Question

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

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

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.

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: \[T_{1,\text{K}} = 0 + 273.15 = 273.15\ \text{K}\]

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: \[v_{2,\text{rms}} = 3v_{1,\text{rms}}\]

Apply the rms speed formula to the initial and final states

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

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, \(T_2\): \[T_2 = \frac{9(3k_\text{B}T_1)}{3k_\text{B}} = 9T_1\]

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.

Key concepts

Gas Laws

Gas laws are a set of rules that relate the pressure, volume, and temperature of a gas to each other. They help us understand how gases behave under different conditions. The most common gas laws used include Boyle's Law, Charles's Law, and Avogadro's Law.

• Boyle's Law: It says that the pressure of a gas is inversely proportional to its volume when temperature is constant. Mathematically, it can be expressed as \( P_1V_1 = P_2V_2 \).
• Charles's Law: This law states that the volume of a gas is directly proportional to its absolute temperature, assuming pressure remains constant. The formula is \( V_1/T_1 = V_2/T_2 \).
• Avogadro's Law: It demonstrates that the volume of a gas is directly proportional to the number of moles of the gas at constant temperature and pressure.
  Thus, gases expand when heated and contract when cooled. Understanding these laws is essential for solving many gas-related problems, including those involving the root mean square speed of gas molecules.

Temperature Conversion

Temperature is a critical factor in calculations related to gases. Different scales like Celsius, Kelvin, and Fahrenheit measure temperature. However, for scientific calculations, especially when dealing with gas laws, we use Kelvin.

• Converting Celsius to Kelvin: The Kelvin scale starts at absolute zero, which is -273.15°C. To convert Celsius to Kelvin, you use the formula: \[ T_\text{K} = T_\text{C} + 273.15 \]This conversion ensures that calculations are in an absolute measure of temperature.
• Importance: Kelvin is crucial in gas law calculations because it puts temperature into absolute terms. Zero Kelvin, theoretically, is the point where all kinetic motion in atoms stops. That's why it's used instead of Celsius, which can measure negative values.

Thus, making sure you convert to Kelvin is a fundamental step in any computation involving gases.

Boltzmann's Constant

Boltzmann's constant is a fundamental physical constant that connects the average kinetic energy of particles in a gas with temperature. It is essential in statistical mechanics and thermodynamics.

• Value: Boltzmann's constant, \( k_\text{B} \), has a value of approximately \(1.38 \times 10^{-23} \text{J/K} \).
• Role in Gas Calculations: It is used in the formula for root mean square speed, allowing us to relate the microscopic properties of particles (like speed and energy) to macroscopic measurements (like temperature and pressure). The formula is:\[ v_\text{rms} = \sqrt{\frac{3k_\text{B}T}{m}} \]
• Applications: Boltzmann's constant appears in many equations, such as the ideal gas law and the kinetic theory of gases, making it a cornerstone of understanding gas behaviors.

Understanding its use helps students grasp how temperature impacts the motion of gas molecules and aids in solving related physics problems.

Temperature in Kelvin

The Kelvin temperature scale is fundamentally important in physics and chemistry. It measures absolute temperature, starting from absolute zero, where theoretically, there would be no thermal motion in the particles.

• Why Kelvin? Kelvin is the standard unit for thermodynamic temperature in the International System of Units (SI). It provides a scale that starts from the lowest conceivable temperature, which makes it very useful for theoretical calculations.
• Relation to Energy: In the context of gases, temperature in Kelvin is directly related to the energy of gas molecules. The higher the Kelvin temperature, the higher the energy and speed of the molecules. This is evident in the root mean square speed formula, where temperature in Kelvin determines the speed of gas particles.
• Practical Application: For scientists and students alike, working in Kelvin gives clarity and consistency in calculations, ensuring that there's no confusion from negative temperatures or conversions.

By using Kelvin, we ensure that the scientific descriptions of gas behavior are accurate and universally applicable.