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Question

One mole of an ideal gas (monoatomic) is taken from A to C along the path ABC. The temperature of the gas is To at A. Then for the process ABC

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Solution

The correct options are

**A** Work done by gas is RTo

**C** Heat absorbed by gas is 112RTo

Work done by gas for AB = Area under the line AB = Po(2Vo−Vo)=PoVo

For 1 mole of an ideal gas, PoVo=RTo

Work done by gas for process BC = 0 (∵ Volume is constant)

Therefore, work done by gas for ABC = Work done by gas for AB + Work done by gas for BC

Thus, total work done = RTo

Internal energy change for ABC can be calculated by adding the internal energy change for AB and BC or directly between A and C because internal energy is point function which depends on final and initial state.

Thus, internal energy change for AC,

ΔU=nCvdt=nCv(TC−TA)

For an ideal monoatomic gas,

Cv=32R

Now AB is a constant pressure process, thus

V∝T

∴VBVA=TBTA

Now VB=2Vo, VA=Vo and TA=To

⇒TB=2To

Now BC is a constant volume process, thus

P∝T

∴PCPB=TCTB

Now PB=Po, PC=2Po and TB=2To

⇒TC=4To

Thus, internal energy for ABC,

ΔU=nCv(TC−TA)

ΔU=1×32R(4To−To)=92RTo

Since Q=U+W

For ABC, Q=92RTo+RTo=112RTo

Work done by gas for AB = Area under the line AB = Po(2Vo−Vo)=PoVo

For 1 mole of an ideal gas, PoVo=RTo

Work done by gas for process BC = 0 (∵ Volume is constant)

Therefore, work done by gas for ABC = Work done by gas for AB + Work done by gas for BC

Thus, total work done = RTo

Internal energy change for ABC can be calculated by adding the internal energy change for AB and BC or directly between A and C because internal energy is point function which depends on final and initial state.

Thus, internal energy change for AC,

ΔU=nCvdt=nCv(TC−TA)

For an ideal monoatomic gas,

Cv=32R

Now AB is a constant pressure process, thus

V∝T

∴VBVA=TBTA

Now VB=2Vo, VA=Vo and TA=To

⇒TB=2To

Now BC is a constant volume process, thus

P∝T

∴PCPB=TCTB

Now PB=Po, PC=2Po and TB=2To

⇒TC=4To

Thus, internal energy for ABC,

ΔU=nCv(TC−TA)

ΔU=1×32R(4To−To)=92RTo

Since Q=U+W

For ABC, Q=92RTo+RTo=112RTo

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