Question

Two closed vessels of an equal volume containing air at pressure ${{P}_{1}}$​ and temperature ${{T}_{1}}$​ are connected to each other through a narrow tube. If the temperature in one of the vessels is now maintained at ${{T}_{1}}$ ​and that in the other at ${{T}_{2}}$​, what will be the pressure in the vessels?
A.​​$\dfrac{2{{P}_{1}}{{T}_{1}}}{{{T}_{1}}+{{T}_{2}}}$
B.​​$\dfrac{{{T}_{1}}}{2{{P}_{1}}{{T}_{2}}}$
C.​​$\dfrac{2{{P}_{1}}{{T}_{2}}}{{{T}_{1}}+{{T}_{2}}}$
D.$\dfrac{2{{P}_{1}}}{{{T}_{1}}+{{T}_{2}}}$

Answer 1

Here you need to use the ideal gas equation. This question comes under hard and conceptual type questions. You need to rearrange the ideal gas equation according to the conditions given in the question and the main core concept here is to equate initial and final mole.

Complete step by step answer:
Initial mole = final mole
According to the ideal gas equation:
$PV=nRT$ …………(i)
${{P}_{1}}=$ Initial pressure of both the vessels
${{T}_{1}}=$ Initial temperature of both the vessels
${{T}_{2}}=$ Changed temperature of one vessel
P = Required pressure of vessel

On rearranging equation (i) and then further solving according to question requirement we will get:
$\begin{aligned} & n=\dfrac{PV}{RT} \\\ & \dfrac{{{P}_{1}}V}{{{T}_{1}}R}+\dfrac{{{P}_{1}}V}{R{{T}_{1}}}=\dfrac{PV}{R{{T}_{1}}}+\dfrac{PV}{R{{T}_{2}}} \\\ & \dfrac{V}{R}\left( \dfrac{{{P}_{1}}}{{{T}_{1}}}+\dfrac{{{P}_{1}}}{{{T}_{1}}} \right)=\dfrac{V}{R}\left( \dfrac{P}{{{T}_{1}}}+\dfrac{P}{{{T}_{2}}} \right) \\\ & \dfrac{2{{P}_{1}}}{{{T}_{1}}}=P\left( \dfrac{1}{{{T}_{1}}}+\dfrac{1}{{{T}_{2}}} \right) \\\ & \dfrac{2{{P}_{1}}}{{{T}_{1}}}=P\left( \dfrac{{{T}_{2}}+{{T}_{1}}}{{{T}_{1}}{{T}_{2}}} \right) \\\ & \dfrac{2{{P}_{1}}{{T}_{1}}{{T}_{2}}}{{{T}_{1}}}=P\left( {{T}_{2}}+{{T}_{1}} \right) \\\ & P=\dfrac{2{{P}_{1}}{{T}_{2}}}{{{T}_{1}}+{{T}_{2}}} \\\ \end{aligned}$

So,the correct option is (C).

Note:
The ideal gas law is also called the general gas equation.
The state of a gas is determined by its pressure, temperature and volume according to following equation:
$PV=nRT$
Here P = pressure
V = volume
R = Universal gas constant
T = temperature
n = number of moles of gas
This ideal gas equation is applicable to gases only not on liquids. Ideal gas is nothing but just a hypothetical concept made by scientists to make calculations and results easier.
An ideal gas can be defined as one in which all collisions between atoms or molecules are perfectly elastic and in which there are no intermolecular attractive forces.