The pressure p, volume V and temperature T for a certain gas are related by p = (AT-BT2/V), where A and B are constants. The work done by the gas when the temperature changes from T1 to T2 while the pressure remains constant, is given by

Question

The pressure p, volume V and temperature T for a certain gas are related by p = (AT-BT2/V), where A and B are constants. The work done by the gas when the temperature changes from T1 to T2 while the pressure remains constant, is given by (A) A(T2-T1)+B(T22 - T21) (B) (A(T2-T1)/v2-v1)-(B(T22 - T21)/v2-v1) (C) A(T2-T1)-B(T22 - T21) (D) (A(T2-T22)/v2-v1)

Tardigrade Admin · Accepted Answer

Given, P=(A T-B T2/V) ⇒ P V =A T-B T2 ⇒ P Δ V =A Δ T-B T Δ T On integrating, we get Work =∫ P d V=A ∫ limitsT1T2 d T-B ∫ limitsT1T2 T d T =A(T2-T1)-(B/2)[(T2)2-(T1)2]

Q. The pressure $p$ , volume $V$ and temperature $T$ for a certain gas are related by $p = \frac{A T - B T ^{2}}{V},$ where $A$ and $B$ are constants. The work done by the gas when the temperature changes from $T_{1}$ to $T_{2}$ while the pressure remains constant, is given by

$A (T_{2} - T_{1}) + B (T_{2}^{2} - T_{1}^{2})$

$\frac{A ( T _{2} - T _{1} )}{v _{2} - v _{1}} - \frac{B ( T _{2}^{2} - T _{1}^{2} )}{v _{2} - v _{1}}$

$A (T_{2} - T_{1}) - B (T_{2}^{2} - T_{1}^{2})$

$\frac{A ( T _{2} - T _{2}^{2} )}{v _{2} - v _{1}}$

Given, $P = \frac{A T - B T ^{2}}{V}$
$\Rightarrow P V = A T - B T^{2}$
$\Rightarrow P Δ V = A Δ T - BT Δ T$
On integrating, we get
Work $= \int P d V = A T_{1} \int T_{2} d T - B T_{1} \int T_{2} T d T$
$= A (T_{2} - T_{1}) - \frac{B}{2} [(T_{2})^{2} - (T_{1})^{2}]$

Q. The pressure p, volume V and temperature T for a certain gas are related by p=VAT−BT2​, where A and B are constants. The work done by the gas when the temperature changes from T1​ to T2​ while the pressure remains constant, is given by

A(T2​−T1​)+B(T22​−T12​)

v2​−v1​A(T2​−T1​)​−v2​−v1​B(T22​−T12​)​

A(T2​−T1​)−B(T22​−T12​)

v2​−v1​A(T2​−T22​)​

Given, P=VAT−BT2​ ⇒PV=AT−BT2 ⇒PΔV=AΔT−BTΔT On integrating, we get Work =∫PdV=AT1​∫T2​​dT−BT1​∫T2​​TdT =A(T2​−T1​)−2B​[(T2​)2−(T1​)2]