Question

A bulb of capacity \[{\text{1d}}{{\text{m}}^{\text{3}}}\] contains\[{\text{1}}{\text{.03}} \times {\text{1}}{{\text{0}}^{{\text{23}}}}\]\[{{\text{H}}_{\text{2}}}\], molecules and pressure exerted by these molecules is \[{\text{101}}{\text{.325}}\]\[{\text{kpa}}\]. Calculate the average square molecular speed and the temperature.
A) \[{\text{4}}{\text{.44}} \times 1{{\text{0}}^{\text{5}}}{\left( {{\text{m/s}}} \right)^{\text{2}}}{\text{;171}}{\text{.27K}}\]
B) \[{\text{8}}{\text{.88}} \times {\text{1}}{{\text{0}}^{\text{5}}}{\left( {{\text{m/s}}} \right)^{\text{2}}}{\text{;71}}{\text{.27K}}\]
C) \[{\text{8}}{\text{.88}} \times {\text{1}}{{\text{0}}^{\text{7}}}{\left( {{\text{m/s}}} \right)^{\text{2}}}{\text{;35}}{\text{.58K}}\]
D) \[{\text{1}}{\text{.224}}{\left( {{\text{m/s}}} \right)^{\text{2}}}{\text{;71}}{\text{.27K}}\]

Answer 1

Hint: As we know that hydrogen is one of the elements in the periodic table. Hydrogen is the first element in the periodic table. The atomic number of hydrogen is \[1\]. The symbol of hydrogen is \[{\text{H}}\]. We need to know that a mole is defined as the given mass of the molecule divided by the molecular mass of the molecule.

Formula used:

\[{\text{moles}}\,{\text{ = }}\dfrac{{{\text{mass}}\,{\text{ofthe}}{\text{molecule}}}}{{{\text{molecular weight of the molecule}}}}\]
The number of atoms of the element is equal to the number of moles of the atom multiplied by Avogadro’s number. The numerical value of Avogadro’s number is \[6.022 \times {10^{23}}\].
\[{\text{The Number Of Atoms = number of moles x 6}}{\text{.022 x 1}}{{\text{0}}^{{\text{23}}}}\]
The ideal gas equation depends on the pressure, temperature, number of moles, volume of the gas molecules in ideal condition.
The ideal gas equation is,
\[{\text{PV = nRT}}\]
The average square molecular speed is
\[{\text{The Average Square Molecular Speed = }}\dfrac{{{\text{3RT}}}}{{\text{M}}}\]
Here, the pressure of the gas is P
The volume of the gas is V
Gas constant is R
The number of moles is n
The molar mass is M.

Complete answer:
The given is
The pressure of the hydrogen gas p is \[{\text{101}}{\text{.325}}\]\[{\text{kpa}}\]
The volume of the gas V is \[{\text{1d}}{{\text{m}}^{\text{3}}}\]
The temperature of the gas T
Gas constant R is \[8.314\]
The molar mass is M
The number of molecules of hydrogen in container is \[{\text{1}}{\text{.03}} \times {\text{1}}{{\text{0}}^{{\text{23}}}}\]
The number of moles we calculated,
The number of molecules of the element is equal to the number of moles of the molecules multiplied by Avogadro’s number. The numerical value of Avogadro’s number is \[6.022 \times {10^{23}}\].
\[{\text{The Number Of Molecules = number of moles x 6}}{\text{.022 x 1}}{{\text{0}}^{{\text{23}}}}\]
\[{\text{number of moles = }}\dfrac{{{\text{The Number Of Molecules}}}}{{{\text{6}}{\text{.022 x 1}}{{\text{0}}^{{\text{23}}}}}}\]
\[{\text{number of moles = }}\dfrac{{{\text{1}}{\text{.03}} \times {\text{1}}{{\text{0}}^{{\text{23}}}}}}{{{\text{6}}{\text{.022 x 1}}{{\text{0}}^{{\text{23}}}}}}\]
\[ = 0.171{\text{moles}}\]
\[0.171{\text{moles}}\] of hydrogen in a container.
We calculated the temperature of the gas in kelvin T,
 \[{\text{PV = nRT}}\]
\[{\text{T = }}\dfrac{{{\text{PV}}}}{{{\text{nR}}}}\]
\[{\text{T = }}\dfrac{{{\text{101}}{\text{.325}} \times 1}}{{0.171 \times 8.314}}\]
\[{\text{T}} = 71.27{\text{k}}\]
The temperature of the gas in kelvin T is \[71.27{\text{k}}\].
The average square molecular speed is
\[{\text{The Average Square Molecular Speed = }}\dfrac{{{\text{3RT}}}}{{\text{M}}}\]
\[{\text{ = }}\dfrac{{{\text{3}} \times 8.314 \times 71.27}}{{2 \times {{10}^{ - 3}}}} = {\text{8}}{\text{.88}} \times {\text{1}}{{\text{0}}^{\text{5}}}{\left( {{\text{m/s}}} \right)^{\text{2}}}\]
The average square molecular speed is \[{\text{8}}{\text{.88}} \times {\text{1}}{{\text{0}}^{\text{5}}}{\left( {{\text{m/s}}} \right)^{\text{2}}}\]
According to above calculation, we conclude is The average square molecular speed is \[{\text{8}}{\text{.88}} \times {\text{1}}{{\text{0}}^{\text{5}}}{\left( {{\text{m/s}}} \right)^{\text{2}}}\] and The temperature of the gas in kelvin T is \[71.27{\text{k}}\].

Hence, option B is correct answer \[{\text{8}}{\text{.88}} \times {\text{1}}{{\text{0}}^{\text{5}}}{\left( {{\text{m/s}}} \right)^{\text{2}}}{\text{;71}}{\text{.27K}}\].

Note:
We need to remember that the atomic number of the atom is nothing but the number of protons or the number of electrons in the atom. The mass number of the atom is nothing but the sum of the number of protons and the number of neutrons in the atom. The same atom number but different mass number of the atom is known as isotopes. Hydrogen having three isotopes. There are protium, deuterium and tritium. All the three having the same atomic number is \[1\]. But the mass numbers of three isotopes are different. The mass number of protium is \[1\]. The mass number of deuterium is \[2\]. The mass number of tritium is \[3\]. These difference mass numbers arise due to the difference in the number of neutrons but the number of protons of all isotopes are the same.