Question

Two thermally insulated vessels 1 and 2 are filled with air at temperature $({{T}_{1}}, {{T}_{2}})$, volume $({{V}_{1}}, {{V}_{2}})$ and pressure $({{P}_{1}}, {{P}_{2}})$ respectively. If the valve joining the two vessels is opened and no work is done then the temperature inside the vessel at equilibrium is then:
$\begin{aligned} & A.{{T}_{1}}+{{T}_{2}} \\\ & B.\dfrac{({{T}_{1}}+{{T}_{2}})}{2} \\\ & C.\dfrac{{{T}_{1}}{{T}_{2}}({{P}_{1}}{{V}_{1}}+{{P}_{2}}{{V}_{2}})}{{{P}_{1}}{{V}_{1}}{{T}_{2}}+{{P}_{2}}{{V}_{2}}{{T}_{1}}} \\\ & D.\dfrac{{{T}_{1}}{{T}_{2}}({{P}_{1}}{{V}_{1}}+{{P}_{2}}{{V}_{2}})}{{{P}_{1}}{{V}_{1}}{{T}_{1}}+{{P}_{2}}{{V}_{2}}{{T}_{2}}} \\\ \end{aligned}$

Answer 1

We have to use the gas equation to find the number of moles in vessel 1 and vessel 2. After opening the valve there is a constant pressure and temperature so we can find the total number of moles by using the gas equation again. On equating the two equations we get the required temperature.

Formula used: $PV=nRT$

Complete step by step solution:
There will be no change in the number of moles if the vessels are joined by valves. Therefore, from gas equation
$PV=nRT$
Number of moles in vessel 1 and 2 is equal to the number of moles during equilibrium
$\dfrac{{{P}_{1}}{{V}_{1}}}{R{{T}_{1}}}+\dfrac{{{P}_{2}}{{V}_{2}}}{R{{T}_{2}}}=\dfrac{P({{V}_{1}}+{{V}_{2}})}{RT}$
$\Rightarrow \dfrac{{{P}_{1}}{{V}_{1}}{{T}_{2}}+{{P}_{2}}{{V}_{2}}{{T}_{1}}}{{{T}_{1}}{{T}_{2}}}=\dfrac{P({{V}_{1}}+{{V}_{2}})}{T}$ (where T is the temperature of the system after opening the valve)
$T=\dfrac{P({{V}_{1}}+{{V}_{2}}){{T}_{1}}{{T}_{2}}}{{{P}_{1}}{{V}_{1}}{{T}_{2}}+{{P}_{2}}{{V}_{2}}{{T}_{1}}}$
Now, according to Boyle’s law (pressure=constant)
Hence, $T=\dfrac{({{P}_{1}}{{V}_{1}}+{{P}_{2}}{{V}_{2}}){{T}_{1}}{{T}_{2}}}{{{P}_{1}}{{V}_{1}}{{T}_{2}}+{{P}_{2}}{{V}_{2}}{{T}_{1}}}$
Therefore the correct option is C.

Note: Thermodynamics is a branch of physics that deals with heat, work, and temperature. Its relationship to energy, radiation, and physical properties of matter defines the functioning of thermodynamics. The four laws of thermodynamics govern the behavior of these quantities. Thermodynamics is applied in different fields of work like chemical engineering, meteorology, mechanical engineering and physical chemistry. Thermodynamics was historically developed to improve the efficiency of engines used at that point of time. The study of chemical compounds and chemical reactions took place under thermodynamics just after its application in mechanical heat engines. A chemical reaction is dependent upon the entropy of the reactant taking part in the reaction. Chemical thermodynamics provides a bulk of knowledge and information of the field of entropy.