Problem 31 - Physics

Question

Two gas cylinders are identical. One contains the monatomic gas argon (Ar), and the other contains an equal mass of the monatomic gas krypton \((\mathrm{Kr}) .\) The pressures in the cylinders are the same, but the temperatures are different. Determine the ratio \(\frac{\overline{\mathrm{KE}}_{\mathrm{Krypton}}}{\overline{\mathrm{KE}}_{\mathrm{Argon}}}\) of the average kinetic energy of a krypton atom to the average kinetic energy of an argon atom.

Step-by-Step Solution

Answer

The ratio \( \frac{\overline{\mathrm{KE}}_{\mathrm{Krypton}}}{\overline{\mathrm{KE}}_{\mathrm{Argon}}} \) is approximately 0.476.

Step 1: Understand Kinetic Energy for Gases

The average kinetic energy per molecule of a monatomic ideal gas is given by the formula \( \overline{\mathrm{KE}} = \frac{3}{2} k_B T \), where \( k_B \) is the Boltzmann constant and \( T \) is the absolute temperature of the gas.

Step 2: Apply Ideal Gas Law for Each Gas

Using the ideal gas law \( PV = nRT \), since the pressures and the volumes of both gases are the same, we know that \( n_{Ar} = n_{Kr} \). Therefore, \( nRT = \frac{m}{M}RT \), where \( m \) is the mass and \( M \) is the molar mass.

Step 3: Relate Mass, Moles, and Temperature

Since the masses \( m \) of argon and krypton in each cylinder are equal, we have \( \frac{m}{M_{Ar}} T_{Ar} = \frac{m}{M_{Kr}} T_{Kr} \). From here, it follows that \( T_{Ar} M_{Ar} = T_{Kr} M_{Kr} \).

Step 4: Find Temperatures Ratio

Solve the proportion \( T_{Ar} M_{Ar} = T_{Kr} M_{Kr} \) for \( \frac{T_{Kr}}{T_{Ar}} \):\[ \frac{T_{Kr}}{T_{Ar}} = \frac{M_{Ar}}{M_{Kr}} \]

Step 5: Calculate Kinetic Energy Ratio

Given \( \overline{\mathrm{KE}} = \frac{3}{2} k_B T \), the ratio of the average kinetic energies is:\[ \frac{\overline{\mathrm{KE}}_{\mathrm{Krypton}}}{\overline{\mathrm{KE}}_{\mathrm{Argon}}} = \frac{T_{Kr}}{T_{Ar}} = \frac{M_{Ar}}{M_{Kr}} \]

Step 6: Use Atomic Masses for Calculation

Use the atomic masses: \( M_{Ar} \approx 39.9 \, \text{u} \) and \( M_{Kr} \approx 83.8 \, \text{u} \). Calculate:\[ \frac{M_{Ar}}{M_{Kr}} = \frac{39.9}{83.8} \approx 0.476 \].

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that relates the pressure, volume, temperature, and amount of gas in a given system. It is expressed as \( PV = nRT \), where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume of the gas.
• \(n\) is the number of moles of gas.
• \(R\) is the ideal gas constant.
• \(T\) is the absolute temperature in Kelvin.

The ideal gas law assumes that the gas behaves ideally, meaning that the gas particles do not interact with one another and there are no intermolecular forces affecting them.
In the context of kinetic energy in gases, the Ideal Gas Law helps in understanding how variables like temperature are directly related to the energy of gas particles. For instance, if two identical cylinders contain different gases but have the same pressure and volume, their behavior under the same conditions illustrates how differences in temperature and moles impact kinetic energy.

Monatomic Gases

Monatomic gases are simply gases which consist of single atoms. Examples include noble gases like Argon (Ar) and Krypton (Kr) that you might come across in exercises involving gas laws. Their unique property is that each particle of the gas is an individual atom, which simplifies the calculations related to kinetic energy.

In systems involving gases, monatomic gases obey the same basic principles as all ideal gases, but their simpler nature makes them easier to study. The kinetic energy of gas particles is crucially related to their motion. For a monatomic ideal gas, the average kinetic energy per molecule is given by the formula \( \overline{\mathrm{KE}} = \frac{3}{2} k_B T \). Here:

• \(k_B\) is Boltzmann's constant.
• \(T\) is the absolute temperature of the gas.

This formula means that even though monatomic gases are simpler, their motion under temperature can be precisely measured, reflecting how energy distributes within the gas.

Temperature and Pressure

Temperature and pressure are two intertwined concepts when discussing gases. The temperature of a gas is related to the average kinetic energy of its particles; that is, how fast the particles are moving. The higher the temperature, the higher the kinetic energy.

Pressure, on the other hand, is the force those gas particles exert when they collide with the walls of their container. Pressure is directly proportional to both the number of molecules and their kinetic energy. Therefore, an increase in temperature leads to an increase in pressure if the volume is kept constant.
When dealing with problems concerning gases, like comparing the kinetic energy of Argon and Krypton, understanding how temperature and pressure interact allows us to see why gases of equal mass but different atomic masses and temperatures can have different kinetic energies. In such calculations, equal pressures in different gases imply that differences in temperature due to diverse atomic masses require a careful calculation of kinetic energy.