A 1.0-mol sample of helium gas has a temperature of 18°C. What is the total kinetic energy of all the gas atoms in the sample?

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Convert the given temperature from Celsius to Kelvin using the formula: $ T(K) = T(°C) + 273.15 $. For this problem, $ T = 18 + 273.15 $.

Recall the formula for the total kinetic energy of an ideal gas: $ KE_{total} = \frac{3}{2} nRT $, where $ n $ is the number of moles, $ R $ is the ideal gas constant (8.314 J/(mol·K)), and $ T $ is the temperature in Kelvin.

Substitute the given values into the formula: $ n = 1.0 $ mol, $ R = 8.314 $ J/(mol·K), and $ T $ (calculated in step 1). The equation becomes $ KE_{total} = \frac{3}{2} (1.0)(8.314)(T) $.

Simplify the expression by multiplying the constants $ \frac{3}{2} \times 8.314 $ and then multiplying by the temperature $ T $ in Kelvin.

The result of the calculation will give the total kinetic energy of all the gas atoms in the sample, expressed in joules (J).

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Key Concepts

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

Kinetic Energy of Gases

The kinetic energy of gas particles is directly related to their temperature. For an ideal gas, the average kinetic energy per particle can be calculated using the formula KE = (3/2)kT, where k is the Boltzmann constant and T is the temperature in Kelvin. This relationship indicates that as temperature increases, the kinetic energy of the gas particles also increases.

Ideal Gas Law

The Ideal Gas Law, represented as PV = nRT, relates the pressure (P), volume (V), and temperature (T) of an ideal gas to the number of moles (n) and the ideal gas constant (R). While this law is not directly used to calculate kinetic energy, it provides a framework for understanding the behavior of gases under various conditions, which is essential for thermodynamic calculations.

Molar Mass and Moles

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole. In this question, the sample of helium gas is given as 1.0 mol. Understanding moles is crucial for calculating the total kinetic energy, as it allows us to determine the total number of gas atoms and subsequently apply the kinetic energy formula to find the total energy of the sample.

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Estimate how many air molecules rebound from a wall in a typical room per second, assuming an ideal gas of N molecules contained in a cubic room with sides of length ℓ at temperature T and pressure P.

(a) Show that the frequency f with which gas molecules strike a wall is ƒ = ( $\overline{υ_{x}}$ /2)(P/kT) ℓ² where $\overline{υ_{x}}$ is the average x component of the molecule’s velocity.

(b) Show that the equation can then be written as ƒ≈ Pℓ² / $the square root of 4 m k T$ where m is the mass of a gas molecule.

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