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Heat and Thermodynamics

Section: Physics

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Question : 32 of 78

Marks: +1, -0

If 2 moles of an ideal monoatomic gas at a temperature of

27^{\circ} \mathrm{C}

is mixed with 4 moles of another ideal monoatomic gas at a temperature of

327^{\circ} \mathrm{C}

, then the temperature of mixture of the two gases is

[AP EAPCET 21 May 2025 S1]

$300^{\circ} \mathrm{C}$
$227^{\circ} \mathrm{C}$
$233^{\circ} \mathrm{C}$
$327^{\circ} \mathrm{C}$

Solution:

Total initial internal energy

U'=U_1\;+U_2=\;\frac{3}{2} n_1 R T_1+\;\frac{3}{2} n_2 R T_2

\;=\;\frac{3}{2} R (n_1 T_1+n_2 T_2)

\;=\;\frac{3}{2} R(2 \times 300+4 \times 600)

\;=4500 R

Total number of moles after mixing

n_{\;\text{total }\;}\;=n_1+n_2=2+4=6

\therefore \;\; U_{\;\text{final }\;}\;=\;\frac{3}{2} n_{\;\text{total }\;} R T_{\;\text{final }\;}

U_{\;\text{final }\;}\;=\;\frac{3}{2} \times 6 \times R T_{\;\text{final }\;}

Since,

U^{'}=U_{\;\text{final }\;}

\;\Rightarrow 4500 R=\;\frac{3}{2} \times 6 \times R \times T_{\;\text{final }\;}

\;T_{\;\text{final }\;} \;\; =500\ \mathrm{K}=(500-273)^{\circ}\ \mathrm{C}=227^{\circ}\ \mathrm{C}

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