– Ideal gas law:
\[
p V=\frac{m}{M} R T
\]
where $M$ is the molar mass.
– Barometric formula:
\[
p=p_{0} \mathrm{e}^{-M g h / R T},
\]
where $p_{0}$ is the pressure at the height $h=0$.
– Van der Waals equation of gas state (for a mole):
\[
\left(p+\frac{a}{V_{M}^{2}}\right)\left(V_{M}-b\right)=R T,
\]
where $V_{M}$ is the molar volume under given $p$ and $T$.
2.1. A vessel of volume $V=301$ contains ideal gas at the temperature $0^{\circ} \mathrm{C}$. After a portion of the gas has been let out, the pressure in the vessel decreased by $\Delta p=0.78 \mathrm{~atm}$ (the temperature remaining constant). Find the mass of the released gas. The gas density under the normal conditions $\rho=1.3 \mathrm{~g} / \mathrm{l}$.
2.2. Two identical vessels are connected by a tube with a valve letting the gas pass from one vessel into the other if the pressure difference $\Delta p \geqslant 1.10 \mathrm{~atm}$. Initially there was a vacuum in one vessel while the other contained ideal gas at a temperature $t_{1}=27^{\circ} \mathrm{C}$ and pressure $p_{1}=1.00 \mathrm{~atm}$. Then both vessels were heated to a temperature $t_{2}=107{ }^{\circ} \mathrm{C}$. Up to what value will the pressure in the first vessel (which had vacuum initially) increase?
2.3. A vessel of volume $V=201$ contains a mixture of hydrogen and helium at a temperature $t=20^{\circ} \mathrm{C}$ and pressure $p=2.0 \mathrm{~atm}$. The mass of the mixture is equal to $m=5.0 \mathrm{~g}$. Find the ratio of the mass of hydrogen to that of helium in the given mixture.
2.4. A vessel contains a mixture of nitrogen $\left(m_{1}=7.0 \mathrm{~g}\right)$ and carbon dioxide $\left(m_{2}=11 \mathrm{~g}\right)$ at a temperature $T=290 \mathrm{~K}$ and pressure $p_{0}=1.0 \mathrm{~atm}$. Find the density of this mixture, assuming the gases to be ideal.
2.5. A vessel of volume $V=7.51$ contains a mixture of ideal gases at a temperature $T=300 \mathrm{~K}: v_{1}=0.10$ mole of oxygen, $v_{2}=0.20$ mole of nitrogen, and $v_{3}=0.30$ mole of carbon dioxide. Assuming the gases to be ideal, find:
(a) the pressure of the mixture;
(b) the mean melar mass $M$ of the given mixture which enters its equation of state $p V=(m / M) R T$, where $m$ is the mass of the mixture.
2.6. A vertical cylinder closed from both ends is equipped with an easily moving pisten dividing the volume inte two parts, each con: taining one mole of air. In equilibrium at $T_{6}=300 \mathrm{~K}$ the volume of the upper part is $n=4.0$ times greater than that of the lower part. At what temperature will the ratio of these volumes be equal to $\eta^{\prime}=3.0$ ?
2.7. A vessel of volume $\boldsymbol{V}$ is evacuated by means of a piston air pump. One piston stroke captures the volume $\Delta V$. How many strokes are needed to reduce the pressure in the vessel $n$ times? The process is assumed to be isothermal, and the gas ideal.
2.8. Find the pressure of air in a vessel being evacuated as a funetion of evacuation time $t$. The vessel volume is $\boldsymbol{V}$, the initial pressure is $P_{0}$. The process is assumed to be isothermal, and the evacuation rate equal to $C$ and independent of pressure.

Note. The evacuation rate is the gas volume being evacuated per unit time, with that volume being measured under the gas pressure attained by that moment.
2.9. A chamber of volume $V=871$ is evacuated by a pump whose evacuation rate (see Note to the foregoing problem) equals $C=$ $=10 \mathrm{l} / \mathrm{s}$. How soon will the pressure in the chanber decrease by $\eta=1000$ times?
2.10. A smooth vertical tube having two different sections is epen from both ends and equipped with two pistons of diflerent areas (Fig.2.1). Each piston slides within a respective tube section. One mole of ideal gas is enclosed between the pistons tied with a non-stretchable thread. The cross. sectional are of the upper piston is $\Delta S=10 \mathrm{~cm}^{2}$ greater than that of the lower one. The combined mass of the two pistons is equal to $m=5.0 \mathrm{~kg}$. The outside air pressure is $p_{0}=1.0 \mathrm{~atm}$. By how many kelvins must the gas between the pistons be heated to shift the pistons through $t=5.0 \mathrm{~cm}$ ?
2.11. Find the maximum attainable temperature of ideal gas in esch of the following processes:
(a) $p=p_{0}-a V^{2}$; (b) $p=p_{0} e^{-t V}$,
where $P_{0,} \alpha$ and $\bar{B}$ are positive constants, and $V$ is the velume of one mole of gas.
2.12. Find the minimum attainable pressure of ideal gas in the process $T=T_{0}+\alpha V^{\prime}$, where $T_{\text {, and }} \alpha$ are positive constants, and $\boldsymbol{V}$ is the volume of one mole of gas. Draw the approximate $p$ vs $\boldsymbol{V}$ plot of this process.
2.13. A tall eylindrical vessel with gaseous nitrogen is located in a uniform gravitational field in which the free-fall scceleration is equal to $\mathrm{g}$. The temperature of the nitrogen varies along the height 76
$h$ so that its density is the same throughout the volume. Find the teaperature gradient $d T / d h$.
2.14. Suppose the prossure $p$ and the density $p$ of air are related as $p / p^{n}=$ const regardless of height ( $n$ is a constant here). Find the corresponding temperature gradient.
2.15. Let us assume that air is under standard conditions close to the Barth’s surface. Presuming that the temperature and the molar mass of air are independent of height, find the air pressure at the height $5.0 \mathrm{~km}$ over the surface and in a mine at the depth $5.0 \mathrm{~km}$ below the surface.
2.16. Assuming the temperature and the molar mass of air, as well as the free-fall acceleration, to be independent of the height, find the difference in heights at which the air densities at the temperature $0^{\circ} \mathrm{C}$ differ
(a) e times; (b) by $\eta=1.0 \%$.
2.17. An ideal gas of molar mass $M$ is centained in a tall vertical eylindrical vessol whose base area is $S$ and height $h$. The temperature of the gas is $T$, its pressure on the bottom base is $P_{y}$. Assuming the tempenture and the free-fall acceleration of to be independent of the height, find the mass of gas in the vessol.
2.18. An ideal gas of golar mase $M$ is contained in a very tall vertical eylindrical vessel in the uniform gravitational field in which the free-fall acceleration equals $\mathrm{g}$. Assuming the gas temperature to be the same and equal to $T$, find the height at which the centre of gravity of the gas is lecated.
2.19. An ideal gas of molar mass $M$ is located in the uniform gravitational field in which the freo-fall acceleration is equal to g. Find the gas pressure as a function of height $h$, if $p=p_{0}$ at $h=0$, and the temperature varies with height as
(a) $T=T_{0}(1-a h) ;$ (b) $T=T_{0}(1+a b)$, where $a$ is a positive constant.
2.20. A horizontal cylinder closed from one end is rotated with a constant angular velocity e about a vertical axis passing through the open end of the cylinder. The outside air pressure is equal to $P_{p}$. the temperature to $T$, and the molar mass of air to M. Find the air presure as a fanction of the distance $r$ from the rotation axis. The molar mass is assumed to be independent of $r$.
2.21. Under what pressure will carbon dioxide have the density $\rho=500 \mathrm{~g} / \mathrm{l}$ at the temperature $T=300 \mathrm{~K}$ ? Carry out the calculations both for an ideal and for a Van der Waals gas.
2.22. One mole of nitregen is contaised in a vessel of volume $\boldsymbol{V}=$ – 1.00 1. Find:
(a) the temperature of the nitrogen at which the pressure can be calculated from an ideal gas law with an error $\eta=10 \%$ (as compared with the pressure calculated from the Van der Waals equation of state);
(b) the gas pressure at this temperature.
2.23. One mole of a certain gas is contained in a vessel of volume $\boldsymbol{V}=0.250 \mathrm{l}$. At a temperature $T_{1}=300 \mathrm{~K}$ the gas pressure is $p_{1}=$
$n$

= 90 atm, and at a temperature $T_{2}=350 \mathrm{~K}$ the pressure is $\mathrm{p}_{2}=$ $=110$ atm. Find the Van der Waals parameters for this gas.
2.24. Find the isothermal compressibility $x$ of a Vas der Waals $g$ as as a function of volume $V$ at temperature $T$.
Note. By definition, $x=-\frac{1}{V} \frac{\partial V}{\partial p}$.
2.25. Making use of the result obtained in the foregoing problem, find at what temperature the isothermal compressibility $x$ of a Van der Waals gas is greater than that of an ideal gas. Examine the case when the molar volume is much greater than the parameter $b$. HEAT CAPACTY