question_answer

Two thermally insulated vessels 1 and 2 are filled with air at temperatures\[({{T}_{1}},{{T}_{2}}),\]volumes \[({{V}_{1}},{{V}_{2}}),\]and pressures\[({{p}_{1}},{{p}_{2}}),\]respectively. If the valve joining the two vessels is opened, the temperature inside the vessel at equilibrium will be                                           [AIEEE 2004]

A)  \[{{T}_{1}}+{{T}_{2}}\]       

B)       \[\frac{({{T}_{1}}+{{T}_{2}})}{2}\]

C)  \[\frac{{{T}_{1}}{{T}_{2}}({{p}_{1}}{{V}_{1}}+{{p}_{2}}{{V}_{2}})}{{{p}_{1}}{{V}_{1}}{{T}_{2}}+{{p}_{2}}{{V}_{2}}{{T}_{1}}}\]

D)  \[\frac{{{T}_{1}}{{T}_{2}}({{p}_{1}}{{V}_{1}}+{{p}_{2}}{{V}_{2}})}{{{p}_{1}}{{V}_{1}}{{T}_{1}}+{{p}_{2}}{{V}_{2}}{{T}_{2}}}\]

Correct Answer: C

Solution :

[c]

From ideal gas equation,

(as P and T will become same)

(moles of gas remain same)

..(i)

Internal energy of the system (air in two vessels) remains same, before and after opening of valve, so

where, f is degree of freedom of gas molecules

From Eq. (i) we can find the equilibrium pressure also.