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Question

The molar heat capacity in a process of diatomic gas if it does a work of $$\dfrac{Q}{4}$$ when a heat of Q is supplied to it is


A
25R
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B
54R
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C
103R
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D
64R
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Solution

The correct option is B $$\dfrac{10}{3}$$R
First law of Thermodynamics we have :-

$$Q=w+ΔU$$   (1)

where,

Q = Heat supplied to the gas

w = Work done by the gas

ΔU = Change in the internal energy of the gas

Given :-

$$w=Q/4$$     (2)

From (1) & (2) we have :-

⟹$$ΔU=3Q/4$$      (3)

Now,

$$ΔU=\dfrac{nRΔT}{γ−1}$$  (4)

where,

n = No. of moles of gas

ΔT = Change in temperature

R = Universal Gas Constant

γ = Specific heat ratio

For a diatomic gas :-

$$γ=7/5$$          ( 5)

Using (3), (4) & (5) we get :-

⟹$$nΔT=\dfrac{3Q}{10R}$$        (6)

Also note that :-

$$Q=nCΔT$$      (7)

where,

C = Molar specific heat for the process

So now replace nΔT in (7) by (6) :-

⟹$$\dfrac{3QC}{10R}=Q$$

⟹$$C=\dfrac{10R}{3}$$

Physics

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