One mole of a rigid diatomic ideal gas…

One mole of a rigid diatomic ideal gas (translation and rotation only, no vibration) undergoes the quasi-static processes listed in List-I. List-II…

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One mole of a rigid diatomic ideal gas (translation and rotation only, no vibration) undergoes the quasi-static processes listed in List-I. List-II gives the molar heat capacity

C

of the gas in the process, in units of

R

. Match List-I with List-II.

List-I
(P) The pressure and volume satisfy

PV^{2}=\text{constant}

(Q) The pressure is directly proportional to the volume,

P\propto V

(R) The pressure and volume satisfy

PV^{3}=\text{constant}

(S) The pressure remains constant

List-II
(1)

1.5

(2)

2

(3)

2.5

(4)

3

(5)

3.5

A. P→1, Q→4, R→2, S→5
B. P→1, Q→4, R→5, S→2
C. P→2, Q→3, R→1, S→5
D. P→1, Q→3, R→2, S→4

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Answer: (A) P→1, Q→4, R→2, S→5

For a rigid diatomic gas

C_V=\tfrac52R

. For

PV^n=\text{const}

dQ=C_VdT+PdV

and

PV=RT

give

dT=\dfrac{d(PV)}{R}

; since

PV=cV^{1-n}

PdV=\dfrac{R\,dT}{1-n}

. Hence

C=C_V+\dfrac{R}{1-n}

. (P)

n=2

C=2.5R-R=1.5R

, which is (1). (Q)

P\propto V

means

n=-1

C=2.5R+0.5R=3R

, which is (4). (R)

n=3

C=2.5R-0.5R=2R

, which is (2). (S) isobaric is

n=0

C=C_P=C_V+R=3.5R

, which is (5). The value

2.5R

in (3) is

C_V

(isochoric), which no process in List-I uses. A finite-difference evaluation of

\Delta Q/\Delta T

for each process confirms these values. The same formula gives infinity for the isothermal case

n=1

and zero for the adiabat

n=\gamma

. Hence P→1, Q→4, R→2, S→5.

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