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Chapter 1: Problem 141

Write the dimensional formula of r.m.s (root mean square) speed. (a) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\) (b) \(\mathrm{M}^{0} \mathrm{~L}^{2} \mathrm{~T}^{-2}\) (c) \(\mathrm{M}^{0} \mathrm{~L}^{1} \mathrm{~T}^{-1}\) (d) \(\mathrm{M}^{1} \mathrm{~L}^{0} \mathrm{~T}^{-1}\)

Short Answer

Expert verified
(c) \(\mathrm{M}^{0} \mathrm{~L}^{1} \mathrm{~T}^{-1}\)

Step by step solution

01

We are given the formula for the rms speed: \[v_{rms} = \sqrt{\frac{3k_BT}{m}}\] where \(v_{rms}\) represents the rms speed, \(k_B\) represents the Boltzmann constant, \(T\) represents the temperature, and \(m\) represents the mass of a single gas particle. #Step 2: Analyze the dimensions of each variable in the formula#

Let's find the dimensional formula for each variable in the rms speed formula: \(v_{rms}\) is a speed and therefore has dimensions of length per time: \(\mathrm{L}^{1}\mathrm{T}^{-1}\) \(k_B\) is the Boltzmann constant with dimensions of energy per temperature: \(\mathrm{M}^{1}\mathrm{L}^{2}\mathrm{T}^{-2}\mathrm{K}^{-1}\) \(T\) has dimensions of temperature: \(\mathrm{K}^{1}\) \(m\) has dimensions of mass: \(\mathrm{M}^{1}\) #Step 3: Substituting the dimensions into the formula#
02

We'll now substitute the dimensions of each variable in the formula: \(v_{rms} = \sqrt{\frac{3k_BT}{m}}\) \(\mathrm{L}^{1}\mathrm{T}^{-1} = \sqrt{\frac{\mathrm{M}^{0}\mathrm{L}^{0}\mathrm{T}^{0}(\mathrm{M}^{1}\mathrm{L}^{2}\mathrm{T}^{-2}\mathrm{K}^{-1})(\mathrm{K}^{1})}{\mathrm{M}^{1}}}\) #Step 4: Simplify and find the dimensional formula of rms speed#

Now we simplify the dimensions in the formula and find the dimensional formula for \(v_{rms}\): \(\mathrm{L}^{1}\mathrm{T}^{-1} = \sqrt{\frac{\mathrm{M}^{0}\mathrm{L}^{2}\mathrm{T}^{-2}}{\mathrm{M}^{1}}}\) \(\mathrm{L}^{1}\mathrm{T}^{-1} = \mathrm{M}^{0}\mathrm{L}^{1}\mathrm{T}^{-1}\) From this simplification, we see that the dimensional formula for rms speed is: \(\mathrm{M}^{0} \mathrm{~L}^{1} \mathrm{~T}^{-1}\) So, the correct answer is: (c) \(\mathrm{M}^{0} \mathrm{~L}^{1} \mathrm{~T}^{-1}\)

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Most popular questions from this chapter

Which of the following symbol of unit does not follow practical norms for the use of SI system? (a) \(\mathrm{Kg}\) (b) \(\mathrm{kg}\). (c) \(\mathrm{k}\) (d) \(\mathrm{A}\)

If $\mathrm{P}=\left[\left(\mathrm{A}^{2} \mathrm{~B}\right) /\left(\mathrm{C}^{3}\right)\right]\( where percentage error in \)\mathrm{A}, \mathrm{B}\( and \)\mathrm{C}$ are respectively \(\pm 2 \% \pm 3 \%\) and \(\pm 5 \%\) then total percentage error in measurement of \(\mathrm{p}\) (a) \(18 \%\) (b) \(14 \%\) (c) \(21 \%\) (d) \(12 \%\)

After rounding of the number 9595 to 3 significant digits the value becomes .............. (a) 9600 (b) 9000 (c) 9590 (d) 9500

If \(\mathrm{A}=\mathrm{b}^{4}\) the fractional error in \(\mathrm{A}\) is $\ldots \ldots \ldots \ldots$ (a) \(\left[(\Delta \mathrm{b})^{4} /(\mathrm{b})\right]\) (b) \([\Delta \mathrm{b} / \mathrm{b}]\) (c) \(4[(\Delta \mathrm{b}) /(\mathrm{b})]\) (d) \((\Delta \mathrm{b})^{4}\)

Which physical quantity is represented by $\mathrm{M}^{1} \mathrm{~L}^{3} \mathrm{~T}^{-3} \mathrm{~A}^{-2}$ ? (a) Resistivity (b) Resistance (c) conductance (d) conductivity
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