Question

A 2L vessel is filled with air ${50^ \circ }C$ and the pressure of 3 ${\text{atm}}$ . The temperature is now raised to ${200^ \circ }C$ . A valve is now opened so that the pressure inside drops to one atm. What will be the fraction of the total number of moles inside, escaped on opening the valve? (Assuming no change in the volume of the container).
A) $7.7$
B) $9.9$
C) $8.9$
D) $0.77$

Answer 1

We know that the ideal gas law expresses the quantitative relation between the four variables that describe the state of the gas. And the four variables are pressure, temperature, number of moles, and volume. In order to solve this question, we must know the relation between all the variables.
Formula used: ${\text{PV = nRT}}$
Here, ${\text{P}}$= pressure
${\text{V}}$ = volume
${\text{n}}$ = number of moles
${\text{R}}$ = Gas constant
${\text{T}}$ = temperature

Complete step by step answer:
First, we have to find the number of moles of gas 1.
We can arrange the following ideal gas law as:
${{\text{n}}_{\text{1}}}{\text{ = }}\dfrac{{{{\text{P}}_{\text{1}}}{{\text{V}}_{\text{1}}}}}{{{\text{R}}{{\text{T}}_{\text{1}}}}}$
By substituting the value we get:
${n_1} = \dfrac{{3 \times 2}}{{0.0082 \times 323}}$
By solving we get the following answer:
${n_1} = 0.226$
In the first part of the question we are provided constant volume i.e. 2L and pressure of the first gas is given 1 atm.
So at the constant volume, we know that:
$\dfrac{{{{\text{P}}_{\text{1}}}}}{{{{\text{P}}_{\text{2}}}}}{\text{ = }}\dfrac{{{{\text{T}}_{\text{1}}}}}{{{{\text{T}}_{\text{2}}}}}$ (equation 1)
Now let us convert the temperature in kelvin from degree Celsius:
${{\text{T}}_{\text{1}}} = 50 + 273 = 323{\text{K}}$
And for second gas:
${{\text{T}}_{\text{2}}} = 200 + 273 = 473{\text{K}}$
Now substituting the value in equation 1
$\dfrac{{{\text{1atm}}}}{{{{\text{P}}_{\text{2}}}}} = \dfrac{{323{\text{K}}}}{{473{\text{K}}}}$
By solving the equation we get the value of the second gas:
${{\text{P}}_{\text{2}}} = 4.39{\text{atm}}$
Now in the second part of the question, the valve is opened and then volume and temperature remain constant:
$\dfrac{{{{\text{P}}_{\text{1}}}}}{{{{\text{P}}_{\text{2}}}}}{\text{ = }}\dfrac{{{{\text{n}}_{\text{2}}}}}{{{{\text{n}}_{\text{1}}}}}$
By substituting we get:
$\dfrac{{1{\text{atm}}}}{{4.39{\text{atm}}}} = \dfrac{{{{\text{n}}_{\text{2}}}}}{{0.226}}$
By solving we get the following value:
${n_2} = 0.77$

Therefore we can conclude that the correct answer to this question is option D

Note: In this type of question we should always focus on the unit conversion like temperature is converted from degree Celsius to kelvin and the gas law constant used should contain the unit of liter as volume and pressure in the atmosphere.