JEE Main · 2004 · 24 Apr 2004 · Offline · Heat & Thermodynamics

Two thermally insulated vessels 1 and 2 are filled with air at temperatures

(T_{1}, T_{2})

, volume

(V_{1}, V_{2})

and pressure

(P_{1}, P_{2})

respectively. If the valve joining two vessels is opened, the temperature inside the vessel at equilibrium will be

(A) $T_{1} + T_{2}$
(B) $(T_{1} + T_{2}) /2$
(C) $\frac{T _{1} T _{2} ( P _{1} V _{1} + P _{2} V _{2} )}{P _{1} V _{1} T _{2} + P _{2} V _{2} T _{1}}$ ✓ Correct
(D) $\frac{T _{1} T _{2} ( P _{1} V _{1} + P _{2} V _{2} )}{P _{1} V _{1} T _{1} + P _{2} T _{2} T _{2}}$

Answer: (C)

Solution

No heat leaves the insulated vessels, so the total internal energy is conserved as the gases mix. The moles in each are

n_{1} = \frac{P _{1} V _{1}}{R T _{1}}

and

n_{2} = \frac{P _{2} V _{2}}{R T _{2}}

, and after mixing the whole amount occupies

V_{1} + V_{2}

at a common temperature

T

. The energy balance

n_{1} C_{v} T_{1} + n_{2} C_{v} T_{2} = (n_{1} + n_{2}) C_{v} T

, with these moles substituted, rearranges to

T = \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}}

JEE Main · 2004 · 24 Apr 2004 · Offline · Single correct option