Question

The internal energy (U), pressure (P) and volume (V) of an ideal gas are related as\[U = 3PV + 4\]. The gas is:
A. Polyatomic only
B. Monoatomic only
C. Either monatomic or diatomic
D. Diatomic only

Answer 1

Hint:The gases can be characterised as monatomic, diatomic or polyatomic based on the degree of freedom of the gaseous molecule. The degree of freedom is the set of axes in which the molecules of the gas are free to have motion.

Formula used:
Ideal gas equation is given as,
\[PV = nRT\]
Here, n is the number of moles of ideal gas, P is the pressure, V is the volume, and T is the temperature.

Complete step by step solution:
The degree of freedom of a molecule depends on atomicity of the molecule, temperature and the molecular structure. The atomicity is the number of atoms present in one molecule of the substance. The relation between the internal energy (U), pressure (P) and the volume (V) of the ideal gas is given as,
\[U = 3PV + 4\]
As the given gas is ideal gas, so it must satisfy the ideal gas equation,
\[PV = nRT\]
Here, n is the number of moles of ideal gas.

If n moles of ideal gas of degree of freedom f is at temperature T then the internal energy is given as,
\[U = \dfrac{{nf}}{2}RT\]
Using the ideal gas equation,
\[U = \dfrac{{nf}}{2}PV\]
Taking unit mole of the ideal gas, i.e. \[n = 1\]
\[\begin{array}{l}\dfrac{f}{2}PV = 3PV + 4\\ \Rightarrow \dfrac{{\dfrac{f}{2}PV}}{{\dfrac{1}{2}PV}} = \dfrac{{3PV + 4}}{{\dfrac{1}{2}PV}}\\ \Rightarrow f = 6 + \dfrac{8}{{PV}}\\ \therefore f > 6\end{array}\]
As the degree of freedom is greater than 6, hence the given ideal gas is a polyatomic gas.

Therefore, the correct option is A, polyatomic only.

Note: The degree of freedom of monoatomic gas is 3. The degree of freedom of diatomic gas is 5. The degree of freedom of polyatomic gas is more than 6.