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Chapter 10: Problem 112

You have a sample of gas at \(0^{\circ} \mathrm{C}\). You wish to increase the \(\mathrm{rms}\) speed by a factor of \(3 .\) To what temperature should the gas be heated?

Short Answer

Expert verified
To increase the root mean square (rms) speed of a gas sample initially at \(0^\circ C\) by a factor of 3, the gas should be heated to approximately \(2458.35\ \text{K}\).

Step by step solution

01

Write down the formula for rms speed

The formula for the rms speed (\(v_\text{rms}\)) for a gas is given by: \[v_\text{rms} = \sqrt{\frac{3k_\text{B}T}{m}}\] where \(v_\text{rms}\) = root mean square speed, \(k_\text{B}\) = Boltzmann's constant, \(T\) = absolute temperature (in Kelvin), and \(m\) = molar mass of the gas. For our problem, let's denote the initial speed as \(v_{1,\text{rms}}\) and the final speed as \(v_{2,\text{rms}}\), and the initial and final temperatures as \(T_1\) and \(T_2\), respectively.
02

Convert initial temperature to Kelvin

To work with the absolute temperature, we need to convert the Celsius temperature to Kelvin. The conversion formula is: \[T_\text{K} = T_\text{C} + 273.15\] In our problem, we have the initial temperature in Celsius, \(T_1 = 0^\circ C\). The corresponding Kelvin temperature is: \[T_{1,\text{K}} = 0 + 273.15 = 273.15\ \text{K}\]
03

Find the relationship between initial and final rms speeds

We are given that we want to increase the rms speed by a factor of 3; hence, we can write: \[v_{2,\text{rms}} = 3v_{1,\text{rms}}\]
04

Apply the rms speed formula to the initial and final states

Using the rms speed formula for initial and final states, we have: \[v_{1,\text{rms}} = \sqrt{\frac{3k_\text{B}T_1}{m}}\] \[v_{2,\text{rms}} = \sqrt{\frac{3k_\text{B}T_2}{m}}\]
05

Substitute the relationship between initial and final speeds and solve for the final temperature

Given the relationship between the initial and final rms speeds, substituting the expressions from Step 4: \[3v_{1,\text{rms}} = \sqrt{\frac{3k_\text{B}T_2}{m}}\] Square both sides of the equation: \[(3v_{1,\text{rms}})^2 = \frac{3k_\text{B}T_2}{m}\] Now substitute the expression for the initial rms speed: \[v_{1,\text{rms}} = \sqrt{\frac{3k_\text{B}T_1}{m}}\] \[(3\sqrt{\frac{3k_\text{B}T_1}{m}})^2 = \frac{3k_\text{B}T_2}{m}\] Solve for the final temperature, \(T_2\): \[T_2 = \frac{9(3k_\text{B}T_1)}{3k_\text{B}} = 9T_1\]
06

Substitute the initial temperature and find the final temperature

Substitute the value of the initial temperature, \(T_1 = 273.15\ \text{K}\): \[T_2 = 9T_1 = 9 \times 273.15\] \[T_2 \approx 2458.35\ \text{K}\] So, the gas should be heated to approximately \(2458.35\ \text{K}\) to increase its rms speed by a factor of 3.

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Most popular questions from this chapter

You have a gas at \(25^{\circ} \mathrm{C}\) confined to a cylinder with a movable piston. Which of the following actions would double the gas pressure? (a) Lifting up on the piston to double the volume while keeping the temperature constant; (b) Heating the gas so that its temperature rises from \(25^{\circ} \mathrm{C}\) to \(50^{\circ} \mathrm{C}\), while keeping the volume constant; (c) Pushing down on the piston to halve the volume while keeping the temperature constant.

Imagine that the reaction $2 \mathrm{CO}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{CO}_{2}(g)$ occurs in a container that has a piston that moves to maintain a constant pressure when the reaction occurs at constant temperature. Which of the following statements describes how the volume of the container changes due to the reaction: (a) the volume increases by \(50 \%,\) (b) the volume increases by \(33 \%\), (c) the volume remains constant, (d) the volume decreases by \(33 \%,(\mathbf{e})\) the volume decreases by $50 \%\(. [Sections 10.3 and 10.4\)]$

Mars has an average atmospheric pressure of 709 pa. Would it be easier or harder to drink from a straw on Mars than on Earth? Explain. [Section 10.2\(]\)

Radon (Rn) is the heaviest (and only radioactive) member of the noble gases. How much slower is the root-mean-square speed of Rn than He at $300 \mathrm{K?}$

A piece of dry ice (solid carbon dioxide) with a mass of \(20.0 \mathrm{~g}\) is placed in a 25.0-L vessel that already contains air at \(50.66 \mathrm{kPa}\) and \(25^{\circ} \mathrm{C}\). After the carbon dioxide has totally sublimed, what is the partial pressure of the resultant \(\mathrm{CO}_{2}\) gas, and the total pressure in the container at \(25^{\circ} \mathrm{C} ?\)
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