Question

An ideal gas mixture filled inside a balloon expands according to the relation\[P{{V}^{{}^{2}/{}_{3}}}=\text{constant}\]. The temperature inside the balloon is
A. Increasing
B. Decreasing
C. Constant
D. Cannot be defined

Answer 1

Hint: This question can be solved using the concept of the ideal gas law, that is, using the equation of the ideal gas law. Thus, we will combine the given equation with the equation of the ideal gas law to represent the equation in terms of pressure and volume.
Formula used:
\[PV=nRT\]

Complete step-by-step solution
The formula that relates the parameters such as the pressure, volume, temperature, and the gas constant along with the number of molecules of the gas is given as follows.
\[PV=nRT\]
Where P represents the pressure, V represents the volume, n represents the number of molecules, R represents the gas constant and T represents the temperature.
This formula is called the Ideal gas law. This law is also called the general gas equation.
From given, we have,
An ideal gas mixture filled inside a balloon expands according to the relation\[P{{V}^{{}^{2}/{}_{3}}}=\text{constant}\].
Now, substitute the expression of the ideal gas law in terms of the pressure. So, we get,
\[\begin{align}
  & PV=nRT \\
 & \Rightarrow P=\dfrac{nRT}{V} \\
\end{align}\]
Thus, the given equation can be represented as follows.
\[\dfrac{nRT}{V}\times {{V}^{{}^{2}/{}_{3}}}=\text{constant}\]
Continue further calculation.
\[\text{nRT}{{\text{V}}^{-{}^{1}/{}_{3}}}=\text{constant}\]
As the number of molecules of the gas and the value of the gas constant are constant, thus, the above equation can be represented in terms of the volume and the temperature as follows.
\[\begin{align}
  & \text{T}{{\text{V}}^{-{}^{1}/{}_{3}}}=\text{constant} \\
 & \text{T }=\text{constant}\times {{\text{V}}^{{}^{1}/{}_{3}}} \\
\end{align}\]
The above equation can be further expressed as follows.
\[T\propto {{\text{V}}^{{}^{1}/{}_{3}}}\]
The above equation represents that, the volume of a gas is directly proportional to the temperature.
$\therefore$ The temperature inside the balloon increases.
As the temperature inside the balloon increases, when an ideal gas mixture filled inside a balloon expands according to the relation\[P{{V}^{{}^{2}/{}_{3}}}=\text{constant}\], thus, the option (A) is correct.

Note: This is a direct question. The formulae used in the above explanation, that is, the ideal gas law equation should be known to solve these types of derivational questions. If they ask for the relation between the pressure and the temperature, we have to follow the same procedure by replacing the volume term with the pressure.