Question

One mole of an ideal gas passes through a process where pressure and volume obey the relation $P =P_{o} \left[1- \frac{1}{2} \left(\frac{V_{0}}{V}\right)^{2}\right] $ .Here $P_o$ and $V_o$ are constants. Calculate the change in the temperature of the gas if its volume changes from $V_o$ to $2 V_o$ .

• A. $\frac{1}{2} \frac{P_{o}V_{o}}{R} $
• B. $\frac{3}{4} \frac{P_{o}V_{o}}{R} $
• C. $\frac{5}{4} \frac{P_{o}V_{o}}{R} $
• D. $\frac{1}{4} \frac{P_{o}V_{o}}{R} $

Answer 1

The Correct Option is C

To find the change in temperature of one mole of an ideal gas when its volume changes from V_o to 2V_o under the given pressure-volume relationship, we need to apply the principles of the ideal gas law and the provided equation:

The given pressure-volume relation is:

P =P_{o} \left[1- \frac{1}{2} \left(\frac{V_{o}}{V}\right)^{2}\right]

We know for an ideal gas, the ideal gas law is given by:

PV = nRT

Since we are dealing with one mole of gas (n=1), the equation becomes:

PV = RT

First, calculate the initial temperature (T_1) when the volume is V = V_o:

Substituting V = V_o into the pressure equation:

P_1 = P_{o} \left[1- \frac{1}{2} \right] = \frac{1}{2} P_{o}

Substitute P_1 and V_o into the ideal gas equation to find T_1:

P_1V_o = RT_1 \Rightarrow \left(\frac{1}{2} P_{o}\right) V_o = RT_1 \Rightarrow T_1 = \frac{1}{2} \frac{P_{o}V_{o}}{R}

Next, calculate the final temperature (T_2) when the volume is V = 2V_o:

Substituting V = 2V_o into the pressure equation:

P_2 = P_{o} \left[1- \frac{1}{2} \left(\frac{V_{o}}{2V_o}\right)^{2}\right] = P_{o} \left[1- \frac{1}{8}\right] = P_{o} \left[\frac{7}{8}\right]

Substitute P_2 and 2V_o into the ideal gas equation to find T_2:

P_2(2V_o) = RT_2 \Rightarrow \left(\frac{7}{8}P_{o}\right)(2V_o) = RT_2 \Rightarrow T_2 = \frac{7}{4} \frac{P_{o}V_{o}}{R}

The change in temperature (\Delta T) is given by:

\Delta T = T_2 - T_1 = \frac{7}{4} \frac{P_{o}V_{o}}{R} - \frac{1}{2} \frac{P_{o}V_{o}}{R}

Simplify to find \Delta T:

\Delta T = \left(\frac{7}{4} - \frac{1}{2}\right) \frac{P_{o}V_{o}}{R} = \left(\frac{7}{4} - \frac{2}{4}\right) \frac{P_{o}V_{o}}{R} = \frac{5}{4} \frac{P_{o}V_{o}}{R}

Thus, the change in temperature is \frac{5}{4} \frac{P_{o}V_{o}}{R}, which corresponds to the correct answer.