Problem 112 - Chemistry The Central Science In Si Units

Question

You have a sample of gas at \(0^{\circ} \mathrm{C}\). You wish to increase the rms speed by a factor of 3 . To what temperature should the gas be heated?

Step-by-Step Solution

Answer

The gas should be heated to 2458.35 K.

Step 1: Understand the Relationship

The root-mean-square (rms) speed of gas molecules, represented by \( v_{rms} \), is related to the temperature, \( T \), of the gas by the equation: \[ v_{rms} = \sqrt{\frac{3kT}{m}} \] where \( k \) is the Boltzmann constant and \( m \) is the mass of a gas particle. From this equation, we can see \( v_{rms} \) is proportional to the square root of the temperature, \( T \).

Step 2: Set up the Proportionality for rms Speed

Since the question asks to increase \( v_{rms} \) by a factor of 3, we can express this as: \[ v'_{rms} = 3v_{rms} \] Since \( v_{rms} \) is proportional to \( \sqrt{T} \), it follows that: \[ \sqrt{T'} = 3\sqrt{T} \] where \( T' \) is the final temperature we need to find.

Step 3: Solve for the New Temperature

To solve for \( T' \), square both sides of the equation from Step 2: \[ T' = 9T \] This shows that the new temperature, \( T' \), must be nine times the initial temperature, \( T \).

Step 4: Substitute the Initial Temperature

The initial temperature is given as \( 0^{\circ} \mathrm{C} \), which is equivalent to \( 273.15 \) K in Kelvin. Substituting this into the equation for \( T' \): \[ T' = 9 \times 273.15 = 2458.35 \text{ K} \] Thus, the gas should be heated to \( 2458.35 \) K.

Key Concepts

Temperature and Gas Laws

The connection between temperature and gas behavior is a fundamental aspect of thermodynamics. When we talk about gases, their molecules are constantly moving, and the speed of these molecules is directly affected by the temperature.

Gases obey certain laws that define how they respond to temperature changes. These are known as gas laws. For instance:

• Charles's Law shows that gas volume increases with higher temperature at constant pressure.
• Boyle's Law states that gas pressure decreases as volume increases, assuming temperature is constant.
• Ideal Gas Law brings these together: \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature.

In physics, temperature specifically affects the speed of gas particles. As the temperature rises, so does the average speed of the molecules. The root-mean-square speed formula \( v_{rms} = \sqrt{\frac{3kT}{m}} \) illustrates this relationship, showing that speed is proportional to the square root of temperature.

Boltzmann Constant

The Boltzmann constant is a vital part of understanding gas behavior at the molecular level. Named after physicist Ludwig Boltzmann, it is denoted by \( k \) and plays a crucial role in connecting temperature with kinetic energy.

The constant appears in the formula for the root-mean-square speed of gas particles: \( v_{rms} = \sqrt{\frac{3kT}{m}} \). It essentially translates the temperature of a substance into energy units, providing a bridge between microscopic and macroscopic phenomena.

• Value: \( k \approx 1.38 \times 10^{-23} \text{ J/K} \)
• Function: Relates the average kinetic energy of particles in a gas to its temperature.
• Applications: Crucial in statistical mechanics and thermodynamics.

Understanding the Boltzmann constant helps explain why heating a gas increases the energy and speed of its particles, illustrating fundamental principles of energy conservation and distribution.

Root-Mean-Square Speed

Root-mean-square (rms) speed is an important concept when discussing gas dynamics. It provides an average measure of the speed of particles within a gas sample.

Root-mean-square speed is calculated using the formula \( v_{rms} = \sqrt{\frac{3kT}{m}} \), which links temperature with speed. Here’s what the terms mean:

• \( v_{rms} \): The rms speed we're calculating.
• \( k \): The Boltzmann constant.
• \( T \): Temperature in Kelvin.
• \( m \): Mass of a single molecule of the gas.

This equation shows that the higher the temperature, the faster the gas particles move. When you want to increase the \( v_{rms} \) by a certain factor, you adjust the temperature accordingly, as shown in the exercise example. Heating a gas by a factor of nine increases the speed threefold. This highlights the importance of temperature control in processes involving gases.