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:
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:
- Statement I is false but Statement II is true
- Both Statement I and Statement II are true
- Both Statement I and Statement II are false
- Statement I is true but Statement II is false
The Correct Option is D
Solution and Explanation
Statement I: The root mean square (RMS) speed of gas molecules is given by: \[ v_{\text{rms}} \propto \sqrt{T} \] Let \(T_1 = -73^\circ C = 200 \, \text{K}\) and \(T_2 = 527^\circ C = 800 \, \text{K}\). The ratio of RMS speeds is: \[ \frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} = \sqrt{\frac{800}{200}} = \sqrt{4} = 2 \] Thus, the RMS speed doubles, and Statement I is true.
Statement II: For an ideal gas, the product of pressure and volume is given by: \[ PV = nRT \] The translational kinetic energy is: \[ \text{Translational KE} = \frac{3}{2} nRT \] Since \(PV \neq \frac{3}{2}nRT\), Statement II is false.
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Concepts Used:
Kinetic energy
Kinetic energy of an object is the measure of the work it does as a result of its motion. Kinetic energy is the type of energy that an object or particle has as a result of its movement. When an object is subjected to a net force, it accelerates and gains kinetic energy as a result. Kinetic energy is a property of a moving object or particle defined by both its mass and its velocity. Any combination of motions is possible, including translation (moving along a route from one spot to another), rotation around an axis, vibration, and any combination of motions.
