For two thermodynamic process temperature and volume, diagrams are drawn. In the first process, it is a straight line having initial and final coordinates as (V0 , T0) and (2 V0 , 2 T0) , whereas in second process it is a rectangular hyperbola having initial and final coordinates (V0, T0) and (2V0, T0/2). Then the ratio of work done in the two processes must be

Question

For two thermodynamic process temperature and volume, diagrams are drawn. In the first process, it is a straight line having initial and final coordinates as (V0 , T0) and (2 V0 , 2 T0) , whereas in second process it is a rectangular hyperbola having initial and final coordinates (V0, T0) and (2V0, T0/2). Then the ratio of work done in the two processes must be <img class="img-fluid question-image" alt="Question" src="https://cdn.tardigrade.in/q/nta/p-id11zfnufcu0xl3a.jpg" /> (A) 1:2 (B) 2:1 (C) 1:1 (D) None of these

Tardigrade Admin · Accepted Answer

The first process is the constant pressure process Hence,

w_{1} = n R (2 T_{0} - T_{0}) = n R T_{0}

Equation of the second process is

T = \frac{c}{V}

Hence,

P = \frac{nRT}{V} = \frac{nRc}{V ^{2}}

∴ w = \int_{V_{0}}^{2 V_{0}} PdV = \frac{nRT _{0}}{2}

∴ \frac{w _{1}}{w _{2}} = 2 : 1

$1 : 2$

$2 : 1$

$1 : 1$

None of these

The first process is the constant pressure process Hence, $w_{1} = n R (2 T_{0} - T_{0}) = n R T_{0}$
Equation of the second process is $T = \frac{c}{V}$
Hence, $P = \frac{nRT}{V} = \frac{nRc}{V ^{2}}$
$∴ w = \int_{V_{0}}^{2 V_{0}} PdV = \frac{nRT _{0}}{2}$
$∴ \frac{w _{1}}{w _{2}} = 2 : 1$

1:2

2:1

1:1

None of these

The first process is the constant pressure process Hence, w1​=nR(2T0​−T0​)=nRT0​ Equation of the second process is T=Vc​ Hence, P=VnRT​=V2nRc​ ∴w=∫V0​2V0​​PdV=2nRT0​​ ∴w2​w1​​=2: 1

$1 : 2$

$2 : 1$

$1 : 1$

The first process is the constant pressure process Hence, $w_{1} = n R (2 T_{0} - T_{0}) = n R T_{0}$
Equation of the second process is $T = \frac{c}{V}$
Hence, $P = \frac{nRT}{V} = \frac{nRc}{V ^{2}}$
$∴ w = \int_{V_{0}}^{2 V_{0}} PdV = \frac{nRT _{0}}{2}$
$∴ \frac{w _{1}}{w _{2}} = 2 : 1$