One mole of a monoatomic gas is taken through a cycle ABCDA as shown in the P-V diagram. Column II gives the characteristics involved in the cycle. Match them with each of the processes given in Column I. [2011]

Question

One mole of a monoatomic gas is taken through a cycle ABCDA as shown in the P-V diagram. Column II gives the characteristics involved in the cycle. Match them with each of the processes given in Column I. [2011]

	Column I		Column II
(A)	Process A → B	(p)	Internal energy decreases
(B)	Process B → C	(q)	Internal energy increases
(C)	Process C → D	(r)	Heat is lost
(D)	Process D → A	(s)	Heat is gained
		(t)	Work is done on the gas

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(1)

(1) Process $A \to B$

This is an isobaric process, P = constant and volume $(V)$ of the gas decreases. Therefore work is done on the gas.

$W = P (3 V - V) = 2 P V$

Also $V$ decreases so temperature at $B$ decreases.

$∴$ Internal energy U decreases.

From, $Q = U + W$ as $U$ and $W$ decrease so $Q$ decreases, that means heat is lost.

(2) Process $B \to C$

This is an isochoric process, V = constant pressure decreases,

$P \propto T$ so temperature also decreases.

$W = 0; Δ U = negative$

so $Δ Q negative$

Hence heat is lost.

(3) Process $C \to D$

This is isobaric, Pressure P = constant $V$ increases and $V \propto T$

so $T$ increases.

Hence $Δ W, Δ U and Δ Q$ are positive, so heat is gained by the gas.

(4) Process $D \to A$

Applying $P V = n R T$

for $D$ $P (9 V) = 1 R T_{D}$ $∴$ $T_{D} = \frac{9 P V}{R}$

for $A$ $3 P (3 V) = 1 R T_{A}$ $∴$ $T_{A} = \frac{9 P V}{R}$

i.e., the process is isothermal. $∴$ $Δ U = 0$

Now, $Δ Q = Δ U + W$ $∴$ $Δ Q = W$

As volume decreases in this process so $W$ is negative i.e., work done on the gas and $Δ Q$ is negative, hence heat is lost.

Solution

1 $A - p, r, t; B - p, r; C - q, s; D - r, t$

2 $A - r, t; B - p, r; C - q, s; D - p, r, t$

3 $A - r, t; B - q, s; C - p, r; D - p, r, t$

4 $A - r, t; B - p, r, t; C - p, r; D - q, s$

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Solution

1 A-p,r,t; B-p,r; C-q,s; D-r,t

2 A-r,t; B-p,r; C-q,s; D-p,r,t

3 A-r,t; B-q,s; C-p,r; D-p,r,t

4 A-r,t; B-p,r,t; C-p,r; D-q,s

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1 $A - p, r, t; B - p, r; C - q, s; D - r, t$

2 $A - r, t; B - p, r; C - q, s; D - p, r, t$

3 $A - r, t; B - q, s; C - p, r; D - p, r, t$

4 $A - r, t; B - p, r, t; C - p, r; D - q, s$