Question

The specific heat at constant volume for a gas is 0.075 cal/g and at constant pressure is 0.125 cal/g. Calculate: (a) the molar mass of gas (b) atomicity of gas (c) no. of atoms, if gas in its 1 mole:
\[
  A.{\text{ a) }}40{\text{ b) }}1.66{\text{ c) }}6.023 \times {10^{23}} \\
  B.{\text{ a) }}30{\text{ b) }}0.66{\text{ c) }}6.023 \times {10^{23}} \\
  C.{\text{ a) }}40{\text{ b) }}1.66{\text{ c) }}5.023 \times {10^{23}} \\
  D.{\text{ None of these}} \\
 \]

Answer 1

Hint: In order to solve the given problem first we will study in brief about the terms given in the problem like specific heat at constant volume of gas and specific heat at constant pressure of gas, we will learn the symbols related to it. In order to find the molar mass of the gas, atomicity of gas and no of atoms we will see the relation of these terms with the specific heat and the formula that relates the two. Further by using these formulas we will find out the value for molar mass, atomicity and no of atoms.

Complete step by step answer:
Let us first study about the specific heat at constant volume of gas and specific heat at constant pressure of gas.
Specific heat at constant pressure and volume for an Ideal Gas reflects, at constant volume, the dimensionless heat capacity; it is normally a temperature function due to intermolecular forces. In general, the particular heats of gases are expressed as basic molar heats. For gases, two specific heats, one for constant volume and one for constant pressure are described.
The representation for specific heat at constant volume of gas is ${C_V}$ .
The representation for specific heat at constant pressure of gas is ${C_P}$ .
Now in order to find the values of the terms given let us see the formula relating these two terms and the asked terms.
a.) The molar mass of gas
The formula relating the molar mass of gas and the specific heat constant is:
${C_P} - {C_V} = \dfrac{R}{M}$
Here, R is the gas constant whose value is around 8 in the SI unit and 800 in the given unit and M is the molar mass of the gas.
Let us substitute the values in order to find the answer.
$
  \because {C_P} - {C_V} = \dfrac{R}{M} \\
   \Rightarrow 0.125 - 0.075 = \dfrac{{800}}{M} \\
   \Rightarrow 0.05 = \dfrac{{800}}{M} \\
   \Rightarrow M = 800 \times 0.05 \\
   \Rightarrow M = 40 \\
 $
Therefore the value of molar mass is 40.

b.) Atomicity of gas
The formula relating the atomicity of gas and the specific heat constant is:
$\gamma = \dfrac{{{C_P}}}{{{C_V}}}$
Here, $\gamma $ is the atomicity of gas whose value is to be found out.
Let us substitute the values in order to find the answer.
$
  \because \gamma = \dfrac{{{C_P}}}{{{C_V}}} \\
   \Rightarrow \gamma = \dfrac{{0.125}}{{0.075}} \\
   \Rightarrow \gamma = 1.66 \\
 $
Therefore the value of atomicity of gas is 1.66.

c.) No. of atoms, if gas in its 1 mole
As we know that the gas is monatomic so the number of atoms of gas in one mole will be the same as Avagadro’s constant.
Which is equal to $6.023 \times {10^{23}}$ atoms.
Hence, the value of the (a) the molar mass of gas (b) atomicity of gas (c) no. of atoms, if gas in its 1 mole are \[{\text{a) }}40{\text{ b) }}1.66{\text{ c) }}6.023 \times {10^{23}}\] respectively.
So, the correct answer is “Option A”.

Note: In order to solve such types of problems students must remember the formulas for some basic and commonly used terms. Also the students must remember the Avagadro’s number and should also know the significance of the Avagadro’s number. The proportionality element that compares the number of constituent particles in a sample to the volume of material in that sample is the Avogadro constant. The reciprocal mole is its SI unit.