Question

If the rms velocity of a gas at \(100 \mathrm{~K}\) is \(10^{4} \mathrm{~cm} \mathrm{sec}^{-1}\), what is the temperature (in \({ }^{\circ} \mathrm{C}\) ) at which the \(\mathrm{rms}\) velocity will be \(3 \times 10^{4} \mathrm{~cm} \mathrm{sec}^{-1} ?\) (a) 900 (b) 627 (c) 327 (d) 1217

Short answer

The temperature is 627 °C.

Step-by-step solution

Understanding RMS Velocity Formula

The root mean square (rms) velocity of a gas is given by the formula \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \(v_{rms}\) is the rms velocity, \(k\) is the Boltzmann constant, \(T\) is the temperature in Kelvin, and \(m\) is the mass of one molecule of the gas.

Set up Proportional Relationship

Since the only changing parameter in our problem is the temperature, and assuming the mass of the molecules does not change, we have the relationship \( \frac{v_{1}^2}{T_{1}} = \frac{v_{2}^2}{T_{2}} \). Here, \(v_{1} = 10^4 \) cm/s, \(T_{1} = 100 \) K, and \(v_{2} = 3 \times 10^4 \) cm/s.

Solve for T2

Rearrange the proportional relationship to solve for \(T_2\):\[ T_2 = \frac{v_{2}^2 \cdot T_1}{v_{1}^2} \]Substituting the given values gives:\[ T_2 = \frac{(3 \times 10^4)^2 \cdot 100}{(10^4)^2} \] \[ T_2 = \frac{9 \times 10^8 \cdot 100}{10^8} = 900 \text{ K} \].

Convert Temperature to Celsius

To convert from Kelvin to Celsius, use the formula \( T_{\circ C} = T_{\text{K}} - 273.15 \). Substituting \( T_2 = 900 \text{ K} \), the temperature in Celsius is \( 900 - 273.15 = 626.85 \), which rounds to \( 627\text{ °C} \).

Key concepts

Kinetic Theory of Gases

The kinetic theory of gases is a theory that explains the behavior of gases, focusing on how gases interact using fundamental principles of physics. Essentially, it describes a gas as a large number of small particles, such as molecules or atoms, which are in constant random motion.
This constant motion is critical to the properties of gases,
being responsible for behaviors like pressure, volume, and temperature. There are several key assumptions in this theory:

• Gas particles are in constant, random motion.
• The volume of the particles themselves is negligible compared to the volume of the gas as a whole.
• There are no forces of attraction or repulsion between the particles.
• Collisions between gas particles and between particles and the walls of the container are perfectly elastic, meaning there is no net loss of kinetic energy.

All these characteristics lead us to important insights regarding gases, one of which is the formula for root mean square (rms) velocity, which helps in understanding the average speed of particles in a gas. The formula reflects that the velocity is directly dependent on the temperature of the gas, according to this theory.

Temperature Conversion

Temperature conversion is an essential part of many scientific calculations, especially when comparing temperatures across different scales. For our purposes, the most common conversions are between Celsius and Kelvin, with Kelvin being the SI unit for temperature in physics.
The conversion from Kelvin to Celsius is straightforward:

• To convert from Kelvin to Celsius, subtract 273.15 from the Kelvin temperature. This is because \[ T_{\circ C} = T_{\text{K}} - 273.15 \]
• For converting Celsius to Kelvin, you instead add 273.15 to the Celsius temperature:\[ T_{\text{K}} = T_{\circ C} + 273.15 \]

This conversion step is crucial in problems like the one given, where you need to determine the temperature in Celsius after calculating it in Kelvin. Especially in physics and chemistry, Kelvin is often used due to its absolute scale, making it necessary to convert back and forth when dealing with real-world temperatures.

Boltzmann Constant

The Boltzmann constant is a fundamental constant in physics that plays a vital role in the field of statistical mechanics and thermodynamics. Its symbol is \( k \), and its value is approximately \( 1.38 \times 10^{-23} \text{ J/K} \). This constant serves as a bridge between the microscopic world of particles and the macroscopic world of quantities like temperature.

• One of the key places it appears is in the formula for the root mean square velocity of gases, where it helps relate the temperature of a gas to the kinetic energy of its molecules:\[ v_{rms} = \sqrt{\frac{3kT}{m}} \],where \( v_{rms} \) is the root mean square velocity, \( k \) is the Boltzmann constant, \( T \) is the absolute temperature, and \( m \) is the mass of a molecule.
• It also appears in the famous Boltzmann distribution, which predicts the distribution of particles over various energy states in thermal equilibrium.

The Boltzmann constant highlights the profound connection between temperature and energy on the microscopic scale, making it an invaluable tool in understanding thermodynamic processes. Its usage helps in translating temperatures into energy terms, allowing for the analysis of gas behaviors and reactions.