Question

The specific heat of a gas is found to be 0.075 calories at constant volume and its formula weight is 40. The atomicity of the gas would be:
A: one
B: two
C: three
D: four

Answer 1

Hint: Atomicity is referred to as the total number of atoms which compose a molecule. In simpler terms, it is the number of atoms that are present in a molecule. For instance, one molecule of oxygen comprises two atoms of oxygen. Thus, the atomicity of oxygen is 2.

Complete step by step solution: Atomicity can be estimated by dividing the molecular mass with the atomic mass.
In the question, we are given the value of specific heat of the gas and we have to calculate the atomicity of the gas. The specific heat capacity (c) of a substance is referred to as the ratio of the heat capacity (C) of the substance and the molecular mass of the substance (M).
$c = \dfrac{C}{M} = \dfrac{1}{M}.\dfrac{{dQ}}{{dT}}$ where dQ = heat required to raise the temperature by a small increment .
In case of an ideal gas (\[PV = nRT\]), the molar heat capacity at constant pressure (Cp) is more in comparison to molar heat capacity at constant volume (CV) by an amount of nR:
${C_p} = {C_V} + nR$ (R = gas constant, n = number of moles of substance)
In the given question we are given the ordinary specific heat of the gas. Thus:
$
  M{C_p} - M{C_V} = R \\
  {C_p} - {C_V} = \dfrac{R}{M} \\
$
(M = Molecular weight of the gas)
${C_V} = 0.075$(Given)
$
  {C_p} - 0.075 = \dfrac{2}{{40}} \\
  {C_p} = 0.075 + 0.05 = 0.125cal \\
$
(R = approx. 2.0 cal/K.mol, formula weight given = 40)
Heat capacity ratio i.e. $\gamma = \dfrac{{{C_p}}}{{{C_V}}} = \dfrac{{0.125}}{{0.075}} = 1.66$ (Hence, the gas is monatomic)

Therefore, the correct answer is Option A i.e. the atomicity of the gas would be one.

Note: The specific heat is having the different formulas from where it can be found out. It can also be calculated by using the relation between it and the gas constant for different types of atoms.The examples of monatomic gases are Helium, Argon, etc. The heat capacity ratio, also known as adiabatic index is 1.44 for the diatomic gas.