Problem 39 - University Physics with Modern Physics

Question

At what temperature is the root-mean-square speed of nitrogen molecules equal to the root-mean-square speed of hydrogen molecules at \(20.0^{\circ} \mathrm{C} ?\) (Hint: The periodic table in Appendix D shows the molar mass (in \(\mathrm{g} / \mathrm{mol}\) ) of each element under the chemical symbol for that element. The molar mass of \(\mathrm{H}_{2}\) is twice the molar mass of hydrogen atoms, and similarly for \(\mathrm{N}_{2} .\) )

Step-by-Step Solution

Answer

The temperature is approximately 4061.46 K.

Step 1: Understand the Problem

We need to find the temperature at which the root-mean-square speed (RMS speed) of nitrogen molecules \( N_2 \) is the same as that of hydrogen molecules \( H_2 \) at \( 20^{\circ} C \). Use the formula for RMS speed: \( v_{rms} = \sqrt{ \frac{3kT}{m}} \), where \( k \) is the Boltzmann constant, \( T \) is the temperature in Kelvin, and \( m \) is the mass of a single molecule.

Step 2: Convert Temperature to Kelvin

Convert \(20.0^{\circ} \text{C} \) to Kelvin. Use the formula \( K = \text{C} + 273.15 \). For \( 20.0^{\circ} \text{C} \), we have:\[ T_{H_2} = 20.0 + 273.15 = 293.15 \text{ K} \]

Step 3: Find Molar Masses

The molar mass of \( H_2 \) is \( 2 \times 1.01 \approx 2.02 \text{ g/mol} \) (since the atomic mass of hydrogen is approximately \( 1.01 \text{ g/mol} \)). For \( N_2 \), the molar mass is \( 2 \times 14.01 \approx 28.02 \text{ g/mol} \) (since the atomic mass of nitrogen is approximately \( 14.01 \text{ g/mol} \)).

Step 4: Calculate the Hydrogen RMS Speed

Using the RMS speed formula and the temperature from Step 2, the RMS speed of \( H_2 \) at \( 293.15 \text{ K} \) is: \[ v_{rms, H_2} = \sqrt{ \frac{3k \times 293.15}{m_{H_2}} } \]Since \( m_{H_2} \) in kg: \( 2.02 \text{ g/mol} = 2.02 \times 10^{-3} \text{ kg/mol} \) and \(k\) is a constant, this step should be kept symbolic.

Step 5: Equate RMS Speeds and Find Temperature for Nitrogen

Set the RMS speed equations equal for \( H_2 \) and \( N_2 \): \[ \sqrt{ \frac{3kT_{N_2}}{m_{N_2}} } = \sqrt{ \frac{3k \times 293.15}{m_{H_2}}} \]Squaring both sides and solving for \( T_{N_2} \), we have:\[ T_{N_2} = 293.15 \times \frac{m_{N_2}}{m_{H_2}} \] Thus, substituting molar masses:\[ T_{N_2} = 293.15 \times \frac{28.02}{2.02} \]

Step 6: Final Calculation

Calculate \(T_{N_2}\) using the ratio obtained:\[\frac{28.02}{2.02} \approx 13.86\]\[T_{N_2} = 293.15 \times 13.86 \approx 4061.46 \text{ K} \]

Key Concepts

Nitrogen Molecules

Nitrogen molecules, represented as \(N_2\), consist of two nitrogen atoms bonded together. This diatomic nature is common for elements like nitrogen, which exists in this form under natural conditions. Each nitrogen atom has an atomic mass of approximately 14.01 grams per mole, making the molar mass of \(N_2\) about 28.02 grams per mole. This is critical information for solving physics problems related to gases, as it influences properties like root-mean-square speed. Understanding molecules and their molar masses helps in manipulating equations and predicting the behavior of gases.

Hydrogen Molecules

Hydrogen molecules, symbolized as \(H_2\), are made up of two hydrogen atoms. With an atomic mass of about 1.01 grams per mole for each hydrogen atom, hydrogen molecules have a molar mass of roughly 2.02 grams per mole. This light atomic weight makes hydrogen behavior distinct under various conditions, especially in kinetic theory problems. Due to their light mass, hydrogen molecules move at high speeds compared to heavier gases at the same temperature, which impacts calculations involving their kinetic energy and speed.

Temperature Conversion

Temperature conversion plays a vital role in physics problems, especially when transitioning from degree Celsius to Kelvin. The conversion formula is straightforward: \(K = C + 273.15\). This conversion is necessary because most physics equations require temperatures in Kelvin, which is an absolute scale starting at absolute zero. It is crucial to perform this conversion accurately to ensure that calculations involving thermal energy or molecular speed are correct and meaningful.

Molar Mass

Molar mass is a fundamental concept in chemistry and physics, representing the mass of one mole of an element or compound, typically in grams per mole (g/mol). It is pivotal for calculating molecular properties like root-mean-square speed. Knowing the molar mass is essential to convert between gram molecular mass and the molecular mass used in equations. For instance, the molar mass is directly used to find the mass of individual molecules in kilograms when dividing by Avogadro's number, forming the basis for kinetic calculations involving gases.

Physics Problem Solving

Physics problem-solving often follows structured steps, where understanding the problem statement is the first crucial step. After identifying what is sought, relevant formulas are used, like the root-mean-square speed formula \(v_{rms} = \sqrt{\frac{3kT}{m}}\). Converting all units to the required form (like Celsius to Kelvin) ensures equations are appropriately used. The given problem demonstrates this by equating the root-mean-square speeds of different gases at different temperatures and solving the resultant equation systematically. This structured approach helps break down complex problems into manageable parts for logical and accurate solutions.