Question

\[1\] mole of a rigid diatomic gas performs a work of \[Q/5\] . When heat \[Q\] is supplied to it. The molar heat capacity of the gas during this transformation is \[xR/8\] . the value of \[x\] is_______.

Answer 1

Hint: Diatomic gases are gases that naturally contain two atoms of the same sort in each of their molecules. For example, oxygen as a gas is made up of molecules with two oxygen atoms in them. Nitrogen and chlorine are two more diatomic gases. Because the atoms within the molecules are not stable on their own, they create covalent connections in order to conserve energy.

Formula Used:
The equation of the first law of thermodynamics is as follows:
\[\Delta Q = \Delta U + W\]
Here, \[\Delta Q\] is change in heat, \[\Delta U\] is change in internal energy, and \[W\] is work.

Complete Step by Step Solution:
Heat capacity is defined as the ratio of heat absorbed by a substance to the temperature change. The amount of material being considered, which is usually a mole, is usually expressed as calories per degree (the molecular weight in grams). The heat capacity in calories per gram is known as specific heat. The specific heat of water, which is one calorie per degree Celsius, is used to define the calorie.
When some amount of heat is provided to a system capable of producing external work, the quantity of heat absorbed is equal to the total of the increase in internal energy of the system owing to a rise in temperature and external work done during expansion, according to the first rule of thermodynamics.
Here, rigid diatomic gas performs a work of \[Q/5\]. So,
\[W = Q/5\]
We know that, for isochoric (constant volume) process,
\[\Delta U = n{C_v}\Delta T\]
And, \[{C_v} = \dfrac{5}{2}R\]
From equation of the first law of thermodynamics,
\[
  \Delta Q = \Delta U + W \\
   \Rightarrow Q = n{C_V}\Delta T + \dfrac{Q}{5} \\
   \Rightarrow Q - \dfrac{Q}{5} = 1 \times \left( {\dfrac{5}{2}R} \right) \times \Delta T \\
 \]
Further solving,
\[Q = \dfrac{{25}}{8}R\Delta T\]
We know that, \[Q = nC\Delta T\]
So,
\[
  \dfrac{{25}}{8}R\Delta T = nC\Delta T \\
   \Rightarrow C = \dfrac{{25}}{8}R \\
 \]
The given molar heat capacity is \[xR/8\]. So,
\[
  \dfrac{{25}}{8}R = \dfrac{{xR}}{8} \\
   \Rightarrow x = 25 \\
 \]
As a result, the value of \[x\] is \[25\] .

Note: Isochoric and isobaric processes are not the same thing.

• • Isochoric process: A process in which the volume is constant throughout.

• Isobaric process: A process in which the pressure is constant throughout.