Relation between volume and pressure of one mole of an ideal gas is gi

1.	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» 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

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»

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