Question

‘n’ moles of an ideal gas undergoes a process A→B as shown in the figure. The maximum temperature of the gas during the process will be:

$fraction numerator 9 straight p subscript straight o straight V subscript straight o over denominator 4 space nR end fraction$
$fraction numerator 3 straight p subscript straight o straight V subscript straight o over denominator 2 nR end fraction$
$9 over 2 fraction numerator straight p subscript straight o straight V subscript straight o over denominator nR end fraction$
$fraction numerator 9 straight p subscript straight o straight V subscript straight o over denominator nR end fraction$

Solution

fraction numerator 9 straight p subscript straight o straight V subscript straight o over denominator 4 space nR end fraction

As, T will be maximum temperature where product of pV is maximum

Equation of line AB, we have
$straight y minus straight y subscript 1 space equals space fraction numerator straight y subscript 2 minus straight y subscript 1 over denominator straight x subscript 2 minus straight x subscript 1 end fraction space left parenthesis straight x minus straight x subscript 1 right parenthesis rightwards double arrow space straight p minus straight p subscript straight o space equals fraction numerator 2 straight p subscript straight o minus straight p subscript straight o over denominator straight V subscript straight o minus 2 straight V subscript straight o end fraction space left parenthesis straight V minus 2 straight V subscript straight o right parenthesis straight p minus straight p subscript straight o space equals space fraction numerator negative straight p subscript straight o over denominator straight V subscript straight o end fraction left parenthesis straight V minus 2 straight V subscript straight o right parenthesis straight p space equals space fraction numerator negative straight p subscript straight o over denominator straight V subscript straight o end fraction straight V squared space plus 3 straight p subscript straight o straight V nRT space equals space fraction numerator negative straight p subscript straight o over denominator straight v subscript straight o end fraction straight V squared space plus space 3 straight p subscript straight o straight V straight T space equals space 1 over nR open parentheses fraction numerator negative straight p subscript straight o straight V squared over denominator straight v subscript straight o end fraction space plus 3 straight p subscript straight o straight V close parentheses For space maximum space temperature fraction numerator partial differential straight T over denominator partial differential straight V end fraction space equals space 0 fraction numerator negative straight p subscript straight o over denominator straight V subscript straight o end fraction left parenthesis 2 straight V right parenthesis space equals space minus space 3 straight p subscript straight o straight V space equals space 3 divided by 2 straight V subscript straight o left parenthesis condition space for space maximum space temperature right parenthesis Thus comma space the space maximum space temperature space of space the space gas space during space the space process space will space be straight T subscript max space equals space 1 over nR open parentheses fraction numerator negative straight p subscript straight o over denominator straight V subscript straight o end fraction space straight x 9 over 4 straight V subscript 0 superscript 2 space plus space 3 straight p subscript straight o space straight x space 3 over 2 straight V subscript straight o close parentheses equals space 1 over nR open parentheses negative 9 over 4 straight p subscript straight o straight V subscript straight o space plus space 9 over 2 straight p subscript straight o straight V subscript straight o close parentheses space equals space 9 over 4 fraction numerator straight p subscript straight o straight V subscript straight o over denominator nR end fraction$

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Mechanical Properties Of Fluids

Some More Questions From Mechanical Properties of Fluids Chapter

What determines the natural frequency of an oscillator?

Is there any relation between uniform circular motion and simple harmonic motion?

What is periodic motion?

Write down the expression for the instantaneous velocity and acceleration of particle vibrating in simple harmonic motion.

How do you define the mean position of particle executing simple harmonic motion?

Define amplitude.

What is the net displacement of the particle executing simple harmonic motion in one complete oscillation?

Are all the periodic motions simple harmonic?

Does the time period of particle vibrating in simple harmonic motion depend upon its displacement?

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