Q. 225.0( 3 Votes )

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Answer :

a) According to the question, conducting separator between both the part is fixed. So, no change in volume will be observed. And as we know work done a gas is PV. Therefore, no work will be done on the left part during the process as the volume is not changing.

b) Given

The vessel is divided into two equal parts.

So, the volume of left part V_{1} and volume of right part V_{2} will half of the total volume.

Initial pressure on each side is p

Number of moles on left side n_{1} =1

Number of moles on right side n_{2}=2

Let initial temperature for the left and right part be T_{1} and T_{2} respectively.

For the left part, applying the ideal gas equation

Similarly, for the right part

c) It is given that the internal energy of the gas is

U=1.5nRT

Where n=number of moles

R=gas constant

T=final equilibrium temperature of both parts

The internal energy of ideal gas is given as

U=nC_{v}T

Where C_{v}=molar specific heat at constant volume

T=temperature.

Total moles n=n_{1}+n_{2}=1+2=3

U=3C_{v}T

Let U_{1} and U_{2} be the internal energy of the left and right part respectively.

So, U_{1}=n_{1}C_{v}T_{1}= C_{v}T_{1}

And U_{2}=n_{2}C_{v}T_{2}=2C_{v}T_{2}

(gas is same on both the part so Cv will be same)

Now

U=U_{1}+U_{2}

3C_{v}T = C_{v}T_{1}+ 2C_{v}T_{2}

3T=T_{1}+2T_{2}

d) As stated in part (a) no work will be done on either part of the vessel as conducting separator is fixed.

So, ΔW=0 for the right part of the vessel.

From first law of thermodynamics, we know that,

ΔQ=ΔU+ΔW

Where ΔQ=heat supplied to the system

ΔU=change in internal energy

ΔW=work done by the system

∴ ΔQ=ΔU

It is given that the internal energy of the gas is

U=1.5nRT

Where n=number of moles

R=gas constant

T=final equilibrium temperature of both parts

ΔU=1.5nRΔT

For the right part, change in internal energy after equilibrium has reached will be due to the change in temperature from T_{2} to T.

ΔU=1.5n_{2}R(T-T_{2})

Thus, heat given to the right part is .

e) Since ΔW is zero as the volume is fixed, therefore, the first law of thermodynamics.

But since heat is given to the right part that means heat is extracted from the left part. So, the internal energy of the left part of will decrease.

Therefore, heat given to the right part will be equal to the negative internal energy of the left part.

ΔQ=-ΔU (left part)

.

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