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Chapter 17: Problem 24

In which gas are the molecules moving faster: hydrogen at \(75 \mathrm{K}\) or sulfur dioxide at \(350 \mathrm{K} ?\)

Short Answer

Expert verified
Hydrogen molecules at \(75 \mathrm{K}\) are moving faster than sulfur dioxide molecules at \(350 \mathrm{K}\)

Step by step solution

01

Identify the given quantities

We know the temperatures of both gases: \(75 \mathrm{K}\) for hydrogen and \(350 \mathrm{K}\) for sulfur dioxide. We can also look for the molar masses in a periodic table: hydrogen's molar mass is roughly \(2 \mathrm{g/mol}\) and sulfur dioxide's molar mass is approximately \(64 \mathrm{g/mol}\).
02

Apply the formula for root mean square speed

The formula for root mean square speed is \(v = \sqrt{\frac{3RT}{M}}\), where R is the gas constant \(R = 8.314 \mathrm{J \cdot K^{-1} \cdot mol^{-1}}\), T is the temperature in Kelvin, and M is the molar mass in kilograms per mole. We apply this formula separately for each gas.
03

Calculate the speed for hydrogen

For hydrogen, we substitute the values into the formula: \(v_H = \sqrt{\frac{3 \cdot 8.314 \cdot 75}{2/1000}} \). Calculating this value gives us \(v_H \approx 1929 \mathrm{m/s}\).
04

Calculate the speed for sulfur dioxide

For sulfur dioxide, we substitute the values into the formula: \(v_{SO_2} = \sqrt{\frac{3 \cdot 8.314 \cdot 350}{64/1000}}\). Calculating this gives us \(v_{SO_2} \approx 483 \mathrm{m/s}\).
05

Compare the speeds

Upon comparing, \(v_H > v_{SO_2}\), thus, the molecules of Hydrogen are moving faster than those of Sulfur Dioxide.

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