Find the minimum attainable pressure of an ideal gas in process T = T0 + α V2 , where T0 and α are positive constants and V is the volume of one mole of gas.

Question

Find the minimum attainable pressure of an ideal gas in process T = T0 + α V2 , where T0 and α are positive constants and V is the volume of one mole of gas. (A) 2 R √α T0 (B) 3 R √α T0 (C) 3 R (D) 3 R √(α T0/2)

Tardigrade Admin · Accepted Answer

Given, T = T0 + α V2 ...(i) For 1 mol of a gas, P V = R T of V = (R T/P) Substituting this value in Eq. (i), we get T = T0 + α ((R T/P))2 = T0 + α (R2 T2/P2) or T P2 = T0 P2 + α R2 T2 or P = √α R T (T - T0)- 1 / 2 ...(ii) After differentiating, we get (d P/d T) = √α R [(T - T0)- 1 / 2 - (1/2) T (T - T0)- 3 / 2] For minimum pressure, (d P/d T) = 0 â´ 0 = √α R [( textT - ( textT)0)- 1 / 2 - (1/2) textT ( textT - ( textT)0)- 3 / 2] After solving, T = 2T0 From Eq. (ii), Pmin = √α R 2 T0 (2 T0 - T0)- 1 / 2 = 2 R √α T0 <img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/p-322aabadhwso.png" />

Q. Find the minimum attainable pressure of an ideal gas in process $T = T_{0} + α V^{2}$ , where $T_{0}$ and $α$ are positive constants and $V$ is the volume of one mole of gas.

$2 R α T_{0}$

$3 R α T_{0}$

$3 R$

$3 R \frac{α T _{0}}{2}$

Q. Find the minimum attainable pressure of an ideal gas in process T=T0​+αV2 , where T0​ and α are positive constants and V is the volume of one mole of gas.

2RαT0​​

3RαT0​​

3R

3R2αT0​​​

Q. Find the minimum attainable pressure of an ideal gas in process $T = T_{0} + α V^{2}$ , where $T_{0}$ and $α$ are positive constants and $V$ is the volume of one mole of gas.

$2 R α T_{0}$

$3 R α T_{0}$

$3 R$

$3 R \frac{α T _{0}}{2}$