Question

Relation between volume and pressure of one mole of an ideal gas is given as \( V^{2}=\left(\frac{P_{0}-P}{a}\right) \), where \( P_{0} \) and \( a \) are positive constants. If maximum temperature attained by the gas during this process is \( \frac{A P_{0}}{B R} \sqrt{\frac{P_{0}}{B a}} \), then value of \( (A+B) \) is equal to . \( A \) and \( B \) are coprime integers) ( \( R \) is universal gas constant)

Answer 1

Given v² = \(\left(\cfrac{p_0-p}a\right)\)

p₀ - p = av²

p = p₀ - av² ......(i)

\(\because\) pv = nRT

p = \(\cfrac{nRT}v\)

\(\cfrac{nRT}v\) = p₀ - av²

T = \(\cfrac{p_0v}{nR}\) - \(\cfrac{av^3}{nR}\)....(ii)

\(\left(\cfrac{dT}{dv}\right)\) = 0

\(\left(\cfrac{dT}{dv}\right)\) = \(\cfrac{p_0}{nR}\) - \(\cfrac{3av^2}{nR}\) = 0

\(\cfrac{3av^2}{nR}\) = \(\cfrac{p_0}{nR}\)

v² = \(\cfrac{p_0}{3a}\)

v = \(\sqrt{\cfrac{p_0}{3a}}\)

there value put in equation (ii)

T_max  = \(\cfrac{p_0v}{nR}\) = \(\cfrac{av^3}{nR}\)

= \(\cfrac{p_0}{nR}\) \(\left(\sqrt{\cfrac{p_0}{3a}}\right)\) - \(\cfrac{a}{nR}\) \(\left({\cfrac{p_0}{3a}}\right)^{3/2}\)

T = \(\sqrt{\cfrac{p_0}{3a}}\) \(\left(\cfrac{p_0}{nR}-\cfrac{a}{nR}\times\cfrac{p_0}{3a}\right)\)

T = \(\sqrt{\cfrac{p_0}{3a}}\) \(\left(\cfrac{3p_0-p_0}{3nR}\right)\)

T = \(\cfrac{2p_0}{3nR}\) \(\sqrt{\cfrac{p_0}{3a}}\)

Give T = \(\cfrac{A\,p_0}{B\,nR}\) \(\sqrt{\cfrac{p_0}{Ba}}\)

then A = 2

B = 3

Then value at A + B equal to

= 2 + 3

= 5