– Number of collisions exercised by gas molecules an a uait area of the wall surface per sait time:
\[
\mathrm{v}=\frac{1}{4} n(v)
\]
where $n$ is the cesegntratios of molecules, and in is their mean velocity.
– Equation of as ideal gas stale:
\[
p=\operatorname{ak} T \text {. }
\]
– Meas energy of molecules:
\[
\text { (e) }=\frac{i}{2} k r \text {, }
\]
stere $i$ is the sum of translational, rotational, and the decble number of vilurstional degrees of treedom.
– Maxwellian distribution:
– Maxwellian diatribution is a reduced lorm:
\[
d N(\omega)=N \frac{4}{\sqrt{\pi}} e^{-21} \omega^{2} d u \text {. }
\]
where $u=d v_{p}, v_{p}$ is the most probable velecity.
\”The most probable, the probable and the roet mean square velecities of molecules:
\[
v_{p}=\sqrt{2 \frac{\pi T}{m}} \quad(v)=\sqrt{\frac{B}{\pi} \frac{k T}{m}}, \quad v_{n v}=\sqrt{3 \frac{k T}{m}} .
\]
– Beltrmann’s formula:
where $U$ is the potential energy of a molecule.
2.62. Modern vacuum pumps permit the pressures down to $p=$ $=4 \cdot 10^{-11} \mathrm{~atm}$ to be reached at room temperatures. Assuming that the gas exhausted is nitrogen, find the number of its molecules per $1 \mathrm{~cm}^{2}$ and the mean distance between them at this pressure.
2.63. A vessel of volume $V=5.01$ contains $m=1.4 \mathrm{~g}$ of nitrogon at a temperature $T=1800 \mathrm{~K}$. Find the gas pressure, taking inte account that $\eta=30 \%$ of molecules are disassociated into atoms at this temperature.
2.64. Under standard conditions the density of the helium and nitrogen mixture equals $p=0.60 \mathrm{~g} / \mathrm{l}$. Find the concentration of helium atoms in the given mixture.
2.65. A parallel beam of nitrogen molecules moving with velocity $p=400 \mathrm{~m} / \mathrm{s}$ impinges on a wall at an angle $\theta=30^{\circ}$ to its normal. The concentration of molecules in the beam $n=0.9 .10^{10} \mathrm{~cm}^{-3}$. Find the pressure exerted by the beam on the wall assuming the molecules to scatter in accordance with the perfectly elastic collision law.
2.66. How many degrees of treedom have the gas molecules, if under standard conditions the gas density is $p-1.3 \mathrm{mg} / \mathrm{cm}^{\prime}$ and the velocity of sound propagation in it is $v=330 \mathrm{~m} / \mathrm{s}$.
2.67. Determine the ratio of the sonic velocity $v$ in a gas to the root mean square velocity of molecules of this gas, if the molecules are
(a) monatomic; (b) rigid diatomic.
2.68. A gas consisting of $N$-atomic molecules has the temperature $r$ at which all degres of freedom (translational, rotational, and vibrational) are excited. Find the mean energy of molecules in such a gas. What fraction of this energy corresponds to that of transletional motion?
2.69. Suppose a gas is heated up to a temperature at which all degrees of freedom (translational, rotational, and vibrational) of its molecules are excited. Find the molar heat capacity of such a gas in the isochoric process, as well as the adiabatic exponent $\gamma$, if the $g$ as consists of
(a) diatomic;
(b) linear $\mathrm{N}$-atomic;
(c) network $N$-atomic molecules.
2.70. An ideal gas consisting of $N$-atomic molecules is expanded isobarically. Assuming that all degrees of freedom (translational, rotational, and vibrational) of the molecules are exeited, find what fraction of heat transferred to the gas in this process is spent to perform the work of expansion. How high is this fraction in the case of a monatemic gas?
2.71. Find the molar mass and the number of degrees of freedom of molecules in a gas if its heat capacities are known: $c_{V}=$ $=0.65 \mathrm{~J} /(\mathrm{g} \cdot \mathrm{K})$ and $\mathrm{c}_{\mathrm{p}}=0.94 \mathrm{~J} /(\mathrm{g} \cdot \mathrm{K})$.
2.72. Find the number of degrees of freedom of molecules in a gas whose molar heat capacity
(a) at constant pressure is equal to $C_{D}=29 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K})$;
(b) is equal to $C=29 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K})$ in the process $p T=$ const.
2.73. Find the adiabatic exponent $\gamma$ for a mixture consisting of $v_{1}$ moles of a monatemic gas and $v_{1}$ moles of gas of rigid diatomic molecules.
2.74. A thermally insulated vessel with gaseous nitrogen at a temperature $t=27^{\circ} \mathrm{C}$ moves with velocity $v=100 \mathrm{~m} / \mathrm{s}$. How much (in per cent) and in what way will the gas pressure change on a sudden stoppage of the vessel?
2.75. Calculate at the temperature $t=17^{\circ} \mathrm{C}$ :
(s) the root mean square velocity and the mean kinetic energy of an exygen molecule in the process of translational motion:
(b) the root mean square velocity of a water droplet of diameter $d=0.10 \mu \mathrm{m}$ suspended in the air.
2.76. A gas consisting of rigid diatemic molecules is expanded idiabatically. How many times has the gas to be expanded to reduce the root mean square velocity of the nolecules $\eta=1.50$ times?
2.77. The mass $m=15 \mathrm{~g}$ of nitrogen is enclosed in a vessel at a temperature $T=300 \mathrm{~K}$. What amount of heat has to be transferred to the gas to increase the root mean square velocity of its molecules ๆ $=2.0$ times?
2.78. The temperature of a gas consisting of rigid diatomic molecules is $T=300 \mathrm{~K}$. Calculate the angular root mean square velocity of a rotating molecule if its moment of inertia is equal to $I=$ $=2.1 \cdot 10^{-0} \mathrm{~g} \cdot \mathrm{cm}^{2}$.
2.79. A gas consisting of rigid diatemie melecules was initially under standard conditions. Then the gas was compressed adiabatically $\eta=5.0$ times. Find the mean kinetic energy of a rotating molecule in the final state.
2.80. How will the rate of collisions of rigid diatomic molecules against the vessel’s wall change, if the gas is expanded adiabatically in times?
2.81. The volume of gas consisting of rigid diatomic molecules was increased $\eta=2.0$ times in a polytropic process with the molar heat capacity $C=R$. How many times will the rate of collisions of molecules against a vessel’s wall be reduced as a result of this process?
2.82. A gas censisting of rigid diatemie molecules was expanded in a polytropic process so that the rate of cellisions of the molecules against the vessel’s wall did not change. Find the molar heat capacity of the gas in this process.
2.83. Calculate the most probable, the mean, and the root mean
square velocities of a molecule of a gas whose density under standard atmespheric pressure is equal to $p=1.00 \mathrm{~g} /$.
2.84. Find the traction of gas molecules whose velocities dilfer by less than $\delta \eta=1.00 \%$ from the value of
(a) the most probable velocity;
(b) the root mean square velecity.
2.85. Determine the gas temperature at which
(a) the root mean square velocity of hydrogen molecules exceeds their most probable velocity by $\Delta v=400 \mathrm{~m} / \mathrm{s}$;
(b) the velocity distribution function $F(v)$ for the exygen molecules will have the maximum value at the velocity $v=420 \mathrm{~m} / \mathrm{s}$.
2.86. In the case of gaseous nitrogen find:
(a) the temperature at which the velocities of the molecules $v_{1}=$ $=300 \mathrm{~m} / \mathrm{s}$ and $v_{1}=600 \mathrm{~m} / \mathrm{s}$ are associated with equal values of the Maxwell distribution function $F(v)$ :
(b) the velocity of the molecules $v$ at which the value of the Maxwell distribution function $F(v)$ for the temperature $T$, will be the same as that for the temperature $\eta$ times higher.
2.87. At what temperature of a nitrogen and exygen mixture do the most probable velocities of nitrogen and oxygen molecules differ by $\Delta v=30 \mathrm{~m} / \mathrm{s}$ ?
2.88. The tempersture of a hydrogen and helium mixture is $T-$ $-300 \mathrm{~K}$. At what value of the molecular velocity $v$ will the Maxwell distribution function $F(v)$ yield the same magnitude for both gases?
2.89. At what temperature of a gas will the number of molecules, whose velocities fall within the given interval from $v$ to $v+d v$, be the greatest? The mass of each melecule is equal to $m$.
2.90. Find the fraction of molecules whose velocity projections on the $x$ axis fall within the interval from $v_{x}$ to $v_{n}+d v_{n}$ while the meduli of perpendicular velocity components fall within the interval from $v_{1}$ to $v_{1}+d v_{1}$. The mass of each molecule is $m$, and the temperature is $T$.
2.91. Using the Maxwell distribution function, calculate the mean velocity projection $\left(w_{n}\right)$ and the mean value of the modulus of this projection $\left(\left|v_{x}\right|\right)$ if the mass of each melecule is equal to $m$ and the gas temperature is $T$.
2.92. From the Maxwell distribution function find $(04)$, the mean value of the squared $v_{3}$ projection of the molecular velocity in a gas at a temperature $T$. The mass of each molecule is equal to $m$.
2.93. Making use of the Maxwell distribution function, calculate the number $v$ of gas molecules reaching a unit area of a wall per unit time, if the concentration of molecules is equal to $n$, the temperature to $T$, and the mass of each molecule is $m$.
2.94. Using the Maxwell distribution function, determine the pressure exerted by gas on a wall, if the gas temperature is $T$ and the concentration of molecules is $n$.
2.95. Making use of the Maxwell distribution function, find $(1 / v)$, the mean value of the reciprocal of the velocity of molecules in an ideal gas at a temperature $\boldsymbol{T}$, if the mass of each melecule is equal to $m$. Compare the value obtained with the reciprocal of the mean velocity.
2.96. A gas consists of molecules of mass $m$ and is at a temperature T. Making use of the Maxwell velocity distribution function, find the corresponding distribution of the molecules over the kinetie energies $\varepsilon$. Determine the most probable value of the kinetic enerzy $e_{p}$. Does $t_{p}$ correspond to the most probable velocity?
2.97. What fraction of monatomic molecules of a gas in a thermal equilibrium possesses kinetic energies differing from the mean value by $\delta \eta=1.0 \%$ and less?
2.98. What fraction of molecules in a gas at a temperature $T$ has the kinetic energy of translational motion exceeding $\varepsilon_{4}$ if $\varepsilon_{4}>$ $\Rightarrow k T$ ?
2.99. The velocity distribution of molecules in a beam coming out of a hole in a vessel is described by the function $F(V)=A v_{0}=\mathrm{m} / \mathrm{aNr}$, where $T$ is the temperature of the gas in the vessel. Find the most probable values of
(a) the velocity of the molecules in the beam; compare the result obtained with the most probable velocity of the molecules in the vessel;
(b) the kinetic energy of the melecules in the beam.
2.100. An ideal gas consisting of molecules of mass $m$ with concentration $n$ has a temperature $T$. Using the Maxwell distribution funetion, find the number of molecules reaching a unit area of a wall at the angles between $\theta$ and $\theta+d \theta$ to its nermal per unit time.
2.10t. From the conditions of the foregoing problem find the number of molecules reaching a unit ares of a wall with the velocities in the interval from $v$ to $v+d v$ per unit time.
2.102. Find the force exerted on a particle by a uniform field if the concentrations of these particles at two levels separated by the distance $\Delta h=3.0 \mathrm{~cm}$ (along the field) differ by $\mathrm{M}=2.0$ times. The temperature of the system is equal to $T=280 \mathrm{~K}$.
2.103. When examining the suspended gamboge droplets under a mieroscope, their average numbers in the layers separated by the distance $h=40 \mu \mathrm{m}$ were found to differ by $\eta=2.0$ times. The environmental temperature is equal to $T=290 \mathrm{~K}$. The diameter of the droplets is $d-0.40 \mu \mathrm{m}$, and their density exceeds that of the surrounding fluid by $\Delta p=0.20 \mathrm{~g} / \mathrm{cm}^{3}$. Find Avogadro’s number from these data.
2.104. Suppose that $\eta_{0}$ is the ratio of the melecular concentration of hydrogen to that of nitrogen at the Earth’s surface, while $n$ is the corresponding ratio at the height $h=3000 \mathrm{~m}$. Find the ratio $\eta / \eta_{0}$ at the temperature $T=280 \mathrm{~K}$, assuming that the temperature and the free fall acceleration are independent of the height.
2.105. A tall vertical vessel contains a gas composed of two kinds of molecules of masses $m_{4}$ and $m_{1}$, with $m_{3}>m_{4}$. The concentrations of these molecules at the bottom of the vessel are equal to $n_{1}$ and $n_{3}$
respectively, with $n_{3}>n_{3}$. Assuming the temperature $T$ and the free-fall acceleration $g$ to be independent of the height, find the height at which the concentrations of these kinds of molecules are equal.
2.106. A very tall vertical cylinder contains carbon dioxide at a certain temperature $\boldsymbol{T}$. Assuming the gravitational field to be uniform, find how the gas pressure on the bottom of the vessel will change when the gas temperature increases in times.
2.107. A very tall vertical cylinder contains a gas at a temperature $\boldsymbol{T}$. Assuming the gravitational field to be uniform, find the mean value of the potential energy of the gas molecules. Does this value depend on whether the gas consists of one kind of molecules or of several kinds?
2.108. A horizental tube of length $l=100 \mathrm{~cm}$ closed from both ends is displaced lengthwise with a constant acceleration w. The tube contains argon at a temperature $T=330 \mathrm{~K}$. At what value of iv will the argon concentrations at the tube’s ends differ by $\eta=1.0 \%$ ?
2.109. Find the mass of a mole of colloid particles if during their centrifuging with an angular velecity $\oplus$ about a vertical axis the concentration of the particles at the distance $r_{2}$ from the rotation axis is ₹) times greater than that at the distance $r_{1}$ (in the same horizontal plane). The densities of the particles and the solvent are equal to $\rho$ and to $\rho_{0}$ respectively.
2.110. A horizental tube with closed ends is rotated with a cens. tant angular velocity $\omega$ about a vertical axis passing through one of its ends. The tube contains carbon dioxide at a temperature $T=$ $=300 \mathrm{~K}$. The length of the tube is $l=100 \mathrm{~cm}$. Find the value at which the ratio of molecular concentrations at the opposite ends of the tube is equal to $\eta=2.0$.
2.111. The potential energy of gas molecules in a certain central field depends on the distance $r$ from the field’s centre as $U(r)=a r^{r}$, where $a$ is a positive constant. The gas temperature is $T$, the concentration of molecules at the centre of the field is $n_{0}$. Find:
(a) the number of molecules located at the distances between $r$ and $r+d r$ from the centre of the field;
(b) the most probable distance separating the molecules from the centre of the field:
(c) the fraction of molecules lecated in the spherical layer between $r$ and $r+d r$;
(d) how many times the concentration of molecules in the centre of the field will change if the temperature decreases in times.
2.112. From the conditions of the foregoing problem find:
(a) the number of molecules whose potential energy lies within the interval from $U$ to $U+d U$ :
(b) the most probable value of the potential energy of a molecule; compare this value with the potential energy of a molecule located at its mest probable distance from the centre of the field.