(c) Calculate the most probable speeds of Cl 2 molecules at 300 K.

Hello, everyone. Today, we have the following problem. A certain amount of roaming gas is placed inside a container at 550 degrees. Kelvin determine the most probable speed of the molecules at this temperature. And choices A 3d are all in meters per second with a being 7.6 B being 9.3 C being 2.4 times 10 to the second. And D being 2.9 times 10 to the second. So we can use the equation to solve for this multiple speed using the multiple speed is equal to the square root of two multiplied by Arbi as a gas constant multiplied by temperature in degrees Kelvin divided by the molar mass. Now, we can find the molar mass of this bromine gas by referring to the periodic table and taking the mass of one bromine atom which is 79.9 g per mole. And multiplying that by two, since we have two atoms of this brome, and that will give us or, and when we actually have that we will need to convert to kilograms for our equation. So we will use the conversion factor and multiply by 1 kg is equal to 10 to the to the third grams, getting rid of units of grams and giving us an ultimate molar mass of 0.1598 kilograms per mole. Now, with our equation, if we rewrite it or if we were to move it over to the right, we would have the most probable speed is equal to the square root of two multiplied by gas constant, which will be 8.314. And usually it'd be in terms of joules per moles times Kelvin. However, there is a conversion factor that joules or units of jewels is equal to kilograms multiplied by meters squared, divided by seconds squared. And this will come into play with our molar mass. So we have 8.3 1 4 kg multiplied by meters squared that will be divided by seconds squared. And then the rest of the gas constant, which is moles times Kelvin, that value will be multiplied by the degrees which is or by the temperature which is 550 degrees Kelvin. And then all of that still under the square root will be divided by the molar mass which is 0.1598 kg per mole. And our units of moles will cancel out as will units of kilograms and temperature such that we get a most probable speed that is equal to 239.2285 m per second. And if we were to convert this or refer to this in terms of significant figures. We would use two significant figures such that we have 2.4 times 10 to the second meters per squared as our answer. And if we go to Andros, Andros c best reflects it value overall, I hope this helped. And until next time.

Ch.10 - Gases

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Problem 82c2

Chapter 10, Problem 82c2

(c) Calculate the most probable speeds of Cl₂ molecules at 300 K.

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Identify the formula for the most probable speed of a gas molecule, which is given by \( v_p = \sqrt{\frac{2kT}{m}} \), where \( v_p \) is the most probable speed, \( k \) is the Boltzmann constant, \( T \) is the temperature in Kelvin, and \( m \) is the mass of a single molecule.

Convert the molar mass of \( \text{Cl}_2 \) from grams per mole to kilograms per molecule. The molar mass of \( \text{Cl}_2 \) is approximately 70.9 g/mol. Use Avogadro's number \( 6.022 \times 10^{23} \) to find the mass of one molecule.

Substitute the values into the formula: \( k = 1.38 \times 10^{-23} \text{ J/K} \), \( T = 300 \text{ K} \), and the calculated mass of a \( \text{Cl}_2 \) molecule.

Calculate the expression under the square root to find the most probable speed.

Ensure the units are consistent throughout the calculation to obtain the speed in meters per second (m/s).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Kinetic Molecular Theory

Kinetic Molecular Theory explains the behavior of gases in terms of particles in constant motion. It posits that the temperature of a gas is directly related to the average kinetic energy of its molecules. This theory helps in understanding how temperature influences the speed of gas molecules, which is crucial for calculating the most probable speed.

Most Probable Speed

The most probable speed of gas molecules is the speed at which the largest number of molecules are moving at a given temperature. It can be calculated using the formula v_mp = sqrt(2kT/m), where v_mp is the most probable speed, k is the Boltzmann constant, T is the temperature in Kelvin, and m is the mass of a molecule. This concept is essential for determining the speed of Cl2 molecules at 300 K.

Molar Mass and Gas Constant

Molar mass is the mass of one mole of a substance, which is critical for converting between grams and moles in calculations. For Cl2, the molar mass is approximately 70.9 g/mol. The gas constant (R) is used in various gas equations and relates to the behavior of gases under different conditions, aiding in the calculation of molecular speeds.

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(c) Calculate the most probable speeds of Cl2 molecules at 300 K.

Verified Solution

Key Concepts

Kinetic Molecular Theory

Most Probable Speed

Molar Mass and Gas Constant

(c) Calculate the most probable speeds of Cl₂ molecules at 300 K.