– Additional (eapillary) presume is a liquid usder an arbitrary surlace (Laplace’s formula):
\[
\Delta p=a\left(\frac{1}{R_{1}}+\frac{1}{R_{1}}\right) \text {. }
\]
where a is the surface tensien of a given liquid.
– Free esergy increment of the surface tayer of a liquid:
\[
\Delta F=a \Delta s \text {, }
\]
where $1 S$ is the ares increment of the surface layer.
– Amesat of hest required to form g anit ams of the liquid warface layer during the isothermal increase of its surlace:
\[
q=-r \frac{d a}{d r} \text {. }
\]
2.160. Find the capillery pressure
(a) in mercury droplets of diameter $d=1.5 \mu \mathrm{m}$;
(b) inside a soap bubble of diameter $d=3.0 \mathrm{~mm}$ if the surface tension of the soap water solution is $\alpha=45 \mathrm{mN} / \mathrm{m}$.
2.161. In the bottom of a vessel with mercury there is a round hole of diameter $d=70 \mathrm{~mm}$. At what maximum thickness of the mercury layer will the liquid still not flow out through this hole?
2.162. A vessel filled with air under pressure $p_{0}$ contains a soap bubble of diameter $d$. The air pressure having been reduced isothermally $n$-fold, the bubble diameter increased $\eta$-fold. Find the surfoce tension of the soap water solution.
2.163. Find the pressure in an air bubble of diameter $d=4.0 \mu \mathrm{m}$, located in water at a depth $h=5.0 \mathrm{~m}$. The atmospheric pressure has the standard value $P_{0}$.
2.164. The diameter of a gas bubble formed at the bottom of a pond is $d=4.0 \mu \mathrm{m}$. When the bubble rises to the surface its diameter increases $n=1.1$ times. Find how deep is the pond at that spot. The atmospheric pressure is standard, the gas expansion is assumed to be isothermal.
2.165. Find the difference in height of mercury columns in two communicating vertical capillaries whose diameters are $d_{1}=$ $=0.50 \mathrm{~mm}$ and $d_{1}=1.00 \mathrm{~mm}$, if the contact angle $\theta=138^{\circ}$.
2.166. A vertical capillary with inside dianeter $0.50 \mathrm{~mm}$ is submerged inte water so that the length of its part protruding over the water surface is equal to $h=25 \mathrm{~mm}$. Find the curvature radius of the meniscus.
2.167. A glass capillary of length $t=110 \mathrm{~mm}$ and inside diameter $d=20 \mu \mathrm{m}$ is submerged vertically into water. The upper end of the capillary is sealed. The outside pressure is standard. To what length $x$ has the capillary to be submorged to make the water levels inside and outside the capillary coincide?
2.168. When a vertical capillary of length $l$ with the sealed upper end was brought in contact with the surface of a liquid, the level of this liquid rose to the height $h$. The liquid density is $p$, the inside diameter of the capillary is $d$, the contact angle is $\theta$, the atmospheric pressure is $P_{0}$. Find the surface tension of the liq̧uid.
2.169. A glass rod of diameter $d_{1}=1.5 \mathrm{~mm}$ is inserted symmetrically inte a glass capillary with inside diameter $d_{1}=2.0 \mathrm{~mm}$. Then the whole arrangement is vertically oriented and brought in contact with the surface of water. To what height will the water rise in the capillary?
2.170. Two vertical plates submerged partially in a wetting liquid form a wedge with a very small angle 84 . The edge of this wedge is vertical. The density of the liquid is $\rho$, its surlace tension is $\alpha$, the contact angle is $\theta$. Find the height $h$, to which the liquid rises, as a function of the distance $x$ from the edge.
2.171. A vertical water jet flows out of a round hole. One of the horizontal sections of the jet has the diameter $d=2.0 \mathrm{~mm}$ while the other section located $t=20 \mathrm{~mm}$ lower has the diameter which is $n=1.5$ times less. Find the volume of the water nowing from the hole each second.
2.172. A water drop falls in air with a uniform velocity. Find the difference between the curvature radii of the drop’s surface at the upper and lower points of the drop separated by the distance $h=2.3 \mathrm{~mm}$.
2.173. A mercury drop shaped as a round tablet of radius $R$ and thickness $h$ is located between two horizontal glass plates. Assuming that $h<A$, find the mass $m$ of a weight which has to be placed en the upper plate to diminish the distance between the plates $n$-times. The contact angle equals $\theta$. Calculate $m$ if $R=2.0 \mathrm{~cm}, h=0.38 \mathrm{~mm}$, $n=2.0$, and $\theta=135^{\circ}$.
2.174. Find the attraction force between two parallel glass plates, separated by a distance $h=0.10 \mathrm{~mm}$, after a water drop of mass $m=70 \mathrm{mg}$ was introduced between them. The wetting is assumed to be complete.
2.175. Two glass discs of radius $R=5.0 \mathrm{~cm}$ were wetted with water and put tegether so that the thickness of the water layer between them was $h=1.9 \mathrm{~mm}$. Asuming the wetting to be complete. find the force that has to be applied at right angles to the plates in erder to pull them apart.
2.176. Twe vertical parallel glass plates are partially submerged in water. The distance between the plates is $d=0.10 \mathrm{~mm}$, and their width is $l=12 \mathrm{~cm}$. Assuming that the water between the plates does not reach the upper edges of the plates and that the wetting is complete, find the force of their mutual attraction.
2.177. Find the lifetime of a soap bubble of radius $\boldsymbol{R}$ connected with the atmesphere through a capillary of length $I$ and inside radius $r$. The surface tension is $a$, the viscosity coefficient of the gas is $\eta$.
2.178. A vertical capillary is brought in contact with the water surface. What amount of heat is liberated while the water rises along the capillary? The wetting is assumed to be complete, the surface tension equals $\alpha$.
2.179. Find the free energy of the surface layer of
(a) a mercury droplet of diameter $d=1.4 \mathrm{~mm}$;
(b) a soap bubble of diameter $d=6.0 \mathrm{~mm}$ if the surface tension of the soap water solution is equal to $a=45 \mathrm{mN} / \mathrm{m}$.
2.180. Find the incremeat of the free energy of the surface layer when twe identical mercury droplets, each of diameter $d=1.5 \mathrm{~mm}$, merge isothermally.
2.181. Find the work to be performed in order to blow a soap bubble of radius $\boldsymbol{A}$ if the outside air pressure is equal to $P_{0}$ and the surlace tension of the soap water solution is equal to $\alpha$.
2.182. A soap bubble of radius $r$ is inflated with an ideal gas. The atmospheric pressure is $p_{4}$, the surface tension of the soop water solution is a. Find the difference between the molar heat capacity of the gas during its heating inside the bubble and the molar heat capacity of the gas under constant pressure, $c-C_{\mathrm{F}}$ :
2.183. Considering the Carnot cycle as applied to a liquid film, show that in an isothermal process the amount of heat required for the formation of a unit area of the surface layer is equal to $9=$ $=-T \cdot d a / d T$, where $d a / d T$ is the temperature derivative of the surface tension.
2.184. The surface of a soap film was increased isothermally by $\Delta \sigma$ at a temperature $T$. Knowing the surface tension of the soap water solution $\alpha$ and the temperature coefficient $d a / d T$, find the increment
(a) of the entropy of the film’s surface layer;
(b) of the internal energy of the surface layer.
2.6. PHASE tRANsFonMations
– Relations between Vas der Waals constants and the parameters of the critical state of a substance:
– Relation betwees the critical parameters for a mele of sulatance:
– Clausius-Clapeyron equation:
\[
\frac{d p}{d r}=\frac{r(1)}{r\left(V_{1}-V_{1}\right)} .
\]
wherg $\mathrm{fig}$ is the specific beat aborted in the tranalormation $\mathrm{f} \rightarrow 2, \mathrm{Vi}_{\mathrm{i}}$ and $\mathrm{V}_{\mathrm{i}}$ are the specific volumes of phasen 1 and 2
2.185. A saturated water vapour is contained in a cylindrical vessel under a weightless piston at a temperature $t=100^{\circ} \mathrm{C}$. As a result of a slow introduction of the piston a small fraction of the vapour $\Delta m=0.70 \mathrm{~g}$ gets condensed. What amount of work was performed over the gas? The vapour is assumed to be ideal, the volume of the liguid is to be neglected.
2.186. A vessel of volume $\boldsymbol{V}=6.0 \mathrm{I}$ contains water tegether with its saturated vapour under a pressure of $40 \mathrm{~atm}$ and at a temperature of $250^{\circ} \mathrm{C}$. The specific volume of the vapour is equal to $V ;=50 \mathrm{l} / \mathrm{kg}$ under these conditions. The total mass of the system water-vapour equals $m=5.0 \mathrm{~kg}$. Find the mass and the volume of the vapour.
2.187. The saturated water vapour is enclosed in a cylinder under a piston and occupies a volume $V_{4}=5.0 \mathrm{I}$ at the temperature $t=$ $=100^{\circ} \mathrm{C}$. Find the mass of the liquid phase formed after the volume under the piston decreased isothermally to $V=1.6 \mathrm{I}$. The saturated vapour is assumed to be ideal.
2.188. A volume eceupied by a saturated rapour is reduced isothermally $n$-fold. Find what fraction $\eta$ of the final volume is occupied by the liquid phase if the specific volumes of the saturated vapour and the liquid phase differ by $N$ times $(N>n)$. Solve the same problem under the condition that the final volume of the substance eorresponds to the midpoint of a herizentsl portien of the isothermal line in the diagram $p, V$.
2.189. An amount of water of mass $m=1.00 \mathrm{~kg}$, boiling at standard atmospheric pressure, turns completely into saturated vapour.
2.199. Find the specific volume of benzene $\left(\mathrm{C}_{4} \mathrm{H}_{6}\right)$ in critical state if its critical temperature $T_{e r}=562 \mathrm{~K}$ and critical pressure $\mathrm{Per}^{-}=$ $=47 \mathrm{~atm}$.
2.200. Write the Van der Waals equation via the reduced parameters $\pi$, $v$, and $\tau$, having taken the cerresponding critical values for the units of pressure, volume, and temperature. Using the equation obtained, find how many times the gas temperature exceeds its critical temperature if the gas pressure is 12 times as high as critical pressure, and the volume of gas is equal to half the critical volume.
2.201. Knowing the Van der Waals constants, find:
(a) the maximum volume which water of mass $m=1.00 \mathrm{~kg}$ can occupy in liquid state;
(b) the maximum pressure of the saturated water vapour.
2.202. Calculate the temperature and density of carbon dioxide in critical state, assuming the gas to be a Vas der Waals one.
2.203. What fraction of the volume of o vessel must liquid ether occupy at room temperature in order to pass inte critical state when critical temperature is reached? Ether has $T_{t r}=467 \mathrm{~K}, P_{e r}=35.5 \mathrm{~atm}$, $M=74 \mathrm{~g} / \mathrm{mol}$.
2.204. Demonstrate that the straight line 1.5 cerresponding to the isothermal-isobarie phase transition cuts the Van der Waals isotherm so that areas $I$ and $I I$ are equal (Fig. 2.5).
2.205. What fraction of water supercooled dows to the temperature $t=-20^{\circ} \mathrm{C}$ under standard pressure turns inte ice when the system passes inte the equilibrium state? At what temperature of the supercooled water does it turn inte ice completely?
2.206. Find the increment of the ice melting temperature in the vieinity of $0^{\circ} \mathrm{C}$ when the pressure is increased by $\Delta p=1.00 \mathrm{~atm}$. The specific volume of ice exceeds that of water by $\Delta V^{\prime}=0.091 \mathrm{~cm}^{3} / \mathrm{g}$.
2.207. Find the specific volume of saturated water vapour under standard pressure if a decrease of pressure by $\Delta p=3.2 \mathrm{kPa}$ is known to decrease the water boiling temperature by $\Delta T=0.9 \mathrm{~K}$.
2.208. Assuming the saturated water vapour to be ideal, find its pressure at the temperature $101.1^{\circ} \mathrm{C}$.
2.209. A small amount of water and its saturated vapour are enclosed in a vessel at a temperature $t=100^{\circ} \mathrm{C}$. How much (in per cent) will the mass of the saturated vapour increase if the temperature of the system goes ap by $\Delta T=1.5 \mathrm{~K}$ ? Assume that the vapour is an ideal gas and the specific volume of water is negligible as compared to that of vapour.
2.210. Find the pressure of saturated vapour as a function of temperature $p(T)$ if at a temperature $T$, its pressure equals $P_{0}$.
Assume that: the specifie latent heat of vaporization $q$ is independent of $T$, the specific volume of liguid is negligible as compared to that of vapour, saturated vapour obeys the equation of state for an ideal gas. Investigate under what conditions these assumptions are permis. sible.
2.211. An ice which was initially under standard conditions was compressed up to the pressure $p=640$ atm. Assuming the lowering of the ice melting temperature to be a linear function of pressure under the given conditions, find what fraction of the ice melted. The specific volume of water is less than that of ice by $\Delta V^{*}=0.09 \mathrm{~cm}^{3} / \mathrm{g}$.
2.212. In the viciaity of the triple point the saturated vapour pressure $P$ of carbon dioxide depends on temperature $T$ as $\log P=$ $=a-b / T$, where $a$ and $b$ are constants. If $p$ is expressed in atmospheres, then for the sublimation process $a-9.05$ and $b-1.80 \mathrm{kK}$, and for the vaporization process $a=6.78$ and $b=1.31 \mathrm{kK}$. Find:
(a) temperature and pressure at the triple point;
(b) the values of the specific latent heats of sublimation, vaporiration, and melting in the vicinity of the triple point.
2.213. Water of mass $m=1.00 \mathrm{~kg}$ is heated from the temperature $t_{1}=10^{\circ} \mathrm{C}$ up to $t_{2}=100^{\circ} \mathrm{C}$ at which it evaporates completely. Find the entropy increment of the system.
2.214. The ice with the initial temperature $t_{1}=0^{\circ} \mathrm{C}$ was first melted, then heated to the temperature $t_{4}=100^{\circ} \mathrm{C}$ and evaporated. Find the increment of the system’s specific entropy.
2.215. A piece of copper of mass $m=90 \mathrm{~g}$ at a temperature $t_{1}=$ $=90^{\circ} \mathrm{C}$ was placed in a calorimeter in which ice of mass $50 \mathrm{~g}$ was at a temperature $-3^{\circ} \mathrm{C}$. Find the entropy increment of the piece of copper by the moment the thermal equilibrium is reached.
2.216. A chunk of ice of mass $m_{1}=100 \mathrm{~g}$ at a temperature $t_{1}=$ $-0^{\circ} \mathrm{C}$ was placed in a calerimeter in which water of mass $m_{3}=$ $=100 \mathrm{~g}$ was at a temperature $t_{\mathrm{s}}$. Asuming the heat capacity of the calorimeter to be negligible, find the entropy increment of the system by the moment the thermal equilibrium is reached. Consider two cases: (a) $t_{1}=60^{\circ} \mathrm{C}$; (b) $t_{2}=94^{\circ} \mathrm{C}$.
2.217. Molten lead of mass $m=5.0 \mathrm{~g}$ at a temperature $t_{4}=327^{\circ} \mathrm{C}$ (the melting temperaturv of lead) was poured into a calorimeter packed with a large amount of ice at a temperature $t_{1}=0^{\circ} \mathrm{C}$. Find the entropy increment of the system lead-ice by the moment the thermal equilibrium is reached. The specific latent heat of melting of lead is equal to $q=22.5 \mathrm{~J} / \mathrm{g}$ and its specific heat capacity is equal to $\epsilon=$ $=0.125 \mathrm{~J} /(\mathrm{g} \cdot \mathrm{K})$.
2.218. A water vapour filling the space under the piston of a cylinder is compressed (or expanded) so that it remains saturated all the time, being just on the verge of condensation. Find the molar heat capacity $C$ of the vapour in this process as a function of temperature $T$, assuming the vapour to be an ideal gas and neglecting the specific volume of water in comparison with that of vapour. Calculate $C$ at a temperature $t=100^{\circ} \mathrm{C}$.
2.219. One mole of water being in equilibrium with a negligible amount of its saturated vapour at a temperature $r_{1}$ was completely converted into saturated vapour at a temperature $T_{2}$. Find the entropy increment of the system. The vapour is assumed to be an ideal gas, the specific volume of the liquid is negligible in comparison with that of the vapour.
2.7. TRANsPont PHENoMENA
– Aelative number of gas molecules traversing the distance s without col. lision:
\[
N / N_{0}=e^{-* / 4}
\]
where $\lambda$ is the meas free pach.
– Mean free path of a gnt molecele:
\[
\lambda=\frac{1}{\sqrt{2 \pi a_{n}}},
\]
where $d$ is the eflective diameter of a molecule, and $s$ is the number of molecules per unit volume.
– Coefficients of dilfusion D, viscosity 7 , and beat ceaductivity $x$ of gaves:
\[
D=\frac{1}{3}(n), n=\frac{1}{3} \text { (n), } n=\frac{1}{3}(n) \text { er } v \text {. }
\]
where $p$ is the gas density, and $\epsilon_{v}$ is its specific beat capacity at ceastant volume.
– Friction force acting on a unit arma of plater dering their metion parallel to each other in a hichly rarefied gas:
\[
F=\frac{1}{6}\left(n p\left|w_{1}-w_{0}\right|\right. \text {. }
\]
where $w_{1}$ and $w_{y}$ are the velocities of the plates.
– Density of a thermal flux translerted between twe walls by bighly rarefied gas:
\[
s=\frac{1}{6}(n) \rho_{v}\left|T_{1}-T_{1}\right| \text {. }
\]
where $T_{1}$ and $T_{4}$ are the temperatures of the walls.
2.220. Calculate what fraction of gas molecules
(a) traverses without collisions the distances exceeding the mean tree path $\lambda$;
(b) has the free path values lying within the interval from $\lambda$ to $2 \lambda$.
2.221. A narrow molecular beam makes its way inte a vessel filled with gas under low pressure. Find the mean free path of molecules if the beam intensity decreases $\eta$-fold over the distance $\Delta l$.
2.222. Let adt be the probability of a gas molecule experiencing a collision during the time interval dt; $\alpha$ is a constant. Find:
(a) the probability of a molecule experiencing no collisions during the time interval $t$;
(b) the mean time interval between successive collisions.
2.223. Find the mean free path and the mean time interval between successive collisions of gaseous nitrogen molecules
(a) under standard conditions:
(b) at temperature $t=0^{\circ} \mathrm{C}$ and pressure $p=1.0 \mathrm{nPa}$ (such a pressure can be reached by means of contemporary vacuum pumps).
2.224. How many times does the mean free path of nitrogen molecules exceed the mean distance between the molecules under standard conditions?
2.225. Find the mean free path of gas melecules under standard conditions if the Van der Waals constant of this gas is equal to $b=$ $=40 \mathrm{ml} / \mathrm{mol}$.
2.226. An acoustic wave propagates through nitrogen under standard conditions. At what frequency will the wavelength be equal to the mean free path of the gas molecules?
2.227. Oxygen is enclosed at the temperature $0^{\circ} \mathrm{C}$ in a vessel with the characteristic dimension $l=10 \mathrm{~mm}$ (this is the linear dimension determining the character of a physical process in question). Find:
(a) the gas pressure below which the mean free path of the molecules $\lambda>l$;
(b) the corresponding molecular concentration and the mean distance between the molecules.
2.228. For the case of nitrogen under standard conditions find:
(a) the mean number of cellisions experienced by each molecule per second;
(b) the total number of collisions occurring between the molecules within $1 \mathrm{~cm}^{3}$ of nitrogen per second.
2.229. How does the mean free path $\lambda$ and the number of collisions of each molecule per unit time v depend on the absolute temperature of an ideal gas undergoing
(a) an isochoric process;
(b) an isobaric process?
2.230. As a result of some process the pressure of an ideal gas increases $n$-fold. How many times have the mean free path $\lambda$ and the number of collisions of each molecule per unit time $v$ changed and how, if the process is
(a) isocheric; (b) isothermal?
2.231. An ideal gas consisting of rigid diatomie molecules goes through an adiabatic process. How do the mean free path $\lambda$ and the number of collisions of each molecule per second $v$ depend in this process on
(a) the volume $V$; (b) the pressure $p$; (c) the temperature $T$ ?
2.232. An ideal gas goes through a polytropic process with exponent $n$. Find the mean free path $\lambda$ and the number of collisions of each molecule per second $\mathrm{y}$ as a function of
(a) the volume $V$; (b) the pressure $p ;$ (c) the temperature $T$. 2.233. Determine the molar heat capacity of a polytropic process through which an ideal gas consisting of rigid diatomic molecules goes and in which the number of collisions between the molecules remains constant
(a) in a unit volume; (b) in the total volume of the gas.
2.234. An ideal gas of molar mass $M$ is enclosed in a vessel of volume $V$ whose thin walls are kept at a constant temperature $\boldsymbol{T}$. At a moment $t=0$ a small hole of area $S$ is opened, and the gas starts escaping inte vacuum. Find the gas concentration $n$ as a funetion of time $t$ if at the initial moment $n(0)=n_{c}$.
2.235. A vessel filled with gas is divided inte two equal parts $I$ and $z$ by a thin heat-insulating partition with two holes. One hole has a small diameter, and the other has a very large diameter (in comparison with the mean free path of molecules). In part 2 the gas is kept at a temperature $\eta$ times higher than that of part $i$. How will the concentration of molecules in part 2 change and how many times after the lange hole is closed?
2.236. As a result of a certain process the viscosity coefficient of an ideal gas increases $a=2.0$ times and its diffusion coefficient $\beta=4.0$ times. How does the gas pressure change and how many times?
2.237. How will a diflusion coefficient $D$ and the viscosity coeffieient $\eta$ of an ideal gas change if its volume increases $n$ times:
(a) isothermally; (b) isobarically?
2.238. An ideal gas consists of rizid diatomic molecules. How will a diflusion coefficient $D$ and viscosity coefficient $\eta$ change and how many times if the gas volume is decreased adiabatically $n=10$ times?
2.239. An ideal gas goes through a polytropic process. Find the polytropic exponent $n$ if in this process the coefficient
(a) of diffusion: (b) of viscosity; (c) of heat conductivity remains constant.
2.240. Knowing the viscosity coefficient of helium under standard conditions, calculate the effective diameter of the helium atom.
2.241. The heat conductivity of helium is 8.7 times that of argon (under standard conditions). Find the ratio of effective diameters of argon and helium atoms.
2.2i2. Under standard conditions helium fills up the space between two long coaxial cylinders. The mean radius of the cylinders is equal to $\boldsymbol{R}$, the gap between them is equal to $\Delta \boldsymbol{A}$, with $\Delta \boldsymbol{R}<\boldsymbol{R}$. The outer eylinder rotates with a fairly low angular velocity $\omega$ about the stationary inner cylinder. Find the moment of friction forces acting on a unit length of the inner cylinder. Down to what magnitude should the helium pressure be lowered (keeping the temperature constant) to decrease the sought moment of triction forces $n=10$ times if $\Delta A=6 \mathrm{~mm}$ ?
2.243. A gas fills up the space between two long ceaxial cylinders of radii $\boldsymbol{R}_{1}$ and $\boldsymbol{R}_{3}$, with $R_{1}<R_{3}$. The outer cylinder rotates with a fairly low angular velocity o about the stationary inser cylinder. The moment of friction forces acting on a unit length of the inser cylinder is equal to $\boldsymbol{N}_{\mathbf{1}}$. Find the viscosity coefficient $\eta$ of the gas taking into account that the friction force acting on a unit area of the cylindrical surface of radius $r$ is determined by the formula $\sigma=$ $=\pi(\partial \omega / \partial r)$.
2.2W. Two identical parallel discs have a common axis and are located at a distance $h$ from each other. The radius of each dise is equal to $a$, with $a>h$. One dise is rotated with a low angular veloc: ity o relative to the other, stationary, dise. Find the moment of friction forces acting on the stationary disc if the viscosity coeffeient of the gas between the dises is equal to $\eta$.
2.245. Solve the foregoing problem, assuming that the discs are located in an ultra-rarefied gas of molar mass $M$, at temperature $T$ and under pressure $p$.
2.246. Making use of Poiseuille’s equation (1.7d), find the mass p of gas flowing per unit time through the pipe of length $l$ and radius a if constant pressures $p_{1}$ and $p_{1}$ are maintained at its ends.
2.247. One end of a rod, enclosed in a thermally insulating sheath, is kept at a temperature $T_{1}$ while the other, at $T_{3}$. The rod is composed of two sections whose lengths are $h_{1}$ and $l_{2}$ and heat conductivity coeffieients $x_{1}$ and $x_{2}$. Find the temperature of the interface.
2.248. Two rods whose lengths are $l_{1}$ and $l_{2}$ and heat conductivity coefficients $x_{1}$ and $x_{2}$ are placed end to end. Find the heat conductivity coefficient of a uniform rod of length $l_{1}+l_{2}$ whose conductivity is the same as that of the system of these two rods. The lateral surfaces of the rods are assumed to be thermally insulated.
2.249. A rod of length $l$ with thermally insulated lateral surface consists of material whose heat conductivity coefficient varies with temperature as $x=a / T$, where $a$ is a constant. The ends of the rod are kept at temperatures $T_{1}$ and $T_{2}$. Find the function $T(x)$, where $x$ is the distance from the end whose temperature is $T_{1}$, and the heat flow density.
2.250. Two chunks of metal with heat capacities $c_{1}$ and $c_{\text {, are }}$ interconnected by a rod of length $l$ and cross-sectional area $S$ and fairly low heat conductivity $x$. The whole system is thermally insulated from the environment. At a moment $t=0$ the temperature difference between the two chunks of metal equals ( $\Delta T$ ). Assuming the heat capacity of the rod to be neglizible, find the temperature difference between the chunks as a function of time.
2.251. Find the temperature distribution in a substance placed between two parallel plates kept at temperatures $T_{1}$ and $T_{2}$. The plate separation is equal to $h$, the heat conductivity coefficient of the substance $x \bar{\propto} V \bar{Y}$.
2.252. The space between two large hurizental plates is filled with helium. The plate separation equals $l=50 \mathrm{~mm}$. The lower plate is kept at a temperature $T_{1}=290 \mathrm{~K}$, the upper, at $T_{2}=$ $=330 \mathrm{~K}$. Find the heat flow density if the gas pressure is close to standard.
2.253. The space between two large parallel plates separated by a distance $l=5.0 \mathrm{~mm}$ is filled with helium under a pressure $p=$ – 1.0 Pa. One plate is kept at a temperature $t_{1}=17^{\circ} \mathrm{C}$ and the other, at a temperature $t_{1}=37^{\circ} \mathrm{C}$. Find the mean inee path of helium atoms and the heat flow density.
2.254. Find the temperature distribution in the space between two coaxial cylinders of radii $R_{1}$ and $R_{2}$ filled with a uniform heat conducting substance if the temperatures of the cylinders are constant and are equal to $T_{1}$ and $T_{2}$ respectively.
2.255. Solve the foregoing problem for the case of two concentric spheres of radii $R_{1}$ and $R_{2}$ and temperatures $T_{1}$ and $T_{2}$.
2.256. A constant electric current flows along a uniform wire with cross-sectional radius $R$ and heat conductivity coefficient $x$. A unit volume of the wire generates a thermal power $w$. Find the temperature distribution across the wire provided the steady-state temperature at the wire surface is equal to $T_{0}$.
2.257. The thermal power of density $w$ is generated uniformly inside a uniform sphere of radius $R$ and heat conductivity coefficient $x$. Find the temperature distribution in the sphere provided the steady-state temperature at its surface is equal to $T_{0}$.