Initial pressure and volume of monoatomic gas is \(P\) and \(V\). It is expanded adiabatically to \(27\) times of initial volume. Find magnitude of change in internal energy.
- \( \dfrac{3}{2}PV \)
- \(PV\)
- \( \dfrac{4}{3}PV \)
- \( \dfrac{PV}{2} \)
The Correct Option is C
Solution and Explanation
Step 1:Use adiabatic relation. \[ P_1V_1^\gamma = P_2V_2^\gamma \] Given \[ V_2 = 27V \] \[ P V^{5/3} = P_2 (27V)^{5/3} \] \[ P_2 = \frac{P}{27^{5/3}} \] \[ 27^{5/3} = (3^3)^{5/3} = 3^5 \] \[ P_2 = \frac{P}{3^5} \]
Step 2:Calculate change in internal energy. \[ \Delta U = \frac{P_2V_2 - P_1V_1}{\gamma-1} \] \[ = \frac{\left(\frac{P}{3^5}\right)(27V) - PV}{\frac{5}{3}-1} \]
Step 3:Simplify the expression. \[ = \frac{\frac{P\cdot27V}{3^5} - PV}{\frac{2}{3}} \] \[ = \frac{\frac{PV}{9} - PV}{\frac{2}{3}} \] \[ = \frac{-\frac{8}{9}PV}{\frac{2}{3}} \] \[ \Delta U = -\frac{4}{3}PV \] Since magnitude is required, \[ \boxed{|\Delta U| = \frac{4}{3}PV} \]
Top JEE Main Chemistry Questions
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- At equilibrium the concentrations of and in a sealed vessel at . What will be for the reaction
- Find out the major products from the following reactions
Cobalt chloride when dissolved in water forms pink colored complex $X$ which has octahedral geometry. This solution on treating with cone $HCl$ forms deep blue complex, $\underline{Y}$ which has a $\underline{Z}$ geometry $X, Y$ and $Z$, respectively, are
- The correct order of atomic radii is:
Top JEE Main Thermodynamics Questions
- If \[ \text{C(diamond)} \rightarrow \text{C(graphite)} + X \, \text{kj mol}^{-1} \] \[ \text{C(diamond)} + \text{O}_2(g) \rightarrow \text{CO}(g) + Y \, \text{kj mol}^{-1} \] \[ \text{C(graphite)} + \text{O}_2(g) \rightarrow \text{CO}(g) + Z \, \text{kj mol}^{-1} \] At constant temperature. Then:
Match List - I with List - II.
Choose the correct answer from the options given below :
- Which of the following statement(s) is correct for the adiabatic process?
- An ideal gas with an adiabatic exponent 1.5, initially at 27°C is compressed adiabatically from 800 cc to 200 cc. The final temperature of the gas is:
- The hydration energies of $ K^+ $ and $ Cl^- $ are $ -x $ and $ -y $ kJ/mol respectively. If lattice energy of KCl is $ -z $ kJ/mol, then the heat of solution of KCl is :
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.