I E Irodov Solutions for Chapter: THERMODYNAMICS AND MOLECULAR PHYSICS, Exercise 1: EQUATION OF THE GAS STATE. PROCESSES

A vessel of volume $V=7.5 \mathrm{L}$, contains a mixture of ideal gases at a temperature $T=300 \mathrm{K}$, $v_1=0.10 \mathrm{mole}$ of oxygen, $v_2=0.20 \mathrm{mole}$ of nitrogen and $v_3=0.30 \mathrm{mole}$ of carbon dioxide. Assuming the gases to be ideal, find

(a) the pressure of the mixture,

(b) the mean molar mass $M$ of the given mixture, which enters its equation of state $pV=\left(\frac{m}{M}\right) RT,$ where, $m$ is the mass of the mixture.

$\left(\text{Use} R=0.082 \mathrm{L} \mathrm{atm} {\mathrm{mol}}^{-1} {\mathrm{K}}^{-1}\text{ and molecular mass, }{M}_{{\mathrm{N}}_2}=28 \mathrm{g} {\mathrm{mol}}^{-1}\text{,} M_{{\mathrm{O}}_2}=32 \mathrm{g} {\mathrm{mol}}^{-1}\text{and }{M}_{{\mathrm{CO}}_2}=44 \mathrm{g} {\mathrm{mol}}^{-1}\right)$

A vertical cylinder closed from both ends is equipped with an easily moving piston dividing the volume into two parts, each containing one mole of air. In equilibrium at $T_0=300 \mathrm{K}$ the volume of the upper part is $\mathrm{\eta }=4.0$ times greater than that of the lower part. At what temperature will the ratio of these volumes be equal to ${\mathrm{\eta }}^{\text{'}}=3.0?$

A vessel of volume $V$ is evacuated by means of a piston air pump. One piston stroke captures the volume $\mathrm{\Delta }V$. How many strokes are needed to reduce the pressure in the vessel $\eta$ times? The process is assumed to be isothermal and the gas, ideal.

Find the pressure of air in a vessel being evacuated as a function of evacuation time $t$. The vessel's volume is $V$ and 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.

A chamber of volume $V=87 \mathrm{L}$ is evacuated by a pump, whose evacuation rate equals $C=10 \mathrm{L} {\mathrm{s}}^{-1}$. How soon will the pressure in the chamber decrease by $\eta =1000$ times?

Given that $p=p_0{e}^{\frac{-Ct}{v}}$, where, $p$ is the pressure as a function of time.

A smooth vertical tube, having two different sections, is open from both ends and equipped with two pistons of different areas. 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 area of the upper piston is $\mathrm{\Delta }S=10 {\mathrm{cm}}^2$, 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 $\mathrm{kelvin}$ must the gas between the pistons be heated, to shift the pistons through $l=5.0 \mathrm{cm}$?

Find the maximum attainable temperature of an ideal gas, in each of the following processes,

(a) $p=p_0-\alpha V^2$,

(b) $p=p_0 {\mathrm{e}}^{-\beta V}$,

where, $p_0, \alpha$ and $\beta$ are the positive constants and $V$ is the volume of one mole of gas.

Find the minimum attainable pressure of ideal gas, in the process $T=T_0+\alpha V^2$, where, $T_0$ and $\alpha$ are positive constants and $V$ is the volume of one mole of gas. Draw the approximate $p$ vs $V$ plot of this process.