately stop moving and the temperature and pressure would become zero.
5. The molecules of a gas do not all move at the same speed, so they have different kinetic energies. However, the average kinetic energy of the molecules is directly proportional to the absolute temperature of the gas. The assumption that at any instant different molecules have different speeds is valid given molecular collisions. Suppose we imagine an instant when they all have the same speed.
The very next instant, their speeds will become different due to intermolecular collisions. As for the second part of this assumption—when the temperature of a gas increases, so does its pressure.
This means that the number of collisions with the walls of the vessel increases or that the molecules start moving faster and their average kinetic energy increases. When the temperature of a gas falls, on the other hand, the average kinetic energy of the molecules decreases (they slow down). This is borne out by the fact that if a gas is cooled enough it becomes a liquid.
Comprehensive Guide To Kinetic Molecular Theory For WBCHSE Class 11 Students
For example, Atomic and molecular sizes are typically of the order of a few angstroms. Assuming that an N2 molecule is spherical in shape with radius (r) = 2 x 10-8 cm, calculate
- The volume of a single N2 molecule, and
- The percentage of space in one mole of N2 gas at stp.
Solution:
1. The volume of a sphere = 4/3πr3, where r is the radius.
The volume of one N2 molecule = 4/3 x 22/7 x (2 x 10-8)3 cm3 = 3.35x 10-23 cm3.
2. To calculate the space, first calculate the total volume of the Avogadro number of N2 molecules.
Volume of 6.022 x 1023 N2 molecules = 3.35 x 10-23 x 6.022 x 1023 = 20.2 cm3.
The volume occupied by 1 mol of the gas at stp = 22.7 L or 22700 cm3.
Empty space = 22700 -20.2 = 22679.8 cm3.
Percentage of empty space = \(\frac{\text { empty volume }}{\text { available volume }} \times 100\)
= \(\frac{22679.8 \times 100}{22700}=99.9\).
This calculation shows that particles of gas occupy only a tiny fraction of the total gaseous volume.
“WBCHSE Class 11, chemistry notes, on kinetic molecular theory, and gas behavior”
Maxwell-Boitzmann distribution of velocities: You have just read that at any given temperature, the molecules of a gas have different speeds and that the speeds and direction of motion of the molecules change constantly. You have also read that the average speed, or average kinetic energy of the molecules, depends on the temperature.
- Maxwell and Boltzmann showed (independently) that though it is not possible to know the individual speeds of the molecules, and though these speeds are constantly changing, the distribution of speeds (of the molecules) is constant at a particular temperature.
- This means that at a particular temperature, a certain fraction of the molecules will have a certain speed. These two scientists predicted the shape of the distribution curve of molecular speeds at a given temperature and the distribution is called the Maxwell-Boitzmann distribution. Some important features of this distribution are as follows.
1. A small fraction of the molecules have very high or very low speeds. We come across this kind of distribution quite often in our everyday lives. Suppose we consider the marks obtained by students (out of 100) in a particular examination. There would be very few students with marks between 95% and 100% and between 0% and 5%. If we were to plot the number of students versus the number of marks, we would get a curve similar to the ones shown in Figure. The maximum number of students would get marks corresponding to the maximum in the curve.
2. The fraction of molecules possessing greater and greater speeds (than the lowest) keeps increasing until the
maximum of the curve is reached and then it keeps falling. The maximum fraction of molecules has the speed corresponding to the peak of the curve. This speed is called the most probable speed (or velocity), ump.
“Kinetic theory of gases, assumptions, postulates, and applications, WBCHSE Chemistry”
3. When temperature increases, the most probable speed increases. The entire distribution shifts to the right as shown by the dotted curve. The area under the curve remains the same as the total number of molecules remains the same. Just the fraction of molecules with higher speeds increases.
Molecular speed: By now, we know that all molecules do not move with the same speed and that it is not possible to determine the individual speeds. From the Maxwell-Boitzmann distribution, however, it is possible to find the speed of the maximum fraction of molecules at a particular temperature for a particular gas. This speed is called the most probable speed and is given by the relation
⇒ \(\alpha=\sqrt{\frac{2 R T}{M}}\), ….. (1)
where R is the gas constant, T is the temperature and M is the molar mass of the gas.
From Equation (1) it is clear that the speed of a gas molecule also depends on its mass.
Another speed often used in calculations is the root mean square speed (urms). It is the square root of the mean of the squares of the speeds of the molecules.
∴ \(u_{\mathrm{ms}}=\sqrt{u^2}=\sqrt{\frac{u_1^2+u_2^2+u_3^2+\ldots}{n}}\),
where u1, u2, u3,… are the speeds of individual molecules and n is the number of molecules.
The root mean square speed is given by the following expression.
⇒ \(u_{\text {mas }}=\sqrt{\frac{3 R T}{M}} \text { or } u_{\text {rus }}=\sqrt{\frac{3 p V}{M}} \text {. }\) …..(2)
Yet another speed that is commonly used is the average speed. It is the simple average of the individual speeds of the molecules.
⇒ \(\bar{u}=\frac{u_1+u_2+u_3+\ldots}{n}\)
The average speed is given by \(\bar{u}=\sqrt{\frac{8 R T}{\pi M}}\) ….(3)
Comparing Equations (1), (2) and (3)
∴ \(\bar{u}=0.921 u_{\mathrm{rms}}, \quad \alpha=0.816 u_{\mathrm{mms}}\)
∴ \(\quad \alpha: \bar{u}: u_{\mathrm{mms}}=1: 1128: 1.224\)
“WBCHSE Class 11 Chemistry, kinetic molecular theory, gas laws, and derivation”
Gas equation in terms of urms: The kinetic theory of gases leads to an equation, the mathematical derivation of which is beyond the scope of this book. However, it is useful to know this equation, which yields an expression relating the average kinetic energy of molecules with the absolute temperature. All the gas laws can be deduced from this equation given here. It is called the kinetic gas equation.
⇒ \(p V=\frac{1}{3} m N \overline{u^2}\) ……(1)
where m = mass of one molecule of gas,
N = number of molecules,
⇒ \(\overline{u^2}\) = mean square speed,
p = pressure exerted by the gas, and
V = volume of the gas.
We will use this relationship and gas laws to derive a relationship between the average translational kinetic energy of a molecule and the temperature of the gas.
The average kinetic energy of a molecule \(E_k=\frac{1}{2} m \overline{u^2}\) at a temperature T.
We also know that pV = RT …..(2) for 1 mole of a gas.
For one mole of the gas, Equation (1) becomes \(p V=\frac{1}{3} N_{\mathrm{A}} m \overline{u^2},\) ….(3)
where NA = Avogadro number (because one mole contains the Avogadro number of molecules).
Comparing Equations (2) and (3), \(\frac{1}{3} N_{\mathrm{A}} m \overline{u^2}=R T\) …..(4)
Dividing both sides by 2, we get
⇒ \(\frac{1}{3} N_{\mathrm{A}} \times \frac{1}{2} m \overline{u^2}=\frac{1}{2} R T\)
⇒ \(\frac{1}{2} m \overline{u^2}=\frac{3}{2} \frac{R T}{N_{\mathrm{A}}} \text {. }\)
⇒ \(E_k=\frac{3}{2} \cdot \frac{R}{N_A} \cdot T\)
= \(\frac{3}{2} k T \text {, }\) …..(5)
where k = Boltzmann constant = \(\frac{R}{N_{\mathrm{A}}}\)
R is the gas constant for one molecule. Equation (4) can be modified as \(\frac{1}{3} N_{\mathrm{A}} m \overline{u^2}=R T\)
= \(\frac{1}{3} M \overline{u^2}=R T\)
or, \(\quad \frac{1}{2} M \overline{u^2}=\frac{3}{2} R T\)
\(\quad \frac{1}{2} M \overline{u^2}\) is the average kinetic energy of 1 mol of the gas molecules.
⇒ \(\underset{\text { (for 1 mol) }}{E_k}=\frac{3}{2} R T\) ….(6)
From Equations (5) and (6) we can see that the average kinetic energy of the gas molecules is proportional to the absolute temperature.
Explanation of gas laws based on the kinetic theory: Boyle’s law According to the kinetic theory, the pressure exerted by a gas depends on the number of collisions between its molecules and the walls of the vessel it is contained in.
The number of collisions depends on the number of molecules and their average speed. For a given mass of a gas at a constant temperature, the number of molecules and the average speed will remain the same.
Now, if the volume of the container is reduced (at constant temperature), the same number of molecules moving at the same average speed will have less space in which to move. Hence, they will collide more often with the walls, which means that the pressure will increase.
“Kinetic molecular theory, relation to ideal gas law, WBCHSE Class 11, chemistry notes”
Kinetic Molecular Theory Assumptions And Derivations For WBCHSE Class 11
If the volume is increased, the molecules will have more space, collisions will decrease and so will pressure. This is under Boyle’s law, according to which the pressure exerted by the gas is inversely proportional to the volume. This can be readily observed from the kinetic gas equation also.
We know that pV = \(\frac{1}{3} m N \overline{u^2}\)
or, \(p=\frac{1}{3} \cdot \frac{1}{V} \cdot m N \overline{u^2}\)
or, \(p \propto \frac{1}{V}\).
Charles’s law According to the kinetic theory, the average kinetic energy (also the speed) of the molecules of a gas is directly proportional to its absolute temperature. When the temperature of a given mass of gas is raised, the average energy of molecules increases and they collide more often and harder with the walls of the container.
“WBCHSE Class 11, chemistry notes, on kinetic theory of gases, and molecular motion”
This naturally causes the pressure to increase. If the pressure is to remain constant (as in Charles’s law), the volume must increase, so that the number of collisions per unit area of the walls decreases enough to compensate for the fact that the collisions are more vigorous at a higher temperature. Therefore, the higher the temperature, the larger the volume. In other words, volume is directly proportional to the absolute temperature.
You can easily deduce what happens when the temperature decreases. V ∝ T