Question

# One mole of gas expands obeying the relation as shown in the P/V diagram. The maximum temperature in this process is equal to

A
P0V0R
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B
3P0V0R
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C
9P0V08R
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D
None of these
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Solution

## The correct option is C 9P0V08RThis question can be solved by maxima and minima.Since the graph shown here is a straight line, we can write it's equation in the form of :y=mx+chere y=P,x=V,m=P2−P1V2−V1=P02−P02V0−V0⇒m=−P02V0c is the intercept and c=3P02We can find c by extra polating the graphNow P=−P02V0V+3P02 ...(1)Also, from equation of state, we know the PV=nRTfor 1molePV=RTor P=RTVSubstituting this in equation (1)RTV=−P02V0V+3P02T=−P02RV0.V2+3P02RVTo find the minimum temperature, we will differentiate the above equationi.e., dTdV=−2P02RV0.V+3P02Rfor dTdV=0⇒P0RV0.V=3P02R⇒V=3V02This is the critical point.d2TdV2=−P0RV0which is negative since pressure and volume can't be negative.⇒ pt. of maximumNow at V=3V02maximum temperature =Tmax=−P02RV09V204+3P02R.3V02⇒Tmax=9P0V08RSo, option (c) is correct option.

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