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Correct Answer: (3p/dm) ½
the root mean square velocity (u_rms) of the molecules in a gas is an important concept in the kinetic theory of gases. this velocity is a measure of the speed of particles within a gas, which can be determined using the equation derived from the kinetic theory. let's analyze each expression provided in the question and identify the one that does not conform to the root mean square velocity. the correct formula for the root mean square velocity, u_rms, for an ideal gas is given by: \[ u_{rms} = \sqrt{\frac{3rt}{m}} \] where: - r is the universal gas constant, - t is the absolute temperature in kelvin, - m is the molar mass of the gas. this formula can also be expressed in terms of pressure (p) and density (d) of the gas, using the ideal gas law, \( pv = nrt \) and the relation \( d = \frac{m}{v} \) where v is the volume. from these relations: \[ p = \frac{nrt}{v} \] \[ p = \frac{drt}{m} \] rearranging it, we get: \[ \frac{rt}{m} = \frac{p}{d} \] thus, \[ u_{rms} = \sqrt{\frac{3p}{d}} \] now, let's examine the given expressions: 1. \( \left(\frac{3rt}{m}\right)^{1/2} \) - this correctly represents the root mean square velocity. 2. \( \left(\frac{3pv}{m}\right)^{1/2} \) - using the ideal gas law, \( pv = nrt \), this expression can be rewritten as \( \left(\frac{3rt}{m}\right)^{1/2} \), which is correct. 3. \( \left(\frac{3p}{d}\right)^{1/2} \) - as shown above, this expression correctly represents the root mean square velocity derived using density. the incorrect expression is: 4. \( \left(\frac{3p}{dm}\right)^{1/2} \) - this expression introduces an additional molar mass term in the denominator, which is not correct according to the kinetic theory of gases. the presence of m in the denominator in this way does not correspond to any derivation from the ideal gas law or the formula for root mean square velocity. hence, this expression does not conform to the root mean square velocity of a gas. therefore, the expression \( \left(\frac{3p}{dm}\right)^{1/2} \) is the correct answer to the question as it does not conform to the standard formula for the root mean square velocity in the kinetic theory of gases.
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