Question

In an adiabatic change, the pressure and temperature of a monatomic gas are related as \[P \propto {T^C}\], where C equals -
A. \[\dfrac{5}{4}\]
B. \[\dfrac{5}{3}\]
C. \[\dfrac{5}{2}\]
D. \[\dfrac{3}{5}\]

Answer 1

Hint: The above problem can be resolved using the fundamentals of the adiabatic process and the adiabatic index's value in the adiabatic process. The monatomic gases are those gaseous molecules that are present in the atmosphere in the form of two atoms collectively. The value of the adiabatic constant for the monatomic gas molecule is fixed, and this value can be used to relate the expression given in the problem. Then, by comparing both the equations., the desired result is obtained.

Complete step by step answer:
Given:
The relation for the pressure and temperature of monatomic gas is,\[P \propto {T^C}\].
Write the mathematical relation for the pressure (P ) and temperature (T) in a adiabatic process as,
\[P \propto {T^{\dfrac{\gamma }{{\gamma - 1}}}}\]
Compare the above expression for the given equation in the problem as,
\[C = \dfrac{\gamma }{{\gamma - 1}}\]
Here, \[\gamma \] denotes the adiabatic index and its value for the monatomic gas is, 5/3
Solve by substituting the values as,
 \[\begin{array}{l}
C = \dfrac{\gamma }{{\gamma - 1}}\\
C = \dfrac{{\dfrac{5}{3}}}{{\dfrac{5}{3} - 1}}\\
C = \dfrac{5}{2}
\end{array}\]
 Therefore, the value of C is equal to 5/2

So, the correct answer is “Option C”.

Note:
 To resolve the given problem, one must try to understand the adiabatic process. The adiabatic process is when the heat interaction within the system or between the different systems is always constant. This possibly means that the thermal energy content of the system is constant. Moreover, the thermal energy transfer only occurs due to the difference in the temperature. In a diatomic molecule, the pressure varies significantly with the temperature during the adiabatic process.