Pressure P , volume V and temperature T of a certain material are related by P = (α T2/V) , where α is constant. Work done by the material when temperature changes from T0 to 2T0 and pressure remains constant is

Question

Pressure P , volume V and temperature T of a certain material are related by P = (α T2/V) , where α is constant. Work done by the material when temperature changes from T0 to 2T0 and pressure remains constant is (A) 3α T02 (B) 5α T02 (C) (3/2)α T02 (D) 7α T02

Tardigrade Admin · Accepted Answer

The given relation is p=(α T2/V) ∴ V=(α T2/P) As pressure is kept constant, dV=((2 α T/P))dT W= displaystyle ∫ p d V = displaystyle ∫ T02 T0 p ((2 α T/p)) d T =2α [(T2/2)]T02 T0=3α T02

Q. Pressure $P$ , volume $V$ and temperature $T$ of a certain material are related by $P = \frac{α T ^{2}}{V},$ where $α$ is constant. Work done by the material when temperature changes from $T_{0}$ to $2 T_{0}$ and pressure remains constant is

$3 α T_{0}^{2}$

$5 α T_{0}^{2}$

$\frac{3}{2} α T_{0}^{2}$

$7 α T_{0}^{2}$

The given relation is $p = \frac{α T ^{2}}{V} ∴ V = \frac{α T ^{2}}{P}$
As pressure is kept constant, $d V = (\frac{2 α T}{P}) d T$
$W = \int p d V = \int_{T_{0}}^{2 T_{0}} p (\frac{2 α T}{p}) d T$
$= 2 α [\frac{T ^{2}}{2}]_{T_{0}}^{2 T_{0}} = 3 α T_{0}^{2}$

Q. Pressure P , volume V and temperature T of a certain material are related by P=VαT2​, where α is constant. Work done by the material when temperature changes from T0​ to 2T0​ and pressure remains constant is

3αT02​

5αT02​

23​αT02​

7αT02​

The given relation is p=VαT2​∴V=PαT2​ As pressure is kept constant, dV=(P2αT​)dT W=∫pdV=∫T0​2T0​​p(p2αT​)dT =2α[2T2​]T0​2T0​​=3αT02​