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Chapter 9: Problem 1279

The rms speed of gas molecules is given by (A) \(2.5 \sqrt{\left[M_{0} /(R T)\right]}\) (B) \(\left.2.5 \sqrt{[}(\mathrm{RT}) / \mathrm{M}_{0}\right]\) (C) \(\left.1.73 \sqrt{[}(\mathrm{RT}) / \mathrm{M}_{0}\right]\) (D) \(1.73 \sqrt{\left[M_{0} /(R T)\right]}\)

Short Answer

Expert verified
The correct formula for the root mean square speed of gas molecules is approximately (C) \(1.73 \sqrt{\frac{RT}{M_0}}\). This is the closest approximation to the correct equation \(v_{rms} = \sqrt{\frac{3RT}{M_0}}\) among the options given.

Step by step solution

01

Identify the correct formula

We are given four options for the rms speed of gas molecules: (A) \(2.5 \sqrt{\frac{M_0}{RT}}\) (B) \(2.5 \sqrt{\frac{RT}{M_0}}\) (C) \(1.73 \sqrt{\frac{RT}{M_0}}\) (D) \(1.73 \sqrt{\frac{M_0}{RT}}\) Our task is to find out which of these options matches the correct equation: \(v_{rms} = \sqrt{\frac{3RT}{M_0}}\). Step 2 - Compare the options to the correct formula
02

Compare the options to the correct formula

Let's compare the given options with the correct rms speed equation: (A) \(2.5 \sqrt{\frac{M_0}{RT}}\) ≠ \(\sqrt{\frac{3RT}{M_0}}\) (B) \(2.5 \sqrt{\frac{RT}{M_0}}\) ≠ \(\sqrt{\frac{3RT}{M_0}}\) (C) \(1.73 \sqrt{\frac{RT}{M_0}}\) ≠ \(\sqrt{\frac{3RT}{M_0}}\) However, we can see that option (C) is just a numerical approximation of the correct equation: Option (C) can be written as follows: \(1.73 \sqrt{\frac{RT}{M_0}} = \sqrt{\frac{3.00 RT}{M_0}}\) which is approximately equal to the original equation given. (D) \(1.73 \sqrt{\frac{M_0}{RT}}\) ≠ \(\sqrt{\frac{3RT}{M_0}}\) Step 3 - Identify the correct option
03

Identify the correct option

Option (C) is the closest approximation to the correct equation, hence the correct formula for the root mean square speed of gas molecules is: \[ \boxed{\text{(C)}\ 1.73 \sqrt{\frac{RT}{M_0}}} \] Note that although this option is not an exact match, it is the closest approximation among the options given.

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Most popular questions from this chapter

To what temperature should the hydrogen at \(327^{\circ} \mathrm{C}\) cooled at constant pressure, so that the root mean square velocity of its molecules become half of its previous value (A) \(-100^{\circ} \mathrm{C}\) (B) \(123^{\circ} \mathrm{C}\) (C) \(0^{\circ} \mathrm{C}\) (D) \(-123^{\circ} \mathrm{C}\)

A diatomic molecule has how many degrees of freedom (For rigid rotator) (A) 4 (B) 3 (C) 6 (D) 5

A sealed container with negligible co-efficient of volumetric expansion contains helium (a monatomic gas) when it is heated from \(200 \mathrm{~K}\) to \(400 \mathrm{~K}\), the average \(\mathrm{K} . \mathrm{E}\). of helium atom is (A) Halved (B) Doubled (C) Unchanged (D) Increased by factor \(\sqrt{2}\)

A diatomic gas molecule has translational, rotational and vibrational degrees of freedom. The $\left(\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{V}}\right)$ is (A) \(1.29\) (B) \(1.33\) C) \(1.4\) (D) \(1.67\)

The average translational kinetic energy of \(\mathrm{O}_{2}\) (molar mass 32 ) molecules at a particular temperature \(0.068 \mathrm{eV}\). The translational kinetic energy of \(\mathrm{N}_{2}\) (molar mass 28 ) molecules in \(\mathrm{eV}\) at the same temperature is (A) \(0.003 \mathrm{eV}\) (B) \(0.068 \mathrm{eV}\) (C) \(0.056 \mathrm{eV}\) (D) \(0.678 \mathrm{eV}\)
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