Question

Calculate the number of moles of hydrogen gas present in \[500{\text{ }}c{m^3}\] of the gas taken at 300 \[K\] and 760 $mm$ pressure. If these samples of hydrogen were found to have a mass equal to \[4.09 \times {10^{ - 2}}{\text{ }}g\] calculate the molar mass of hydrogen.

Answer 1

Hint: Ideal gas equation gives the relation of pressure, volume, temperature and number of moles. In this solution by substituting the values of three parameters we can easily calculate the desired one.

Step by step solution:
As given in the question:
Volume of gas is \[500{\text{ }}c{m^3}\]
As we know \[1{\text{ }}m = 100{\text{ }}cm\]
Then \[1{\text{ }}{m^3} = {\text{ }}{10^6}{\text{ }}c{m^3}\]
So Volume of gas will be $500 \times {10^{ - 6}}{\text{ }}c{m^3}$ which is equal to $5 \times {10^{ - 4}}{\text{ }}c{m^3}$
Temperature is 300 K
Pressure is 760 mm which is equal to 101325 pa
Let assume the ideal gas behaviour:
Ideal gas equation: $PV = nRT$
Where R is the universal gas constant, which has the value of 8.314 $\dfrac{{Pa.{m^3}}}{{mol.K}}$
On substituting the values in the equation,
$101325 \times 5 \times {10^{ - 4}} = n \times 8.314 \times 300$
So $n = \dfrac{{101325 \times 5 \times {{10}^{ - 4}}}}{{8.314 \times 300}}$
$n = 0.02$ mole
From the definition of mole
$Number\;of\;mole = \dfrac{{Mass}}{{Molecular\;mass}}$
Now on substituting the values of moles and mass we can find the value of molecular mass as:
$0.02 = \dfrac{{4.09 \times {{10}^{ - 2}}}}{M}$
$M = \dfrac{{4.09 \times {{10}^{ - 2}}}}{{0.02}}$
$M = 2.045\;\dfrac{{gram}}{{mole}}$
Hence the molar mass of hydrogen is 2.045 gram/mole, which is approximately equal to what is given in periodic table (2.01568)

Additional information: The molar mass, also known as molecular weight, is the sum of the total mass in grams of all the atoms that make up a mole of a particular molecule. The unit used to measure is grams per mole.

Note: The gas constant also known as the molar gas constant, universal gas constant, or ideal gas constant.
It is denoted by the symbol R.
When using the ideal gas equation, the unit of universal gas constant must be taken carefully. Because the value of R changes with the unit of pressure, temperature and volume.