Question

A gas expand adiabatically such that
$P{V}^{a-\frac{1}{2}}=$constant. Which of the following is/are true about the nature of the gas.

• A. for monoatomic, $a=\frac{13}{6}$
• B. for diatomic, $a=\frac{17}{10}$
• C. for diatomic, $a=\frac{19}{10}$
• D. for monoatomic, $a=\frac{11}{6}$

Answer 1

Answer: (A), (C)

The correct options are
A for monoatomic, $a=\frac{13}{6}$
C for diatomic, $a=\frac{19}{10}$

Given: $P{V}^{a-\frac{1}{2}}=constant$ ...(1)
For adiabatic process, $P{V}^{\gamma }$ = constant
..(2)
equating both the equations we get,
$a-\frac{1}{2}=\gamma$
$a=\gamma +\frac{1}{2}$

for monoatomic gas, $\gamma =\frac{5}{3}$
$\Rightarrow a=\frac{5}{3}+\frac{1}{2}$
$\Rightarrow a=\frac{13}{6}$

for diatomic gas, $\gamma =\frac{7}{5}$
$\Rightarrow a=\frac{7}{5}+\frac{1}{2}$
$\Rightarrow a=\frac{19}{10}$