Question

A closed tank has two compartments A and B, both filled with oxygen (assumed to be an ideal gas). The partition separating the two compartments is fixed and is a perfect heat insulator (Figure 1). If the old partition is replaced by a new partition, which can slide and conduct heat does not allow the gas to leak across (Figure 2), the volume (in $m^3$) of the compartment A after the system attains equilibrium is ______________

Answer 1

Hint: We can solve this question by using many formulas like $n={\displaystyle \frac{PV}{TR}}$, ${\displaystyle \frac{P_A}{T_A}}={\displaystyle \frac{P_B}{T_B}}$, where P is the pressure, V is the volume, R is the gas constant, T is the temperature, n is the number of moles.

Complete step-by-step answer: So according to Figure 1, in compartment A, the pressure is 5 bar, the volume is 1 $m^3$, and temperature 400 K. This can be written as:
$P_A=5$
$V_A=1$
$T_1=400$
In compartment B, the pressure is 1 bar, the volume is 3 $m^3$, and temperature 400 K. This can be written as:
$P_B=1$
$V_B=3$
$T_B=300$
So, from the given values we can calculate the number of moles for both the compartments by using the formula $n={\displaystyle \frac{PV}{TR}}$, we get
$n_A={\displaystyle \frac{\text{ 5 x 1}}{400\text{ x R}}}={\displaystyle \frac{5}{400R}}$
$n_B={\displaystyle \frac{\text{ 1 x 3}}{300\text{ x R}}}={\displaystyle \frac{3}{300R}}$
The total volume will be = 1 + 3 = 4 $m^3$
In Figure 2, let us assume, the volume of A is x and volume of B is (4 – x).
We know that:
${\displaystyle \frac{P_A}{T_A}}={\displaystyle \frac{P_B}{T_B}}$, because it will attain equilibrium.
This can be written as:
${\displaystyle \frac{n_A\text{ x R}}{V_A(new)}}={\displaystyle \frac{n_B\text{ x R}}{V_B(new)}}$
Putting, the values in this, we get:
${\displaystyle \frac{5}{400(x)}}={\displaystyle \frac{3}{300(x-4)}}$
For solving x, we get:
$5(4-x)=4x$
$V_A=x={\displaystyle \frac{20}{9}}=2.22$
The volume of A after equilibrium will be 2.22 $m^3$.

Note: The formulas that we have used in the question, $n={\displaystyle \frac{PV}{TR}}$, and ${\displaystyle \frac{P_A}{T_A}}={\displaystyle \frac{P_B}{T_B}}$ can only be used because the condition was mentioned that the system is an ideal system, if the system is real then we cannot use these formulas.