Question

The rms velocity of hydrogen is \(\sqrt{7}\) times the rms velocity of nitrogen. If \(\mathrm{T}\) is the temperature of the gas: (a) \(\mathrm{T}\left(\mathrm{H}_{2}\right)=\mathrm{T}\left(\mathrm{N}_{2}\right)\) (b) \(\mathrm{T}\left(\mathrm{H}_{2}\right)>\mathrm{T}\left(\mathrm{N}_{2}\right)\) (c) \(\mathrm{T}\left(\mathrm{H}_{2}\right)<\mathrm{T}\left(\mathrm{N}_{2}\right)\) (d) \(\mathrm{T}\left(\mathrm{H}_{2}\right)=\sqrt{7 \mathrm{~T}\left(\mathrm{~N}_{2}\right)}\)

Short answer

(c) \(\mathrm{T}(\mathrm{H}_{2})<\mathrm{T}(\mathrm{N}_{2})\)

Step-by-step solution

Understanding the Problem

The problem is asking us to compare the temperatures of hydrogen (\(H_2\)) and nitrogen (\(N_2\)) based on their root-mean-square (rms) velocities. We are given that the rms velocity of \(H_2\) is \(\sqrt{7}\) times that of \(N_2\). We need to determine which of the provided options correctly describes the relationship between the temperatures of these gases.

Using the RMS Velocity Formula

The root-mean-square velocity \(v_{\text{rms}}\) for a gas is given by the equation \[v_{\text{rms}} = \sqrt{\frac{3kT}{m}}\]where \(T\) is the temperature, \(m\) is the molar mass of the gas, and \(k\) is the Boltzmann constant. We need to compare the rms velocities for both hydrogen and nitrogen provided this equation.

Expressing RMS Velocities for Hydrogen and Nitrogen

For hydrogen, \(v_{\text{rms}}(H_2) = \sqrt{\frac{3kT(H_2)}{m_{H_2}}}\).For nitrogen, \(v_{\text{rms}}(N_2) = \sqrt{\frac{3kT(N_2)}{m_{N_2}}}\).We know that \[v_{\text{rms}}(H_2) = \sqrt{7} \times v_{\text{rms}}(N_2).\]

Substituting and Squaring Both Sides

Substitute these expressions into the equation for the relationship given in the problem:\[\sqrt{\frac{3kT(H_2)}{m_{H_2}}} = \sqrt{7} \times \sqrt{\frac{3kT(N_2)}{m_{N_2}}}.\]Square both sides to eliminate the square roots:\[\frac{3kT(H_2)}{m_{H_2}} = 7 \times \frac{3kT(N_2)}{m_{N_2}}.\]

Simplifying the Equation

Cancel out the constant terms: \(3k\) appears on both sides, so it can be cancelled. This leaves us with \[\frac{T(H_2)}{m_{H_2}} = 7 \times \frac{T(N_2)}{m_{N_2}}.\]Use the known molar masses of hydrogen \(m_{H_2} = 2 \, \text{g/mol}\) and nitrogen \(m_{N_2} = 28 \, \text{g/mol}\) to simplify further:\[\frac{T(H_2)}{2} = 7 \times \frac{T(N_2)}{28}.\]

Solving for Temperature Relation

Simplify \(\frac{7}{28}\) to \(\frac{1}{4}\), giving\(\frac{T(H_2)}{2} = \frac{T(N_2)}{4}.\)Multiply both sides by 4 to solve for \(T(H_2)\):\[2T(H_2) = T(N_2).\]Divide both sides by 2:\[T(H_2) = \frac{T(N_2)}{2}.\]This equation shows that the temperature of hydrogen is half the temperature of nitrogen.

Key concepts

Kinetic Theory of Gases

The kinetic theory of gases is a fundamental scientific theory that helps explain gas properties by understanding their molecular composition and behavior. Gases are made up of a large number of small particles, which we assume to be in random motion. When studying gases, this theory simplifies analysis by representing gas molecules as perfectly elastic spheres that undergo collisions with one another and the walls of their container.

• Gases consist of a vast number of molecules that move in random directions.
• Molecular collisions are elastic, meaning that the total kinetic energy remains constant before and after the collision.
• The velocities of these molecules are distributed according to a statistical distribution, known as the Maxwell-Boltzmann distribution.

This theory explains several macroscopic properties of gases, such as pressure and temperature. It also forms the basis for deriving equations such as the root-mean-square (RMS) speed, which helps understand the relationship between a gas's kinetic energy and its temperature.

Root-Mean-Square (RMS) Speed

RMS speed, or root-mean-square speed, is a measure used in the kinetic theory to quantify the average speed of particles in a gas. This concept is particularly important because, while individual gas particle speeds can vary significantly, the RMS speed provides a consistent measure of their kinetic behavior.

• The RMS speed is calculated using the formula: \[v_{\text{rms}} = \sqrt{\frac{3kT}{m}}\] where \(k\) is the Boltzmann constant, \(T\) is the temperature, and \(m\) is the molar mass.
• RMS speed represents the square root of the average of the squares of all the speeds in a gas.
• It allows us to link the microscopic motions of molecules to macroscopic properties like temperature and energy.

Understanding RMS speed is crucial for comparing velocities of different gases under varying conditions, such as temperature changes, as seen in this problem with hydrogen and nitrogen gases.

Gas Laws

Gas laws are a series of equations that describe the behavior of gases in various conditions of temperature, pressure, and volume. These laws are founded on the principles of the kinetic theory of gases and they provide a framework for understanding how gases behave under different environments.

• Boyle's Law states that the pressure of a gas is inversely proportional to its volume, as long as temperature is constant: \[P_1V_1 = P_2V_2\]
• Charles's Law explains that the volume of a gas is directly proportional to its temperature, at constant pressure: \[\frac{V_1}{T_1} = \frac{V_2}{T_2}\]
• Avogadro's Law describes how equal volumes of gases, at the same temperature and pressure, contain an equal number of molecules.

These laws combine into the ideal gas law, \( PV = nRT \), where \( n \) is the number of moles of gas and \( R \) is the ideal gas constant. The ideal gas law provides a valuable tool for predicting the behavior of gases.

Temperature and Pressure Relationship

The relationship between temperature and pressure is a crucial aspect of understanding gases. This relationship is embedded in the concept that gas pressure increases with an increase in temperature when volume is constant. This direct relationship arises because higher temperatures result in greater kinetic energy of gas molecules.

• An increase in temperature causes molecules to move more vigorously, increasing collision frequency with the container walls, hence raising the pressure.
• Conversely, lowering the temperature slows down particle movement, reducing pressure.

In the exercise with hydrogen and nitrogen gases, even though hydrogen has a higher RMS velocity due to its lower molar mass, it requires a lower temperature to maintain a stable system when compared to nitrogen. This reciprocal relationship highlights the complexities involved in the interplay of these fundamental properties. Understanding this relationship is vital for calculating temperature changes in response to pressure shifts and is a core part of calculating RMS speeds in different gases.