States of Matter - Result Question 47
47. A closed tank has two compartments $A$ and $B$, both filled with oxygen (assumed to be ideal gas). The partition separating the two compartments is fixed and is a perfect heat insulator (Fig. 1). If the old partition is replaced by a new partition which can slide and conduct heat but does not allow the gas to leak across (Fig. 2), the volume (in $m^{3}$ ) of the compartment $A$ after the system attains equilibrium is _____
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Answer:
Correct Answer: 47. (2.22)
Solution:
Given $p _1=5$ bar, $V _1=1 m^{3}, T _1=400 K$
So, $\quad n _1=\dfrac{5}{400 R}$ (from $p V=n R T$ )
Similarly, $p _2=1$ bar, $V _2=3 m^{3}, T _2=300 K, n _2=\dfrac{3}{300 R}$
Let at equilibrium the new volume of $A$ will be $(1+x)$
So, the new volume of $B$ will be $(3-x)$
Now, from the ideal gas equation.
$ \dfrac{p _1 V _1}{n _1 R T _1}=\dfrac{p _2 V _2}{n _2 R T _2} $
and at equilibrium (due to conduction of heat)
$ \begin{aligned} \dfrac{p _1}{T _1} & =\dfrac{p _2}{T _2} \\ \text { So, } \quad \dfrac{V _1}{n _1} & =\dfrac{V _2}{n _2} \text { or } V _1 n _2=V _2 n _1 \end{aligned} $
After putting the values
$ \begin{aligned} & (1+x) \times \dfrac{3}{300 R}=(3-x) \times \dfrac{5}{400 R} \text { or }(1+x)=\dfrac{(3-x) 5}{4} \\ & \text { or } 4(1+x)=15-5 x \text { or } 4+4 x=15-5 x \text { or } x=\dfrac{11}{9} \end{aligned} $
Hence, new volume of $A$ i.e., $(1+x)$ will comes as $1+\dfrac{11}{9}=\dfrac{20}{9}$ or $2.22$
BETA
