Question

The density of gas at \[{\mathbf{27}}^\circ \] Celsius and \[{\mathbf{1}}{\text{ }}{\mathbf{atm}}\]is\[\;{\mathbf{d}}\]. Pressure remaining constant, the temperature at which its density becomes \[{\mathbf{0}}.{\mathbf{75}}{\text{ }}{\mathbf{d}}\;\]is
A. \[\;\;\;{\mathbf{20}}^\circ C\]
B. \[{\mathbf{30}}^\circ C\]
C. \[\;{\mathbf{400}}K\]
D. \[{\mathbf{300}}K\]

Answer 1

Hint: In the given question we have to find the temperature of the given gas at particular density and pressure. Ideal gas law is used to determine the temperature of gases. Please keep in mind that the unit of the temperature obtained will be in absolute temperature scale i.e kelvin.

Complete Step by step answer: The ideal gas law is often written in an empirical form, \[PV{\text{ }} = {\text{ }}nRT\] and where P, V and T stands for the pressure, volume and temperature whereas n is the amount of substance; and R is the ideal gas constant. It is the same for all gases. The state of an amount of gas might be determined by using its pressure, volume, along with the temperature.
Given values according to question
\[d{\text{ }}1 = d\]
\[d{\text{ }}2 = 0.75d\]
At \[P1 = 1atm\] \[T1 = 27^\circ \] that is \[C = 300K\]
And at \[P2 = 1atm\] T2=?
We have to find the temperature
As we know that, \[PV{\text{ }} = {\text{ }}nRT\]
or \[PV = \left( {\dfrac{m}{M}} \right)RT\]
By rearranging we get,
\[PM = \left( {\dfrac{m}{V}} \right)RT = dRT\]
so, In condition (1) we have : \[PV = dR300\]
as In condition (2) we have : \[PV = 0.75(d) (R)(T)\]
now, by Dividing condition (1) by (2), we get:
\[1 = \dfrac{{\left( {300} \right)}}{{0.75T}}\]
or \[T = 400K\]
Or we can solve it as
\[P = MdRT\]
When, \[P\] is constant then \[\] = constant
Therefore, \[d \times 300 = 0.75d \times T\]
So\[{\text{ }}T = \dfrac{{300}}{{0.75}} = 400\;K\]

Hence, option (c) 400 K is the right answer.

Note: The ideal gas law was derived from the first principles by using the kinetic theory of gases, which includes several types of simplifying assumptions that are made, main among which are that the molecules of the gas are point masses and possessing mass but not having significant volume and undergo only elastic collisions with each other and the sides of the container in which both linear momentum and kinetic energy are conserved.