Which of the assumptions the kinetic molecular theory best explains the observation that a gas can be compressed?

The velocity of gas molecules is proportional to their kelvin temperature.

What are the main assumptions of kinetic molecular theory quizlet?

Gases consist of large numbers of tiny particles that are far apart relative to their size. Collisions between gas particles and between particles and container walls are elastic collisions (there is no net loss of total kinetic energy). You just studied 5 terms!

What does the kinetic molecular theory describes?

Kinetic Molecular Theory states that gas particles are in constant motion and exhibit perfectly elastic collisions. Kinetic Molecular Theory can be used to explain both Charles’ and Boyle’s Laws. The average kinetic energy of a collection of gas particles is directly proportional to absolute temperature only.

What is the basic assumption of kinetic theory?

Kinetic theory is based on an atomic model of matter. The basic assumption of kinetic theory is that the measurable properties of gases, liquids, and solids reflect the combined actions of countless numbers of atoms and molecules.

What assumption of the kinetic molecular theory explains why a gas can expand to fill a container?

Gases are in rapid motion, and they undergo elastic collisions with each other and the walls of the container; that is, momentum and energy is transfered not lost during collisions. Gases expand spontaneously to fill any container (rapid motion).

Which of the Assumption of the kinetic molecular theory best explain the observation that a gas can be compressed?

Gases can be compressed because most of the volume of a gas is empty space. If we compress a gas without changing its temperature, the average kinetic energy of the gas particles stays the same. There is no change in the speed with which the particles move, but the container is smaller.

What are the three assumptions of the kinetic theory of matter quizlet?

Explains how particles in matter behave by making three assumptions: 1) All matter is made of small particles. 2) These particles are in constant, random motion. 3) These particles are colliding with each other and the walls of their container.

What are the main ideas of the kinetic molecular theory?

The five main postulates of the KMT are as follows: (1) the particles in a gas are in constant, random motion, (2) the combined volume of the particles is negligible, (3) the particles exert no forces on one another, (4) any collisions between the particles are completely elastic, and (5) the average kinetic energy of …

Which of the following is not an assumption of the kinetic-molecular theory?

The average kinetic energy is proportional to the temperature of the ideal gases. Gases do not put any kind of force of attraction or repulsion on the other molecules. So, options a,b,c,d are the assumptions of kinetic molecular theory. But the theory does not relate the speed of the molecule with the pressure value.

What are three basic assumptions in kinetic theory?

Following are the kinetic theory of gases assumptions: All gases are made up of molecules that are constantly and persistently moving in random directions. The separation between the molecules is much greater than the size of molecules. When a gas sample is kept in a container, the molecules of the sample do not exert any force on the walls of the container during the collision.

What are the four statements of the kinetic molecular theory?

The kinetic-molecular theory of gases can be stated as four postulates: 1. A gas consists of molecules in constant random motion . 2. Gas molecules influence each other only by collision; they exert no other forces on each other. 3. All collisions between gas molecules are perfectly elastic; all kinetic energy is conserved.

What are the postulates of the kinetic theory?

Postulates of the kinetic theory of matter. The postulates of kinetic theory of matter are used to explain the existence of matter in different phases (i.e. solid, liquid and gas), and how it is able to change from one phase to the next. The kinetic theory of matter also helps us to understand the macroscopic properties of matter.

What does the kinetic molecular theory say about?

Kinetic molecular theory (also known as particle theory) states that all matter is made up particles and these particles are always in motion. Kinetic molecular theory is useful in describing the properties of solids, liquids and gases at the molecular level.

Related

The laws that describe the behavior of gases were well established long before anyone had developed a coherent model of the properties of gases. In this section, we introduce a theory that describes why gases behave the way they do. The theory we introduce can also be used to derive laws such as the ideal gas law from fundamental principles and the properties of individual particles.

The kinetic molecular theory of gases explains the laws that describe the behavior of gases. Developed during the mid-19th century by several physicists, including the Austrian Ludwig Boltzmann (1844–1906), the German Rudolf Clausius (1822–1888), and the Englishman James Clerk Maxwell (1831–1879), who is also known for his contributions to electricity and magnetism, this theory is based on the properties of individual particles as defined for an ideal gas and the fundamental concepts of physics. Thus the kinetic molecular theory of gases provides a molecular explanation for observations that led to the development of the ideal gas law. The kinetic molecular theory of gases is based on the following five postulates:

1. A gas is composed of a large number of particles called molecules (whether monatomic or polyatomic) that are in constant random motion.
2. Because the distance between gas molecules is much greater than the size of the molecules, the volume of the molecules is negligible.
3. Intermolecular interactions, whether repulsive or attractive, are so weak that they are also negligible.
4. Gas molecules collide with one another and with the walls of the container, but these collisions are perfectly elastic; that is, they do not change the average kinetic energy of the molecules.
5. The average kinetic energy of the molecules of any gas depends on only the temperature, and at a given temperature, all gaseous molecules have exactly the same average kinetic energy.

Although the molecules of real gases have nonzero volumes and exert both attractive and repulsive forces on one another, for the moment we will focus on how the kinetic molecular theory of gases relates to the properties of gases we have been discussing. In Section 10.8, we explain how this theory must be modified to account for the behavior of real gases.

Postulates 1 and 4 state that gas molecules are in constant motion and collide frequently with the walls of their containers. The collision of molecules with their container walls results in a momentum transfer (impulse) from molecules to the walls (Figure \(\PageIndex{2}\)).

The momentum transfer to the wall perpendicular to \(x\) axis as a molecule with an initial velocity \(u_x\) in \(x\) direction hits is expressed as:

\[\Delta p_x=2mu_x \label{6.7.1}\]

The collision frequency, a number of collisions of the molecules to the wall per unit area and per second, increases with the molecular speed and the number of molecules per unit volume.

\[f\propto (u_x) \times \Big(\dfrac{N}{V}\Big) \label{6.7.2}\]

The pressure the gas exerts on the wall is expressed as the product of impulse and the collision frequency.

\[P\propto (2mu_x)\times(u_x)\times\Big(\dfrac{N}{V}\Big)\propto \Big(\dfrac{N}{V}\Big)mu_x^2 \label{6.7.3}\]

At any instant, however, the molecules in a gas sample are traveling at different speed. Therefore, we must replace \(u_x^2\) in the expression above with the average value of \(u_x^2\), which is denoted by \(\overline{u_x^2}\). The overbar designates the average value over all molecules.

The exact expression for pressure is given as :

\[P=\dfrac{N}{V}m\overline{u_x^2} \label{6.7.4}\]

Finally, we must consider that there is nothing special about \(x\) direction. We should expect that \(\overline{u_x^2}= \overline{u_y^2}=\overline{u_z^2}=\dfrac{1}{3}\overline{u^2}\). Here the quantity \(\overline{u^2}\) is called the mean-square speed defined as the average value of square-speed (\(u^2\)) over all molecules. Since \(u^2=u_x^2+u_y^2+u_z^2\) for each molecule, \(\overline{u^2}=\overline{u_x^2}+\overline{u_y^2}+\overline{u_z^2}\). By substituting \(\dfrac{1}{3}\overline{u^2}\) for \(\overline{u_x^2}\) in the expression above, we can get the final expression for the pressure:

\[P=\dfrac{1}{3}\dfrac{N}{V}m\overline{u^2} \label{6.7.5}\]

Because volumes and intermolecular interactions are negligible, postulates 2 and 3 state that all gaseous particles behave identically, regardless of the chemical nature of their component molecules. This is the essence of the ideal gas law, which treats all gases as collections of particles that are identical in all respects except mass. Postulate 2 also explains why it is relatively easy to compress a gas; you simply decrease the distance between the gas molecules.

Postulate 5 provides a molecular explanation for the temperature of a gas. Postulate 5 refers to the average translational kinetic energy of the molecules of a gas \((\overline{e_K})\), which can be represented as and states that at a given Kelvin temperature \((T)\), all gases have the same value of

\[\overline{e_K}=\dfrac{1}{2}m\overline{u^2}=\dfrac{3}{2}\dfrac{R}{N_A}T \label{6.7.6}\]

where \(N_A\) is the Avogadro's constant. The total translational kinetic energy of 1 mole of molecules can be obtained by multiplying the equation by \(N_A\):

\[N_A\overline{e_K}=\dfrac{1}{2}M\overline{u^2}=\dfrac{3}{2}RT \label{6.7.7}\]

where \(M\) is the molar mass of the gas molecules and is related to the molecular mass by \(M=N_Am\).

By rearranging the equation, we can get the relationship between the root-mean square speed (\(u_{\rm rms}\)) and the temperature.

The rms speed (\(u_{\rm rms}\)) is the square root of the sum of the squared speeds divided by the number of particles:

\[u_{\rm rms}=\sqrt{\overline{u^2}}=\sqrt{\dfrac{u_1^2+u_2^2+\cdots u_N^2}{N}} \label{6.7.8}\]

where \(N\) is the number of particles and \(u_i\) is the speed of particle \(i\).

The relationship between \(u_{\rm rms}\) and the temperature is given by:

\[u_{\rm rms}=\sqrt{\dfrac{3RT}{M}} \label{6.7.9}\]

In this equation, \(u_{\rm rms}\) has units of meters per second; consequently, the units of molar mass \(M\) are kilograms per mole, temperature \(T\) is expressed in kelvins, and the ideal gas constant \(R\) has the value 8.3145 J/(K•mol).

The equation shows that \(u_{\rm rms}\) of a gas is proportional to the square root of its Kelvin temperature and inversely proportional to the square root of its molar mass. The root mean-square speed of a gas increase with increasing temperature. At a given temperature, heavier gas molecules have slower speeds than do lighter ones.

The rms speed and the average speed do not differ greatly (typically by less than 10%). The distinction is important, however, because the rms speed is the speed of a gas particle that has average kinetic energy. Particles of different gases at the same temperature have the same average kinetic energy, not the same average speed. In contrast, the most probable speed (vp) is the speed at which the greatest number of particles is moving. If the average kinetic energy of the particles of a gas increases linearly with increasing temperature, then Equation 6.7.8 tells us that the rms speed must also increase with temperature because the mass of the particles is constant. At higher temperatures, therefore, the molecules of a gas move more rapidly than at lower temperatures, and vp increases.

Note

At a given temperature, all gaseous particles have the same average kinetic energy but not the same average speed.

Example \(\PageIndex{1}\)

The speeds of eight particles were found to be 1.0, 4.0, 4.0, 6.0, 6.0, 6.0, 8.0, and 10.0 m/s. Calculate their average speed (\(v_{\rm av}\)) root mean square speed (\(v_{\rm rms}\)), and most probable speed (\(v_{\rm m}\)).

Given: particle speeds

Asked for: average speed (\(v_{\rm av}\)), root mean square speed (\(v_{\rm rms}\)), and most probable speed (\(v_{\rm m}\))

Strategy:

Use Equation 6.7.6 to calculate the average speed and Equation 6.7.8 to calculate the rms speed. Find the most probable speed by determining the speed at which the greatest number of particles is moving.

Solution:

The average speed is the sum of the speeds divided by the number of particles:

\[v_{\rm av}=\rm\dfrac{(1.0+4.0+4.0+6.0+6.0+6.0+8.0+10.0)\;m/s}{8}=5.6\;m/s\]

The rms speed is the square root of the sum of the squared speeds divided by the number of particles:

\[v_{\rm rms}=\rm\sqrt{\dfrac{(1.0^2+4.0^2+4.0^2+6.0^2+6.0^2+6.0^2+8.0^2+10.0^2)\;m^2/s^2}{8}}=6.2\;m/s\]

The most probable speed is the speed at which the greatest number of particles is moving. Of the eight particles, three have speeds of 6.0 m/s, two have speeds of 4.0 m/s, and the other three particles have different speeds. Hence \(v_{\rm m}=6.0\) m/s. The \(v_{\rm rms}\) of the particles, which is related to the average kinetic energy, is greater than their average speed.

At any given time, what fraction of the molecules in a particular sample has a given speed? Some of the molecules will be moving more slowly than average, and some will be moving faster than average, but how many in each situation? Answers to questions such as these can have a substantial effect on the amount of product formed during a chemical reaction, as you will learn in Chapter 14 "Chemical Kinetics". This problem was solved mathematically by Maxwell in 1866; he used statistical analysis to obtain an equation that describes the distribution of molecular speeds at a given temperature. Typical curves showing the distributions of speeds of molecules at several temperatures are displayed in Figure \(\PageIndex{1}\). Increasing the temperature has two effects. First, the peak of the curve moves to the right because the most probable speed increases. Second, the curve becomes broader because of the increased spread of the speeds. Thus increased temperature increases the value of the most probable speed but decreases the relative number of molecules that have that speed. Although the mathematics behind curves such as those in Figure \(\PageIndex{1}\) were first worked out by Maxwell, the curves are almost universally referred to as Boltzmann distributions, after one of the other major figures responsible for the kinetic molecular theory of gases.

We now describe how the kinetic molecular theory of gases explains some of the important relationships we have discussed previously.

• Pressure versus Volume: At constant temperature, the kinetic energy of the molecules of a gas and hence the rms speed remain unchanged. If a given gas sample is allowed to occupy a larger volume, then the speed of the molecules does not change, but the density of the gas (number of particles per unit volume) decreases, and the average distance between the molecules increases. Hence the molecules must, on average, travel farther between collisions. They therefore collide with one another and with the walls of their containers less often, leading to a decrease in pressure. Conversely, increasing the pressure forces the molecules closer together and increases the density, until the collective impact of the collisions of the molecules with the container walls just balances the applied pressure.
• Volume versus Temperature: Raising the temperature of a gas increases the average kinetic energy and therefore the rms speed (and the average speed) of the gas molecules. Hence as the temperature increases, the molecules collide with the walls of their containers more frequently and with greater force. This increases the pressure, unless the volume increases to reduce the pressure, as we have just seen. Thus an increase in temperature must be offset by an increase in volume for the net impact (pressure) of the gas molecules on the container walls to remain unchanged.
• Pressure of Gas Mixtures: Postulate 3 of the kinetic molecular theory of gases states that gas molecules exert no attractive or repulsive forces on one another. If the gaseous molecules do not interact, then the presence of one gas in a gas mixture will have no effect on the pressure exerted by another, and Dalton’s law of partial pressures holds.

Example \(\PageIndex{2}\)

The temperature of a 4.75 L container of N2 gas is increased from 0°C to 117°C. What is the qualitative effect of this change on the

1. average kinetic energy of the N2 molecules?
2. rms speed of the N2 molecules?
3. average speed of the N2 molecules?
4. impact of each N2 molecule on the wall of the container during a collision with the wall?
5. total number of collisions per second of N2 molecules with the walls of the entire container?
6. number of collisions per second of N2 molecules with each square centimeter of the container wall?
7. pressure of the N2 gas?

Given: temperatures and volume

Asked for: effect of increase in temperature

Strategy:

Use the relationships among pressure, volume, and temperature to predict the qualitative effect of an increase in the temperature of the gas.

Solution:

1. Increasing the temperature increases the average kinetic energy of the N2 molecules.
2. An increase in average kinetic energy can be due only to an increase in the rms speed of the gas particles.
3. If the rms speed of the N2 molecules increases, the average speed also increases.
4. If, on average, the particles are moving faster, then they strike the container walls with more energy.
5. Because the particles are moving faster, they collide with the walls of the container more often per unit time.
6. The number of collisions per second of N2 molecules with each square centimeter of container wall increases because the total number of collisions has increased, but the volume occupied by the gas and hence the total area of the walls are unchanged.
7. The pressure exerted by the N2 gas increases when the temperature is increased at constant volume, as predicted by the ideal gas law.

Exercise \(\PageIndex{2}\)

A sample of helium gas is confined in a cylinder with a gas-tight sliding piston. The initial volume is 1.34 L, and the temperature is 22°C. The piston is moved to allow the gas to expand to 2.12 L at constant temperature. What is the qualitative effect of this change on the

1. average kinetic energy of the He atoms?
2. rms speed of the He atoms?
3. average speed of the He atoms?
4. impact of each He atom on the wall of the container during a collision with the wall?
5. total number of collisions per second of He atoms with the walls of the entire container?
6. number of collisions per second of He atoms with each square centimeter of the container wall?
7. pressure of the He gas?

Answer: a. no change; b. no change; c. no change; d. no change; e. decreases; f. decreases; g. decreases

Kinetic-Molecular Theory of Gases: https://youtu.be/9f83XAYfXAg

• The kinetic molecular theory of gases provides a molecular explanation for the observations that led to the development of the ideal gas law.
• Average kinetic energy:\[\overline{e_K}=\dfrac{1}{2}m{u_{\rm rms}}^2=\dfrac{3}{2}\dfrac{R}{N_A}T,\]
• Root mean square speed: \[u_{\rm rms}=\sqrt{\dfrac{u_1^2+u_2^2+\cdots u_N^2}{N}},\]
• Kinetic molecular theory of gases: \[u_{\rm rms}=\sqrt{\dfrac{3RT}{M}}.\]

The behavior of ideal gases is explained by the kinetic molecular theory of gases. Molecular motion, which leads to collisions between molecules and the container walls, explains pressure, and the large intermolecular distances in gases explain their high compressibility. Although all gases have the same average kinetic energy at a given temperature, they do not all possess the same root mean square (rms) speed (vrms). The actual values of speed and kinetic energy are not the same for all particles of a gas but are given by a Boltzmann distribution, in which some molecules have higher or lower speeds (and kinetic energies) than average.