Question

When a sample of ideal gas is changed from an initial state to a final state, various curves can be plotted for the process like P-V curve, V-T curve,P-T curve etc.
For example, P-V curve for a fixed amount of an ideal gas at constant temperature is a rectangular hyperbola, V-T curve for a fixed amount of an ideal gas at constant volume is again a straight line. However, the shapes may vary if the constant parameters are also changed.
Now, answer the questions :
Which of the following statement(s) is/are correct regarding a fixed amount of ideal gas undergoing the following process?

• A. Root mean square (RMS) speed of gas molecules increases during the process $A\to B$
• B. Density of the gas increases during the process $A\to B$.
• C. Such a graph is not possible.
• D. If $P_B=4P_A$, then $V_A=4V_B$ (Where $P_A,V_A,P_B$ & $V_B$ represent pressure and volume values at states A and B).

Answer 1

The correct option is B Density of the gas increases during the process $A\to B$.

$\text{(a.) The root mean square speed is directly proportional to the square root of temperature}$ $(\because V_{RMS}\alpha \sqrt{T})$.

$\text{During the process A to B, the temperature decreases. Hence, the RMS speed decreases.}$

$\text{Thus, option (a) is incorrect.}$

$\text{(b). The expression for the density of gas is}$ $d_{gas}=\frac{PM}{RT}$. $\text{During the process A to B, the pressure increases, and}$ $\text{temperature decreases. The density of gas increases.}$

$\text{Thus, option (b) is correct.}$

$\text{(c). If during the process,}$ $P\alpha \frac{1}{V^{1/2}}$ and $\because PT=Constant\therefore P\left(\frac{PV}{nR}\right)=Constant\therefore P^{2}V=constant$ ot $P{V}^{1/2}=Constant$.

$\text{Hence, The graph is possible. Thus option (c) is incorrect.}$

$\text{(d). For the process,}$ $P^{2}V=constant$. $Thuswhen$ $P_B=4P_A$, then $V_A=16V_B$. $\text{Thus option (d) is incorrect.}$