A Molecular Description
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:
- A gas is composed of a large number of particles called molecules (whether monatomic or polyatomic) that are in constant random motion.
- Because the distance between gas molecules is much greater than the size of the molecules, the volume of the molecules is negligible.
- Intermolecular interactions, whether repulsive or attractive, are so weak that they are also negligible.
- 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.
- 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.
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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.
[fpropto (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.
[Ppropto (2mu_x)times(u_x)timesBig(dfrac{N}{V}Big)propto Big(dfrac{N}{V}Big)mu_x^2 label{6.7.3}]
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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}moverline{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}moverline{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}moverline{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_Aoverline{e_K}=dfrac{1}{2}Moverline{u^2}=dfrac{3}{2}RT label{6.7.7}]
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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.
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