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Question

Answers

(A) $\dfrac { { 2P }_{ 1 }{ T }_{ 1 } }{ { T }_{ 1 }+{ T }_{ 2 } } $

(B) $\dfrac { { 2P }_{ 1 } }{ { T }_{ 1 }+{ T }_{ 2 } } $

(C) $\dfrac { { 2P }_{ 1 }{ T }_{ 2 } }{ { T }_{ 1 }+{ T }_{ 2 } } $

(D) None of these

Answer

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Given in the question , that there are two vessels, each of volume V at pressure ${ P }_{ 1 }$ and Temperature${ T }_{ 1 }$, and they have been connected together by a narrow tube.

So, we can name each vessel as A and B respectively.

After connecting, vessel B is changing it’s temperature to ${ T }_{ 2 }$and therefore we need to find the final pressure of both the vessels.

For that, we need to use Ideal Gas Equation,

Which is given by PV=nRT

Where P = pressure

T= temperature

R= universal gas constant

V=volume of the gas

Initially,

For vessel A , ${ P }_{ 1 }{ V }_{ 1 }={ nRT }_{ 1 }$ and for vessel B, ${ P }_{ 1 }{ V }_{ 1 }={ nRT }_{ 1 }$ (since the volume are same in both the vessels , we are taking it as V)

Finally, when both the vessels are connected together, then the temperature of vessel B changes to${ T }_{ 2 }$.

For vessel A , ${ P }_{ 2 }{ V }_{ 1 }={ nRT }_{ 1 }$and for vessel B, ${ P }_{ 2 }{ V }_{ 1 }={ nRT }_{ 2 }$(since the volume are same in both the vessels , we are taking it as V)

We need to find the expression of pressure${ P }_{ 2 }$,

After rearranging the ideal gas equation, we will get $\dfrac { PV }{ T } $= nR, which is a constant.

n is the number of moles which remain constant in both vessels and R is the universal gas constant.

We know that when the vessels are connected together by a narrow tube, it attains an equilibrium state, and so, the sum of initial conditions of both the vessels should be equal to the sum of final conditions of both the vessels.

Hence, $\dfrac { { P }_{ 1 }{ V } }{ { T }_{ 1 } } +\dfrac { { P }_{ 1 }{ V } }{ { T }_{ 1 } } =\dfrac { { P }_{ 2 }{ V } }{ { T }_{ 1 } } +\dfrac { { P }_{ 2 }{ V } }{ { T }_{ 2 } } \\ \Rightarrow \dfrac { { P }_{ 1 }\left( 2V \right) }{ { T }_{ 1 } } =\quad { P }_{ 2 }{ V }\quad \left( \dfrac { 1 }{ { T }_{ 1 } } +\dfrac { 1 }{ { T }_{ 2 } } \right) \\ \Rightarrow \dfrac { { 2P }_{ 1 } }{ { T }_{ 1 } } =\quad { P }_{ 2 }\left( \dfrac { 1 }{ { T }_{ 1 } } +\dfrac { 1 }{ { T }_{ 2 } } \right) \\ \Rightarrow \quad \dfrac { { 2 }P_{ 1 } }{ { T }_{ 1 } } ={ { P }_{ 2 } }\left( \dfrac { { T }_{ 2 }+{ T }_{ 1 } }{ { T }_{ 1 }{ T }_{ 2 } } \right) \\ \Rightarrow { \quad 2P }_{ 1 }={ { P }_{ 2 } }\left( \dfrac { { T }_{ 2 }+{ T }_{ 1 } }{ { T }_{ 2 } } \right) \\ \Rightarrow \quad { P }_{ 2 }=\dfrac { { 2P }_{ 1 }{ T }_{ 2 } }{ \left( { T }_{ 1 }+{ T }_{ 2 } \right) } $