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We have $0.5 \mathrm{~g}$ of hydrogen in a cubic chamber of size $3 \mathrm{~cm}$ kept at NTP. The gas in the chamber is compressed keeping the temperature constant till a final pressure of $100 \mathrm{~atm}$. Is one justified in assuming the ideal gas law, in the final state? (Hydrogen molecules can be consider as spheres of radius $1 Å$ ).
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Consider hydrogen molecules as spheres of radius $1 Å$. So, $r=1 Å=$ radius $=10^{-10} \mathrm{~m}$ Volume of hydrogen molecules $=\frac{4}{3} \pi r^3$ $\begin{aligned} \text { Volume of } 1 \text { molecule } &=\frac{4}{3}(3.14)\left(10^{-10}\right)^3 \\ &=4 \times 10^{-30} \mathrm{~m}^3 \end{aligned}$
Number of moles is $0.5 \mathrm{~g}$ of $\mathrm{H}_2$ gas
$$
\begin{aligned}
&=\frac{\text { Mass }}{\text { Molecular mass }} \\
&=\frac{0.5}{2}=0.25 \text { mole }\left(\mathrm{H}_2 \text { has } 2 \text { mole }\right)
\end{aligned}
$$
Molecules of $\mathrm{H}_2$ present
$=$ Number of moles of $\mathrm{H}_2$ present $\times 6.023 \times 10^{23}$
$=0.25 \times 6.023 \times 10^{23}$
$\therefore$ Volume of $\mathrm{H}_2$ molecules
$=$ number of molecules $\times$ volume of each molecule
$=0.25 \times 6.023 \times 10^{23} \times 4 \times 10^{-30} \mathrm{~m}^3$
$=6.023 \times 10^{23} \times 10^{-30} \approx 6 \times 10^{-7} \mathrm{~m}^3$
Now for ideal gas at constant temperature is considered to be constant
$$
p_i V_i=p_f V_f
$$
$\begin{aligned} V_f &=\left(\frac{p_i}{p_f}\right) V_i=\left(\frac{1}{100}\right)\left(3 \times 10^{-2}\right)^3 \\ &=\frac{27 \times 10^{-6}}{10^2}=2.7 \times 10^{-7} \mathrm{~m}^3 \end{aligned}$
$\left(\therefore\right.$ Volume of cube $V_i=(\text { Side })^3$ and $p_i=1$ atm at NTP. $)$ Hence on compression the volume of the gas is of the order of the molecular volume [from Eq. (i) and Eq. (ii)]. If the intermolecular forces will play the role as in kinetic theory of gas, molecules do not interact each other so gas will deviate from ideal gas behaviour.
Number of moles is $0.5 \mathrm{~g}$ of $\mathrm{H}_2$ gas
$$
\begin{aligned}
&=\frac{\text { Mass }}{\text { Molecular mass }} \\
&=\frac{0.5}{2}=0.25 \text { mole }\left(\mathrm{H}_2 \text { has } 2 \text { mole }\right)
\end{aligned}
$$
Molecules of $\mathrm{H}_2$ present
$=$ Number of moles of $\mathrm{H}_2$ present $\times 6.023 \times 10^{23}$
$=0.25 \times 6.023 \times 10^{23}$
$\therefore$ Volume of $\mathrm{H}_2$ molecules
$=$ number of molecules $\times$ volume of each molecule
$=0.25 \times 6.023 \times 10^{23} \times 4 \times 10^{-30} \mathrm{~m}^3$
$=6.023 \times 10^{23} \times 10^{-30} \approx 6 \times 10^{-7} \mathrm{~m}^3$
Now for ideal gas at constant temperature is considered to be constant
$$
p_i V_i=p_f V_f
$$
$\begin{aligned} V_f &=\left(\frac{p_i}{p_f}\right) V_i=\left(\frac{1}{100}\right)\left(3 \times 10^{-2}\right)^3 \\ &=\frac{27 \times 10^{-6}}{10^2}=2.7 \times 10^{-7} \mathrm{~m}^3 \end{aligned}$
$\left(\therefore\right.$ Volume of cube $V_i=(\text { Side })^3$ and $p_i=1$ atm at NTP. $)$ Hence on compression the volume of the gas is of the order of the molecular volume [from Eq. (i) and Eq. (ii)]. If the intermolecular forces will play the role as in kinetic theory of gas, molecules do not interact each other so gas will deviate from ideal gas behaviour.
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