– The first law of thermedyanics:
\[
Q=\Delta U+A \text {. }
\]
where $\Delta V$ is the jecrement of the internal energ of the votem.
– Work performed by ore:
\[
A=\int P d V .
\]
– Internal energy of an ideal gas:
\[
U=\frac{\mathrm{m}}{M} C_{v} r=\frac{\mathrm{m}}{M} \frac{R T}{\gamma-1}=\frac{p^{V}}{\gamma-1} .
\]
– Molar heat eapacity in a pelytropie proces $6 \mathrm{Vn}^{\mathrm{n}}=$ eonat):
\[
c=\frac{R}{\gamma-1}-\frac{R}{n-1}=\frac{(n-7) R}{(n-1)(\gamma-1)} .
\]
– Internal energy of one mole of a Vas der Waals gas:
\[
U=C_{V} T-\frac{a}{V_{M}} \text {. }
\]
2.26. Demonstrate that the interval energy $U$ of the air in a room is independent of temperature provided the outside pressure $P$ is constant. Calculate $U$, if $p$ is equal to the nermal atmospheric pressure and the room’s volume is equal to $V=40 \mathrm{~m}^{2}$.
2.27. A thermally insulated vessel containing a gas whose molar mass is equal to $\boldsymbol{M}$ and the ratio of specifie hents $C_{p} / C_{v}=\gamma$ moves with a velocity $v$. Find the gas temperature increment resulting from the sudden steppage of the vessel.
2.28. Two thermally insulated vessels $I$ and 2 are flled with air and connected by a short tube equipped with a valve. The volumes of the vessels, the pressures and temperatures of air in them are known $\left(V_{1}, P_{1}, T_{1}\right.$ and $\left.V_{n}, P_{2}, T_{2}\right)$. Find the air temperature and pressure established after the opening of the valve.
2.29. Gaseous hydrogen contained initially under standard conditions in a sealed vessel of volume $V=5.01$ was cooled by $\Delta T=$
– $55 \mathrm{~K}$. Find how much the internal energy of the gas will change and what amount of heat will be lost by the gas.
2.30. What amount of heat is to be transferred to nitregen in the isobaric heating process for that gas to perform the work $A=2.0 \mathrm{~J}$ ?
2.31. As a result of the isobaric heating by $\Delta T=72 \mathrm{~K}$ one mole of a certain ideal gas obtains an amount of heat $Q=1.60 \mathrm{~kJ}$. Find the work performed by the gas, the increment of its internal energy, and the value of $\gamma=c_{\gamma} / c_{y}$.
2.32. Twe moles of a certain ideal gas at a temperature $T_{4}=300 \mathrm{~K}$ were cooled isochorically so that the gas pressure reduced $n=2.0$ times. Then, as a result of the isobaric process, the gas expanded till its temperature got back to the initial value. Find the total amount of heat absorbed by the gas in this process.
2.33. Calculate the value of $\gamma=C_{p} / C_{\gamma}$ for a gaseous mixture consisting of $y_{1}=2.0$ moles of oxygen and $y_{2}=3.0$ meles of carbon diexide. The gases are assumed to be ideal.
2.34. Find the specific heat capacities $c_{y}$ and $c_{\text {p }}$ for a gaseous mixture consisting of $7.0 \mathrm{~g}$ of nitrogen and $20 \mathrm{~g}$ of argon. The gases are assumed to be ideal.
2.35. One mole of a certain ideal gas is contained under a weightless piston of a vertical cylinder at a temperature $\boldsymbol{T}$. The space over the piston opens inte the atmesphere. What work has to be performed in order to increase isothermally the gas volume under the piston $n$ times by slowly raising the piston? The friction of the piston against the cylinder walls is negligibly small.
2.36. A piston cas freely move inside a horizontal cylinder closed from both ends. Initially, the piston soparates the inside space of the cylinder inte two equal parts each of volume $V_{6}$, in which an ideal $\mathrm{gas}$ is contained under the same pressure $\mathrm{p}_{g}$ and at the same temperature. What work has to be performed in order to increase isothermally the volume of one part of gas $\eta$ times compared to that of the other by slowly meving the piston?
2.37. Three moles of an ideal gas being initially at a tomperature $r_{4}=273 \mathrm{~K}$ were isothermally expanded $n=5.0$ times its initial volume and then isocherically heated so that the pressure in the final state became equal to that in the initial state. The total amount of heat transferred to the gas during the process equals $Q=80 \mathrm{~kJ}$. Find the ratio $\gamma-C_{
u} / C_{v}$ for this gas.
2.38. Draw the approximate plots of isochoric, isobaric, isothermal, and adiabatic processes for the case of an ideal gas, using the following variables:
(a) $P, T$; (b) $\boldsymbol{Y}, \boldsymbol{T}$.
2.39. One mole of exygen being initially at a temperature $T,=$ – $290 \mathrm{~K}$ is adiabatically compressed to increase its pressure $\eta=$ – 10.0 times. Find:
(a) the gas temperature after the compression;
(b) the work that has been performed on the gas.
2.40. A certain mass of nitrogen was compressed $\eta=5.0$ times
(in terms of volume), first adiabatically, and then isothermally. In both cases the initial state of the gas was the same. Find the ratio of the respective works expended in each compression.
2.41. A heat-conducting piston can freely move inside a closed thermally insulated cylinder with an ideal gas. In equilibrium the pisten divides the cylinder into two equal parts, the gas temperature being equal to $T_{4}$. The piston is slowly displaced. Find the gas temperature as a function of the ratio $n$ of the volumes of the greater and smaller sections. The adiabatic exponent of the gas is equal to $\gamma$.
2.42. Find the rate $v$ with which helium flows out of a thermally insulated vessel inte vacuum through a sanll hole. The flow rate of the gas inside the vessel is assumed to be negligible under these conditions. The temperature of helium in the vessel is $T=1,000 \mathrm{~K}$.
2.43. The volume of one mole of in ideal gas with the adiabatic exponent $\gamma$ is varied according to the $\operatorname{la} w=a / T$, where $a$ is a constant. Find the amount of heat ebtained by the gas in this process if the gas temperature increased by $\Delta T$.
2.44. Demonstrate that the process in which the work performed by an ideal gas is proportional to the corresponding increment of its internal energy is described by the equation $p^{*}=$ censt, where $n$ is a constant.
2.45. Find the molar heat capacity of an ideal gas in a polytropie process $p^{*}=$ const if the adiabatic exponent of the gas is equal to $\gamma$. At what values of the polytropic constant $n$ will the heat capacity of the gas be negative?
2.46. In a certain polytropic process the volume of argon was inereased $a=4.0$ times. Simultaneously, the pressure decreased $\beta=8.0$ times. Find the molar heat capacity of argon in this process, assuming the gas to be ideal.
2.47. One mole of argon is expanded polytropically, the polytropic constant being $n=1.50$. In the process, the gas temperature changes by $\Delta T=-26 \mathrm{~K}$. Find:
(a) the amount of heat obtained by the gas;
(b) the work performed by the gas.
2.48. An ideal gas whose adiabatic exponent equals $\gamma$ is expanded according to the law $p=a V$, where $a$ is a coastant. The initial volume of the gas is equal to $V_{4}$. As a result of expansion the volume increases $\eta$ times. Find:
(a) the increment of the internal energy of the gas;
(b) the work performed by the gas;
(c) the molar heat capacity of the gas in the process.
2.49. An ideal gas whose adiabatic exponent equals $\gamma$ is expanded so that the amount of heat transferred to the gas is equal to the decrease of its internal energy. Find:
(a) the molar heat capacity of the gas in this process;
(b) the equation of the process in the variables $T, V$;
(c) the work performed by one mole of the gas when its volume increases $\eta$ times if the initial temperature of the gas is $T_{0}$.
2.50. One mele of an ideal gas whose adiabatic expenent equals $\gamma$ undergoes a process in which the gas pressure relates to the temperature as $p=a T$, where $a$ and $a$ are constants. Find:
(a) the work performed by the gas if its temperature gets an increment $\Delta T_{\text {; }}$
(b) the molar heat capacity of the gas in this process; at what value of $a$ will the hest capacity be negative?
2.51. An ideal gas with the adiabatic exponent $\gamma$ undergoes a process in which its internal energy relates to the volume as $U=\mathrm{aV}^{\mathrm{a}}$, where $a$ and $a$ are constants. Find:
(a) the work performed by the gas and the amount of heat to be transferred to this gas to increase its internal energy by $\Delta U$;
(b) the melar heat capacity of the gas in this process.
2.52. An ideal gas has a molar heat capacity $C_{V}$ at censtant volume. Find the molar heat capacity of this gas as a function of its volume $\boldsymbol{V}$, if the gas undergoes the following process:
(a) $T=T_{0} \mathrm{e}^{a V} ;$ (b) $p=p_{0} e^{a V}$, where $T_{8}, P_{6}$, and $\alpha$ are censtants.
2.53. One mole of an ideal gas whose adiabatic exponent equals $\gamma$ undergoes a process $p=p_{0}+\alpha / V$, where $p_{0}$ and $\alpha$ are positive constants. Find:
(a) heat capacity of the gas as a function of its volume;
(b) the internal energy increment of the gas, the work performed by it, and the amount of heat transferred to the gas, if its volume increased from $V_{1}$ to $V_{\text {, }}$.
2.54. One mole of an ideal gas with heat capacity at constant presure $C_{\text {, }}$ undergoes the process $T=T_{0}+a V$, where $T_{\text {, and }} a$ are constants. Find:
(a) heat capacity of the gas as a function of its volume;
(b) the amount of heat transferred to the gas, if its volume increased frem $V_{1}$ to $V_{1}$.
2.55. For the case of an ideal gas find the equation of the process (in the variables $T, V$ ) in which the molar heat capacity varies as:
(a) $c=C_{v}+a r$;
(b) $c=c_{V}+\beta V_{\text {; }}$
(c) $C=C_{v}+a p$. where $\alpha, \beta$, and $a$ are constants.
2.56. An ideal gas has an adiabatic expenent $\gamma$. In some process its molar heat capacity varies as $C=a / T$, where $a$ is a constant. Find:
(a) the work performed by one mole of the gas during its heating from the temperature $T$, to the temperature $\eta$ times higher;
(b) the equation of the process in the variables $p, \boldsymbol{V}$.
2.57. Find the work performed by one mole of a Van der Waals gas during its isothermal expansion from the volume $V_{1}$ to $V_{2}$ at a temperature $T$.
2.58. One mole of exyzen is expanded from a velume $V_{1}=$ -1.00 I te $V_{1}=5.0 \mathrm{I}$ at a constant temperature $T=280 \mathrm{~K}$. Calculate:
(a) the increment of the internal energy of the gas:
(b) the amount of the absorbed heat.

The gas is assumed to be a Van der Waals gas.
2.59. For a Van der Waals gas find:
(a) the equation of the adiabatic curve in the variables $T, V$;
(b) the difference of the molar heat capacities $C_{v}-C_{v}$ as a function of $T$ and $V$.
2.60. Twe thermally insulated vesels are intereonnected by a tube equipped with a valve. One vessel of volume $V_{1}=101$ contains $v=2.5$ moles of carbon dioxide. The other vessel of volume $V_{1}=$ $100 \mathrm{I}$ is evacuated. The valve having been opened, the gas adiabatically expanded. Assuming the gas to obey the Van der Waals equation, find its temperature change accompanying the expansion.
2.61. What amount of heat has to be transferred to $v=3.0$ moles of carbon dioxide to keep its temperature constant while it expands inte vacuum trom the volume $V_{1}=5.0 \mathrm{I}$ to $V_{2}=10 \mathrm{I}$ The gas is assumed to be a Van der Waals gas.