Question

During an experiment an ideal gas is found to obey an additional law $V{P^2} = const$. The gas is initially at temperature T and volume V, when it expands to volume 2V, the resulting temperature is ${T_2}$​:

(a) $\dfrac{T}{2}$
(b) 2T
(c) $\sqrt 2 T$
(d) $\dfrac{T}{{\sqrt 2 }}$

Answer 1

In order to solve this numerical we should understand the law of ideal gas that is PV=nRT. From the above question using the given condition we can find the required change in temperature.

Complete step by step answer:
Given data:
Initial volume=V
Final volume=2V
Initial temperature=${T_1}$
Final temperature=${T_2}$
From the law of ideal gas equation is given as
$PV = nRT$ …….. (1)
$\ P = \dfrac{{nRT}}{V} \\\ \$…………… (2)
The above condition is given that
$V{P^2} = const$……. (3)
The equation (1) can be written as
$VP \times P = const$…….. (4)
That is $nRT \times P = const$
Substitute equation 2 in above
$\dfrac{{{{(nRT)}^2}}}{V} = const$
$\dfrac{{{T^2}}}{V} = const$
Hence the volume V expands to 2V and temperature ${T_1}$ expands to ${T_2}$
$\ {T_2} = \sqrt {\dfrac{{2V}}{V}} \\\ \Rightarrow {T_2} = \sqrt 2 {T_1} \\\ \$

Note: Whenever we need to solve this type of question we need to remember all types of gas laws with respect to temperature, pressure and volume. As per the given data in the question we can apply the respective gas law.