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

Fig. 2

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Assertion and Reason

Read the following questions and answer as per the direction given below :

(a) Statement I is correct; Statement II is correct; Statement II is the correct explanation of Statement I

(b) Statement I is correct; Statement II is correct; Statement II is not the correct explanation of Statement I

(c) Statement I is correct; Statement II is incorrect

(d) Statement I is incorrect; Statement II is correct

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Solution:

  1. Given $p _1=5$ bar, $V _1=1 m^{3}, T _1=400 K$

So, $\quad n _1=\frac{5}{400 R}$

(from $p V=n R T$ )

Similarly, $p _2=1$ bar, $V _2=3 m^{3}, T _2=300 K, n _2=\frac{3}{300 R}$

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.

$$ \frac{p _1 V _1}{n _1 R T _1}=\frac{p _2 V _2}{n _2 R T _2} $$

and at equilibrium (due to conduction of heat)

$$ \begin{aligned} \frac{p _1}{T _1} & =\frac{p _2}{T _2} \ \text { So, } \quad \frac{V _1}{n _1} & =\frac{V _2}{n _2} \text { or } V _1 n _2=V _2 n _1 \end{aligned} $$

After putting the values

$$ \begin{aligned} & (1+x) \times \frac{3}{300 R}=(3-x) \times \frac{5}{400 R} \text { or }(1+x)=\frac{(3-x) 5}{4} \ & \text { or } 4(1+x)=15-5 x \text { or } 4+4 x=15-5 x \text { or } x=\frac{11}{9} \end{aligned} $$

Hence, new volume of $A$ i.e., $(1+x)$ will comes as $1+\frac{11}{9}=\frac{20}{9}$ or 2.22



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