Question

A ratio of Boyle’s temperature and critical temperature of a gas is:
A. \[\dfrac{8}{27}\]
B. $\dfrac{27}{8}$
C. $\dfrac{1}{2}$
D. $\dfrac{2}{1}$

Answer 1

Hint: Temperature is defined as the measure heat present in a system which is expressed in a comparative scale and shown by a device known as a thermometer. It is measured in Celsius, Kelvin and Fahrenheit scale. It basically tells us the degree of hotness and coldness of a body.

Complete step-by-step answer: Boyle’s temperature is defined as the temperature at which a non ideal gas acts as an ideal gas over a range of pressure. At this temperature for a real gas, the attractive forces and the repulsive forces that are acting on the gas particles become balanced. Boyle’s temperature shows a relationship with Van Der Waals constant that are a, b. It is denoted as ${{T}_{B}}$ .
Let us see the equation of Boyle’s temperature-
${{T}_{B}}=\dfrac{a}{bR}$
Where, ${{T}_{B}}$ is the Boyle’s temperature
$a,b$ are Van Der Waals constant
$R$ is universal gas constant
Critical temperature is defined as the maximum temperature where a substance can exist as a liquid. Above this temperature, a substance can no longer be liquefied. It is denoted as ${{T}_{C}}$ .On increasing the temperature, it becomes difficult to liquefy a substance because the kinetic energy of the particle increases that can form vapour.
Let us see the equation of critical temperature-
${{T}_{C}}=\dfrac{8a}{27bR}$
Where, ${{T}_{C}}$ is the Critical temperature
$a,b$ are Van Der Waals constant
$R$ is universal gas constant
As we have discussed above that ${{T}_{B}}=\dfrac{a}{bR}$
The equation of critical temperature is ${{T}_{C}}=\dfrac{8a}{27bR}$
Hence, on substituting the value of ${{T}_{B}}$ to ${{T}_{C}}$ we get,
${{T}_{C}}=\dfrac{8}{27}{{T}_{B}}$
Hence, the ratio of Boyle’s temperature to critical temperature is :
$\dfrac{{{T}_{B}}}{{{T}_{C}}}=\dfrac{\dfrac{a}{bR}}{\dfrac{8a}{27bR}}$
On cancelling out the values we get,
$\dfrac{{{T}_{B}}}{{{T}_{C}}}=\dfrac{27}{8}$

Therefore, the correct option is B.

Note: We should know the meaning of ideal gas and non ideal gas. Ideal gas is defined as the theoretical gas that moves randomly but does not interact with each other whereas non ideal gases are defined as the gases that occupy space and interact with each other.