Question

When temperature of an ideal gas is increased from \(27^{\circ} \mathrm{C}\) to \(227^{\circ} \mathrm{C}\), its rms speed changed from \(400 \mathrm{~ms}^{-1}\) to \(\mathrm{V}_{\mathrm{s}}\). The \(\mathrm{V}_{\mathrm{s}}\) is (A) \(516 \mathrm{~ms}^{-1}\) (B) \(746 \mathrm{~ms}^{-1}\) (C) \(310 \mathrm{~ms}^{-1}\) (D) \(450 \mathrm{~ms}^{-1}\)

Short answer

The correct answer is (A) \(516 \mathrm{~ms}^{-1}\). This is found by using the equation \(\frac{v_{rms2}}{v_{rms1}} = \sqrt{\frac{T_2}{T_1}}\), converting temperatures to Kelvins, and substituting the given values.

Step-by-step solution

Understand the rms speed and temperature relationship

For an ideal gas, the rms speed is related to its temperature by the equation: \(v_{rms} = \sqrt{\dfrac{3kT}{m}}\) where \(v_{rms}\) is the rms speed, \(k\) is the Boltzmann constant, \(T\) is the temperature in Kelvins, and \(m\) is the mass of a single gas molecule. The ratio of the rms speeds at two different temperatures can be found by dividing the equation at the final temperature by the equation at the initial temperature: \(\dfrac{v_{rms2}}{v_{rms1}} = \sqrt{\dfrac{3kT_2/m}{3kT_1/m}}\) The mass of a single gas molecule and the Boltzmann constant are the same for both temperatures, so the constants can cancel out, leaving: \(\dfrac{v_{rms2}}{v_{rms1}} = \sqrt{\dfrac{T_2}{T_1}}\)

Convert the temperatures to Kelvins

To solve the equation, first convert the given temperatures from Celsius to Kelvins: Initial temperature: \(T_1 = 27°C + 273.15 = 300.15 K\) Final temperature: \(T_2 = 227°C + 273.15 = 500.15 K\)

Substitute the given values into the equation

Now, substitute the given values and the converted temperatures into the equation: \(\dfrac{v_{rms2}}{400 m/s} = \sqrt{\dfrac{500.15 K}{300.15 K}}\)

Solve for the final rms speed (Vs)

Rearrange the equation to find the final rms speed \(v_{rms2}\): \(v_{rms2} = 400 m/s \times \sqrt{\dfrac{500.15 K}{300.15 K}}\) Calculate the value of \(v_{rms2}\): \(v_{rms2} \approx 516.14 m/s\)

Match the answer to the given options

The calculated value of the final rms speed (Vs) matches the option (A) \(516 \mathrm{~ms}^{-1}\). So, the correct answer is (A) \(516 \mathrm{~ms}^{-1}\).

Key concepts

rms speed

Root mean square (rms) speed is a statistical measure used to describe the speed of particles in a gas. Within the context of ideal gas behavior, it is an important way to connect the microscopic motion of molecules with macroscopic observations, like temperature. The formula for rms speed is given by:\[ v_{rms} = \sqrt{\dfrac{3kT}{m}} \]

• v_rms: The rms speed we want to calculate.
• k: Boltzmann constant, which helps bridge the gap between macroscopic and microscopic physics.
• T: Absolute temperature of the gas in Kelvins.
• m: Mass of an individual gas molecule.

The important relationship here is that the rms speed changes proportionally to the square root of the temperature. This makes rms speed quicker to compute once temperatures are known, and critical for understanding kinetic theories of gases.

temperature conversion

Temperature conversion from Celsius to Kelvin is a crucial step in many physics problems, particularly those involving gases. Kelvin is the absolute temperature scale used in science because it starts at absolute zero, the theoretical point where molecular motion ceases. Converting Celsius to Kelvin is straightforward:\[ T_{K} = T_{°C} + 273.15 \]

• The conversion ensures calculations align with theoretical expectations about temperature and molecular speed.
• Since calculations often reflect absolute energy levels (as in kinetic energy), using Kelvin offers a more intuitive understanding.

Remember, understanding the Kelvin scale is essential for bridging theoretical physics to practical applications, especially when considering changes in state or reactions to heating.

Kelvin scale

The Kelvin scale is the SI base unit for temperature and is integral within scientific communities across the globe. Named after physicist William Thomson, 1st Baron Kelvin, it starts at absolute zero, a theoretical temperature where all kinetic energy, and thus molecular motion, ceases.

• Absolute Zero: 0 Kelvin (K), equivalent to -273.15°C, is the lowest possible temperature.
• Units of Kelvin: Every Kelvin unit represents the same temperature interval as one degree Celsius; however, Kelvin is used for scientific precision.
• No Negative Values: Kelvin temperature never dips below zero, reflecting the concept's foundation in absolute physical limits.

Kelvin scale is pivotal in thermodynamics and physics as it offers a true zero reference point that aids in defining the absolute measures of thermodynamic principles.

Boltzmann constant

The Boltzmann constant (\( k \)) is a crucial physical constant that bridges macroscopic and microscopic physics using statistical mechanics. It establishes a relation between the temperature of a system and the kinetic energy of molecules. The value is:\[ k = 1.38 \times 10^{-23} \text{ J/K} \]

• Role in Kinetic Theory: Incorporated into equations like that for rms speed, it underscores the impact of atomic-level interactions on observable quantities such as temperature and pressure.
• Energy Per Molecule: Facilitates calculations involving energy at an individual molecular level rather than bulk properties.
• Entropy and Thermodynamics: Foundations in understanding entropy changes with temperature changes within any defined system.

For students and scientists, the Boltzmann constant is indispensable in calculations that necessitate precision on extremely small scales, interfacing quantum mechanics with everyday observations.