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

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.