Question 125: Thermodynamics & KTG

Question

In an insulated vessel, 0.05 kg steam at 373 K and 0.45 kg of ice at 253 K are mixed. Then, find the final temperature of the mixture.

Accepted Answer

$$ 0.05 \mathrm{~kg} 	ext { steam at } 373 \mathrm{~K} \xrightarrow{\mathrm{Q}_1}0.05 \mathrm{~kg} 	ext { water at } 373 \mathrm{~K} 	ext {. } $$
$$ 0.05 \mathrm{~kg} 	ext { water at } 373 \mathrm{~K} \xrightarrow{\mathrm{Q}_2} 0.05 \mathrm{~kg} 	ext { water at } 273 \mathrm{~K}$$
$$ 0.45 \mathrm{~kg} 	ext { ice at } 253 \mathrm{~K} \xrightarrow{\mathrm{Q}_3} 0.45 \mathrm{~kg} 	ext { ice at } 273 \mathrm{~K} $$
$$ 0.45 \mathrm{~kg} 	ext { ice at } 273 \mathrm{~K} \xrightarrow{\mathrm{Q}_4} 0.45 \mathrm{~kg} 	ext { water at } 273 \mathrm{~K}$$
$$ \begin{aligned} &\begin{aligned} & \Rightarrow \mathrm{Q}_1=m \mathrm{~L}=50(540)=27000 \mathrm{cal}=27 \mathrm{kcal} (\mathrm{Q}=m \mathrm{~L}) \ & \mathrm{Q}_2=m s \Delta t=(50)(1)(100)=500 \mathrm{cal}=5 \mathrm{kcal} \ & \mathrm{Q}_3=m s \Delta t=(4.50)(0.5)(20)=4500 \mathrm{cal}=4.5 \mathrm{kcal} \ & \mathrm{Q}_4=(m \mathrm{~L})=450 	imes 580=36000 \mathrm{cal}=36 \mathrm{kcal} \,\, [fusion]\end{aligned}\ &	ext { } \end{aligned} $$
Since, $Q_1+Q_2>Q_3$ but $Q_1+Q_2
Therefore temperature of mixture $=273 \mathrm{~K}$
(or $0^{\circ} \mathrm{C}$ )
Method II
Heat lost by steam to covert into $0^{\circ} \mathrm{C}$ water
$$ \begin{gathered} \mathrm{H}_{\mathrm{L}}=0.05 	imes 540+0.05 	imes 10 	imes 1 \ {\left[\mathrm{Q}=\mathrm{H}_{\mathrm{L}}=m \mathrm{~L}+m s \Delta tight]} \ \mathrm{H}_{\mathrm{L}}=27+5=32 \mathrm{kcal} \end{gathered} $$
Heat required by ice to change into $0^{\circ} \mathrm{C}$ water.
$$ \begin{aligned} \mathrm{H}_{\mathrm{g}} & =450(20) \frac{1}{2}+450 	imes 80 \ & =4500+36000=40500 \mathrm{cal} \ & =40.5 \mathrm{kcal} \end{aligned} $$
$\Rightarrow$ To convert ice, we have $450 	imes \frac{1}{2}=225 \mathrm{cal}$.
But we need 32000 cal.
Therefore, ice will remain as ice. Hence final temperature is $0^{\circ} \mathrm{C}$.

Thermodynamics & KTG

In an insulated vessel, 0.05 kg steam at 373 K and 0.45 kg of ice at 253 K are mixed. Then, find the final temperature of the mixture.

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