Question

In which of the following cases does the gas law $\dfrac{{PV}}{T} = {\text{constant}}$ holds true?
A. isothermal changes only
B. adiabatic changes only
C. both isothermal and adiabatic changes
D. neither isothermal or adiabatic changes

Answer 1

Hint: Ideal gas equation is given as $PV = nRT$
Where $P$ is the pressure, $V$is the volume, $n$ is the number of moles and $R$ is a universal gas constant. Ideal gas equation holds good for all types of changes

Complete step by step answer:
Ideal gas equation is given as $PV = nRT$
Where $P$ is the pressure, $V$is the volume, $n$ is the number of moles and $R$ is a universal gas constant.
Ideal gas equation is obeyed by ideal gases. The conditions for ideal gas is that
1 there should be no intermolecular forces
2. molecules are considered as point masses
3 all collisions are perfectly elastic
Rearrange the ideal gas equation. Then we get
$\dfrac{{PV}}{T} = nR$
That is $\dfrac{{PV}}{T} = $constant.
An isothermal process is a process in which temperature remains constant.
Adiabatic change is the process in which the heat exchange will be zero.
Ideal gas equation holds good for all types of changes. Therefore, this equation is true for both adiabatic and isothermal changes.
So, the answer is option C.

Additional information:
The relation $\dfrac{{PV}}{T} = {\text{constant}}$ is not valid in an open system. An open system is a system in which there is interaction with the surroundings. Both matter and energy exchange can take place between the system and the surrounding. Since there is exchange of matter we cannot consider the number of moles as a constant and thus $\dfrac{{PV}}{T}$ will not be a constant .
Whereas in a closed system $\dfrac{{PV}}{T} = {\text{constant}}$ is valid. In a closed system there is no transfer of matter therefore the number of moles will remain constant for a closed system and thus $\dfrac{{PV}}{T}$ will be constant.

Note: The number of moles is directly proportional to mass. It is given by the equation $n = \dfrac{m}{M}$ where $m$ is the total mass of the gas and $M$ the molar mass. Hence, we have considered mass as a constant in order to get the right hand side of the equation $\dfrac{{PV}}{T} = nR$ as a constant.