Question

One mole of an ideal monatomic gas undergoes a process described by the equation $P{V^n} =$Constant. The heat capacity of the gas during this process is
A. R
B. 2R
C. 3R
D. none of these

Answer 1

The process that obeys the relation $P{V^n} =$Constant is called polytropic process. Monoatomic gas means a gas where the atoms of the gas are not bonded to each other.
Formula Used:
$C = \left( {{C_V} + \dfrac{R}{{1 - n}}} \right)$

Complete step by step answer:
A polytropic process is a thermodynamic process that obeys the relation between pressure and volume. The relation is given as $P{V^n} = C$
For monoatomic gas, atom has 3 translational degrees of freedom
$\therefore$avg. energy per atom is $E = \dfrac{3}{2}{N_A}{K_B}T$ (for total molecule)
$\therefore$molar heat capacity at constant volume,
${C_V} = \dfrac{{dE}}{{dT}} = \dfrac{3}{2}{N_A}{K_B}$
$= \dfrac{3}{2}R$ . . . (1)
where $R = {N_A}{K_B}$
${N_A}$represents Avogadro’s number
${K_B}$represents Boltzmann’s constant
Heat capacity during polytropic process is given by
$C = \left( {{C_v} + \dfrac{R}{{1 - n}}} \right)$
Where, ${C_v}$ is specific heat capacity at constant volume.
$n$ is a polytropic index.
Substituting the value of ${C_v}$from equation (1) to the above equation, we get
$C = \left( {\dfrac{{3R}}{2} + \dfrac{R}{{1 - n}}} \right)$
As the value of $n$is not given the value of$\;C$cannot be determined due to data insufficiency. So the answer is option (D) none of these.

Note: Degree of freedom for monatomic or diatomic gas will never be given in the question. So make sure that you know them. From this question, we can observe that, sometimes just the name of a law can be given while sometimes a formula is given and the law would be asked. So it is important that we know the important laws and can understand it even just as a formula.