Question

An ideal gas undergoes a state change according to a PV diagram. What is the value of \[Vx\]?

A. \[\dfrac{{V_o}}{2}\]
B. \[V_o\]
C. \[2V_o\]
D. \[\dfrac{{V_o}}{4}\]

Answer 1

Hint: In this question, a PV graph is given. A relation between pressure and the volume of an ideal gas is given. An ideal gas is a hypothetical gas and its behavior is described by kinetic molecular theory of gases.

Complete answer:
An ideal gas law is simply the combination of simple gas law such as Boyle’s law, Charles law and Avogadro’s law. The ideal gas equation is \[PV = nRT\]
Let’s first understand each gas law:
1. Boyle’s law- This law describes the inverse relation between the pressure and volume at a constant temperature and a fixed volume of a gas.
\[P \propto \dfrac{1}{V}\]
2. Charles’s law- This law describes the directly proportional relation between the volume and temperature of a gas where pressure and amount of a gas is kept constant.
\[V \propto T\]
3. Avogadro’s law- This law describes that volume of a gas is directly proportional to the amount of a gas at a constant pressure and temperature.
\[V \propto n\]
By combining all the above law, we get ideal gas equation i.e. \[PV = nRT\]
According to this question, the relation between pressure and volume of a gas is given, so we use Boyle’s law which states that the pressure exerted by a given mass of a gas is inversely proportional to the volume if the temperature and amount of a gas is kept constant.
\[P \propto \dfrac{1}{V}\]
We can also write it as: \[{P_1}{V_1} = {P_2}{V_2}\]
In this question, it is given:
Initial pressure and volume,
\[{P_1}\, = {P_o}\]
\[{V_1} = \,{V_o}\]
Final pressure and volume,
\[{P_2}\, = 2{P_o}\]
\[{V_2}\, = \,{V_x}\]
Put all the values in this equation \[{P_1}{V_1} = {P_2}{V_2}\]
We get,
\[{P_o}\,{V_o}\, = 2{P_o}\,{V_x}\]
\[{V_x} = \dfrac{{{V_o}}}{2}\]

Hence, the correct answer is option (A).

Note:
There is another law which makes the ideal gas equation i.e. Amonton’s law which state that pressure of a gas is directly proportional to temperature at a constant volume and amount of a gas i.e. \[P \propto T\]. It must be noted that no such ideal gas exists in reality. It is just a hypothetical gas which is used to study real gases.