Question

Given below are two statements : 
Statement I : The temperature of a gas is $-73^{\circ} C$ When the gas is heated to $527^{\circ} C$, the root mean square speed of the molecules is doubled.
Statement II : The product of pressure and volume of an ideal gas will be equal to translational kinetic energy of the molecules.
In the light of the above statements, choose the correct answer from the options given below:

• A. Statement I is false but Statement II is true
• B. Both Statement I and Statement II are true
• C. Both Statement I and Statement II are false
• D. Statement I is true but Statement II is false

Answer 1

The Correct Option is D

To evaluate the given statements, let's analyze each one separately:

1. Statement I: The temperature of a gas is \( -73^{\circ} C \). When the gas is heated to \( 527^{\circ} C \), the root mean square (RMS) speed of the molecules is doubled.
   • The RMS speed of gas molecules is given by the formula: \(v_{\text{rms}} = \sqrt{\frac{3kT}{m}}\), where \( k \) is the Boltzmann constant, \( T \) is the absolute temperature in Kelvin, and \( m \) is the mass of a molecule.
   • The initial temperature \( T_1 = -73^{\circ} C = 200 \, K \), and the final temperature \( T_2 = 527^{\circ} C = 800 \, K \).
   • The relation for RMS speed when temperature changes is: \(v_{\text{rms, 2}} = \sqrt{\frac{T_2}{T_1}} \cdot v_{\text{rms, 1}}\)
   • If the RMS speed is doubled, \(\frac{v_{\text{rms, 2}}}{v_{\text{rms, 1}}} = 2\), then: \(2 = \sqrt{\frac{T_2}{T_1}}\)
   • Squaring both sides: \(4 = \frac{T_2}{T_1}\)
   • This implies: \(T_2 = 4 \cdot T_1\), substituting the values gives: \(T_2 = 4 \cdot 200 = 800 \, K\), which matches the given final temperature of \( 800 \, K \).
   • This confirms that Statement I is true.
2. Statement II: The product of pressure and volume of an ideal gas will be equal to the translational kinetic energy of the molecules.
   • According to the ideal gas law, the product of pressure and volume is given by \(PV = nRT\).
   • The translational kinetic energy of the gas is given by: \(KE = \frac{3}{2}nRT\).
   • Clearly, \(PV\) is not equal to the translational kinetic energy \(KE\).
   • Thus, Statement II is false.

By analyzing both statements, the correct answer is:

Statement I is true but Statement II is false.